From 4256e369a54fbda996c98b0db8bf854046a53a49 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Fri, 22 Dec 2023 19:27:30 +0000 Subject: [PATCH] build based on 44ccbf6 --- dev/.documenter-siteinfo.json | 2 +- dev/index.html | 244 +++++++++++++++++----------------- dev/search_index.js | 2 +- 3 files changed, 124 insertions(+), 124 deletions(-) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 4ca94dd..d33cadc 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-22T18:37:46","documenter_version":"1.2.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2023-12-22T19:27:26","documenter_version":"1.2.1"}} \ No newline at end of file diff --git a/dev/index.html b/dev/index.html index fe57401..7d598f5 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,24 +1,24 @@ -Home · Dysts.jl

Dysts.jl

Full list of all dynamical systems:

Dysts.Aizawa — Function
   Aizawa()

A torus-like attractor related to the forced Lorenz system.

Stats

  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.3501297618370773
  • Lyapunov spectrum (estimated): Any[0.08947878317195473, 0.020305496211675024, -0.3090729926541944]
  • Maximum Lyapunov exponent (estimated): 0.13489555530106362
  • Multiscale Entropy: 0.514512246123007
  • Non-autonomous: false
  • Period: 2.5837
  • Pesin entropy: 0.10978427938362978

Citation

Aizawa, Yoji, and Tatsuya Uezu (1982). Topological Aspects in Chaos and in 2 k-Period Doubling Cascade. Progress of Theoretical Physics 67.3 (1982): 982-985. See also Langford (1984). Numerical studies of torus bifurcations. Numerical Methods for Bifurcation Problems. Birkhauser, Basel, 1984. 285-295.

Original Python Code from the dysts package

class Aizawa(DynSys):
+Home · ChaoticDynamicalSystemsLibrary.jl
ChaoticDynamicalSystemsLibrary.jl
ChaoticDynamicalSystemLibrary.jl implements a collection of chaotic dynamical systems in the Julia SciML ecosystem.
The package is a partial port of the dysts Python package developed by William Gilpin. It implements many (but not all) of the same systems, but does not provide any functionality to simulate them. In contrast to dysts, the main focus of this package is not to be a benchmark for general time-series ML models, but only to provide a collection of ODEs that can be used with the DifferentialEquations.jl ecosystem.
William Gilpin deserves most of the credit for this package. He is the original author of the dysts, and without dysts this package would not exist. –-
Full list of all dynamical systems:
ChaoticDynamicalSystemsLibrary.Aizawa — Function
   Aizawa()
A torus-like attractor related to the forced Lorenz system.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.3501297618370773
  • Lyapunov spectrum (estimated): Any[0.08947878317195473, 0.020305496211675024, -0.3090729926541944]
  • Maximum Lyapunov exponent (estimated): 0.13489555530106362
  • Multiscale Entropy: 0.514512246123007
  • Non-autonomous: false
  • Period: 2.5837
  • Pesin entropy: 0.10978427938362978
Citation
Aizawa, Yoji, and Tatsuya Uezu (1982). Topological Aspects in Chaos and in 2 k-Period Doubling Cascade. Progress of Theoretical Physics 67.3 (1982): 982-985. See also Langford (1984). Numerical studies of torus bifurcations. Numerical Methods for Bifurcation Problems. Birkhauser, Basel, 1984. 285-295.
Original Python Code from the dysts package
class Aizawa(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d, e, f):
         xdot = x * z - b * x - d * y
         ydot = d * x + y * z - b * y
         zdot = c + a * z - 0.333333333333333333 * z ** 3 - x ** 2 - y ** 2 - e * z * x ** 2 - e * z * y ** 2 + f * z * x ** 3
-        return xdot, ydot, zdot
source
Dysts.AnishchenkoAstakhov — Function
   AnishchenkoAstakhov()
Stochastic resonance in forced oscillators.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0916403758729327
  • Lyapunov spectrum (estimated): Any[0.039777554890210017, 0.0008785544495802295, -0.44025308728606666]
  • Maximum Lyapunov exponent (estimated): 0.10107726791527546
  • Multiscale Entropy: 0.7612343892027325
  • Non-autonomous: false
  • Period: 6.814
  • Pesin entropy: 0.040715925817307534
Citation
Anishchenko, et al. Nonlinear dynamics of chaotic and stochastic systems: tutorial and modern developments. 2007.
Original Python Code from the dysts package
class AnishchenkoAstakhov(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.AnishchenkoAstakhov — Function
   AnishchenkoAstakhov()
Stochastic resonance in forced oscillators.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0916403758729327
  • Lyapunov spectrum (estimated): Any[0.039777554890210017, 0.0008785544495802295, -0.44025308728606666]
  • Maximum Lyapunov exponent (estimated): 0.10107726791527546
  • Multiscale Entropy: 0.7612343892027325
  • Non-autonomous: false
  • Period: 6.814
  • Pesin entropy: 0.040715925817307534
Citation
Anishchenko, et al. Nonlinear dynamics of chaotic and stochastic systems: tutorial and modern developments. 2007.
Original Python Code from the dysts package
class AnishchenkoAstakhov(DynSys):
     def rhs(self, X, t):
         x, y, z = X
         mu, eta = self.mu, self.eta
         xdot = mu * x + y - x * z
         ydot = -x
         zdot = -eta * z + eta * np.heaviside(x, 0) * x ** 2
-        return (xdot, ydot, zdot)
source
Dysts.Arneodo — Function
   Arneodo()
A modified Lotka-Volterra ecosystem, also known as the ACT attractor.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.198607392233255
  • Lyapunov spectrum (estimated): Any[0.24285661379003065, -0.0006168721448237418, -1.2169777901647385]
  • Maximum Lyapunov exponent (estimated): 0.23737468667051287
  • Multiscale Entropy: 0.7388404221540498
  • Non-autonomous: false
  • Period: 3.1641
  • Pesin entropy: 0.2428749592514372
Citation
Arneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional Volterra equations. Physics Letters A 79, 259–263 (1980)
Original Python Code from the dysts package
class Arneodo(DynSys):
+        return (xdot, ydot, zdot)
source
ChaoticDynamicalSystemsLibrary.Arneodo — Function
   Arneodo()
A modified Lotka-Volterra ecosystem, also known as the ACT attractor.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.198607392233255
  • Lyapunov spectrum (estimated): Any[0.24285661379003065, -0.0006168721448237418, -1.2169777901647385]
  • Maximum Lyapunov exponent (estimated): 0.23737468667051287
  • Multiscale Entropy: 0.7388404221540498
  • Non-autonomous: false
  • Period: 3.1641
  • Pesin entropy: 0.2428749592514372
Citation
Arneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional Volterra equations. Physics Letters A 79, 259–263 (1980)
Original Python Code from the dysts package
class Arneodo(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d):
         xdot = y
         ydot = z
         zdot = -a * x - b * y - c * z + d * x ** 3
-        return xdot, ydot, zdot
source
Dysts.ArnoldBeltramiChildress — Function
   ArnoldBeltramiChildress()
An exact solution of Euler's equation for inviscid flow in 3D. Often referred to as the ABC flow.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.612281260580053
  • Lyapunov spectrum (estimated): Any[1.47222087191395, -0.01824039630029517, -2.378562986862767]
  • Maximum Lyapunov exponent (estimated): 0.3667300080060703
  • Multiscale Entropy: 1.2759526834024162
  • Non-autonomous: false
  • Period: 2.8508
  • Pesin entropy: 1.47222087191395
Citation
V. I. Arnold, Journal of Applied Mathematics and Mechanics 30, 223 (1966).
Original Python Code from the dysts package
class ArnoldBeltramiChildress(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.ArnoldBeltramiChildress — Function
   ArnoldBeltramiChildress()
An exact solution of Euler's equation for inviscid flow in 3D. Often referred to as the ABC flow.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.612281260580053
  • Lyapunov spectrum (estimated): Any[1.47222087191395, -0.01824039630029517, -2.378562986862767]
  • Maximum Lyapunov exponent (estimated): 0.3667300080060703
  • Multiscale Entropy: 1.2759526834024162
  • Non-autonomous: false
  • Period: 2.8508
  • Pesin entropy: 1.47222087191395
Citation
V. I. Arnold, Journal of Applied Mathematics and Mechanics 30, 223 (1966).
Original Python Code from the dysts package
class ArnoldBeltramiChildress(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         dx = a * np.sin(z) + c * np.cos(y)
@@ -28,7 +28,7 @@
 
     @staticjit
     def _postprocessing(x, y, z):
-        return np.sin(x), np.cos(y), np.sin(z)
source
Dysts.ArnoldWeb — Function
   ArnoldWeb()
A quasi-integrable system that transitions to Chirikov diffusive dynamics when perturbed.
Stats
  • Embedding dimension: 5
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 22.554733912627043
  • Lyapunov spectrum (estimated): Any[0.06266308160064184, 0.0032177165561558833, 0.0011281526899086728, -0.00233619507966634, -0.0039196281630406]
  • Maximum Lyapunov exponent (estimated): 0.06514413995171335
  • Multiscale Entropy: 0.620621428966386
  • Non-autonomous: true
  • Period: 67.25
  • Pesin entropy: 0.06706808893310089
Citation
Froeschle, C., Guzzo, M. & Legga, E (2000). Graphical evolution of the Arnold web: From order to chaos. Science 289.
Original Python Code from the dysts package
class ArnoldWeb(DynSys):
+        return np.sin(x), np.cos(y), np.sin(z)
source
ChaoticDynamicalSystemsLibrary.ArnoldWeb — Function
   ArnoldWeb()
A quasi-integrable system that transitions to Chirikov diffusive dynamics when perturbed.
Stats
  • Embedding dimension: 5
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 22.554733912627043
  • Lyapunov spectrum (estimated): Any[0.06266308160064184, 0.0032177165561558833, 0.0011281526899086728, -0.00233619507966634, -0.0039196281630406]
  • Maximum Lyapunov exponent (estimated): 0.06514413995171335
  • Multiscale Entropy: 0.620621428966386
  • Non-autonomous: true
  • Period: 67.25
  • Pesin entropy: 0.06706808893310089
Citation
Froeschle, C., Guzzo, M. & Legga, E (2000). Graphical evolution of the Arnold web: From order to chaos. Science 289.
Original Python Code from the dysts package
class ArnoldWeb(DynSys):
     @staticjit
     def _rhs(p1, p2, x1, x2, z, t, mu, w):
         denom = 4 + np.cos(z) + np.cos(x1) + np.cos(x2)
@@ -41,7 +41,7 @@
 
     @staticjit
     def _postprocessing(p1, p2, x1, x2, z):
-        return p1, p2, np.sin(x1), np.sin(x2), np.cos(z)
source
Dysts.AtmosphericRegime — Function
   AtmosphericRegime()
A model of regime transition in atmospheric flows. Contains widely-separated timescales due a quadratic term, which itself arises from a conservation law.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1859596374283137
  • Lyapunov spectrum (estimated): nothing
  • Maximum Lyapunov exponent (estimated): 0.010543274932274862
  • Multiscale Entropy: 0.4278776303819699
  • Non-autonomous: false
  • Period: 89.59496756697331
  • Pesin entropy: 0.3055574134566322
Citation
Tuwankotta (2006). Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity
Original Python Code from the dysts package
class AtmosphericRegime(DynSys):
+        return p1, p2, np.sin(x1), np.sin(x2), np.cos(z)
source
ChaoticDynamicalSystemsLibrary.AtmosphericRegime — Function
   AtmosphericRegime()
A model of regime transition in atmospheric flows. Contains widely-separated timescales due a quadratic term, which itself arises from a conservation law.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1859596374283137
  • Lyapunov spectrum (estimated): nothing
  • Maximum Lyapunov exponent (estimated): 0.010543274932274862
  • Multiscale Entropy: 0.4278776303819699
  • Non-autonomous: false
  • Period: 89.59496756697331
  • Pesin entropy: 0.3055574134566322
Citation
Tuwankotta (2006). Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity
Original Python Code from the dysts package
class AtmosphericRegime(DynSys):
     @staticjit
     def _rhs(
         x, y, z, t, alpha, beta, mu1, mu2, omega, sigma
@@ -49,14 +49,14 @@
         xdot = mu1 * x + sigma * x * y
         ydot = mu2 * y + omega * z + alpha * y * z + beta * z ** 2 - sigma * x ** 2
         zdot = mu2 * z - omega * y - alpha * y ** 2 - beta * y * z
-        return xdot, ydot, zdot
source
Dysts.BeerRNN — Function
   BeerRNN()
A two-neuron minimal model nervous system.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0303797735862443
  • Lyapunov spectrum (estimated): Any[0.016274571192515633, -0.00011803510272572572, -0.5318869010933206]
  • Maximum Lyapunov exponent (estimated): 0.038767800131815223
  • Multiscale Entropy: 0.5244807630539362
  • Non-autonomous: false
  • Period: 181.58
  • Pesin entropy: 0.01647451962951782
Citation
Beer, R. D. (1995). On the dynamics of small continuous-time recurrent neural networks. Adapt. Behav., 3(4), 4692013509.
Original Python Code from the dysts package
class BeerRNN(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.BeerRNN — Function
   BeerRNN()
A two-neuron minimal model nervous system.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0303797735862443
  • Lyapunov spectrum (estimated): Any[0.016274571192515633, -0.00011803510272572572, -0.5318869010933206]
  • Maximum Lyapunov exponent (estimated): 0.038767800131815223
  • Multiscale Entropy: 0.5244807630539362
  • Non-autonomous: false
  • Period: 181.58
  • Pesin entropy: 0.01647451962951782
Citation
Beer, R. D. (1995). On the dynamics of small continuous-time recurrent neural networks. Adapt. Behav., 3(4), 4692013509.
Original Python Code from the dysts package
class BeerRNN(DynSys):
     @staticjit
     def _sig(x):
         return 1.0 / (1.0 + np.exp(-x))
 
     def rhs(self, X, t):
         Xdot = (-X + np.matmul(self.w, self._sig(X + self.theta))) / self.tau
-        return Xdot
source
Dysts.BelousovZhabotinsky — Function
   BelousovZhabotinsky()
A reduced-order model of the BZ reaction that exhibits period doubling. The bifurcation parameter for controlling the onset of chaos is kf. The system undergoes regular cycling when kf=3e-4, and chaotic oscillations when kf=3.5e-4
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0206505961796535
  • Lyapunov spectrum (estimated): Any[114.64513768787073, -0.4669861532822692, -5529.048679334775]
  • Maximum Lyapunov exponent (estimated): 132.08766434104578
  • Multiscale Entropy: 0.8764191281600929
  • Non-autonomous: false
  • Period: 0.018257
  • Pesin entropy: 114.64513768787073
Citation
Gyorgyi and Field (1992). A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction. Nature.
Original Python Code from the dysts package
class BelousovZhabotinsky(DynSys):
+        return Xdot
source
ChaoticDynamicalSystemsLibrary.BelousovZhabotinsky — Function
   BelousovZhabotinsky()
A reduced-order model of the BZ reaction that exhibits period doubling. The bifurcation parameter for controlling the onset of chaos is kf. The system undergoes regular cycling when kf=3e-4, and chaotic oscillations when kf=3.5e-4
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0206505961796535
  • Lyapunov spectrum (estimated): Any[114.64513768787073, -0.4669861532822692, -5529.048679334775]
  • Maximum Lyapunov exponent (estimated): 132.08766434104578
  • Multiscale Entropy: 0.8764191281600929
  • Non-autonomous: false
  • Period: 0.018257
  • Pesin entropy: 114.64513768787073
Citation
Gyorgyi and Field (1992). A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction. Nature.
Original Python Code from the dysts package
class BelousovZhabotinsky(DynSys):
     @staticjit
     def _rhs(
         x,
@@ -92,7 +92,7 @@
         xdot = c1 * x * ybar + c2 * ybar + c3 * x ** 2 + c4 * rf + c5 * x * z - kf * x
         zdot = (c6 / z0) * rf + c7 * x * z + c8 * z * v + c9 * z - kf * z
         vdot = c10 * x * ybar + c11 * ybar + c12 * x ** 2 + c13 * z * v - kf * v
-        return xdot * t0, zdot * t0, vdot * t0
source
Dysts.BickleyJet — Function
   BickleyJet()
A zonal jet passing between two counter rotating vortices. A steady Hamiltonian with a time-dependent perturbation.
Stats
  • Embedding dimension: 3
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 4.066710025133773
  • Lyapunov spectrum (estimated): Any[1.9651794377408363e-7, 2.5409880958463764e-9, -9.631983584415862e-8]
  • Maximum Lyapunov exponent (estimated): 3.0318672544949706e-6
  • Multiscale Entropy: 0.733566708397195
  • Non-autonomous: false
  • Period: 87826.57
  • Pesin entropy: 1.99059498764899e-7
Citation
Hadjighasem, Karrasch, Teramoto, Haller (2016). A Spectral Clustering Approach to Lagrangian Vortex Detection Phys Rev E.
Original Python Code from the dysts package
class BickleyJet(DynSys):
+        return xdot * t0, zdot * t0, vdot * t0
source
ChaoticDynamicalSystemsLibrary.BickleyJet — Function
   BickleyJet()
A zonal jet passing between two counter rotating vortices. A steady Hamiltonian with a time-dependent perturbation.
Stats
  • Embedding dimension: 3
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 4.066710025133773
  • Lyapunov spectrum (estimated): Any[1.9651794377408363e-7, 2.5409880958463764e-9, -9.631983584415862e-8]
  • Maximum Lyapunov exponent (estimated): 3.0318672544949706e-6
  • Multiscale Entropy: 0.733566708397195
  • Non-autonomous: false
  • Period: 87826.57
  • Pesin entropy: 1.99059498764899e-7
Citation
Hadjighasem, Karrasch, Teramoto, Haller (2016). A Spectral Clustering Approach to Lagrangian Vortex Detection Phys Rev E.
Original Python Code from the dysts package
class BickleyJet(DynSys):
     @staticjit
     def _rhs(y, x, z, t, ell, eps, k, omega, sigma, u):
         sechy = 1 / np.cosh(y / ell)
@@ -106,7 +106,7 @@
     def _postprocessing(self, x, y, z):
         km = np.min(self.k)
         sm = np.min(self.sigma)
-        return x, np.sin(km * y), np.sin(self.omega * z * km * sm)
source
Dysts.Blasius — Function
   Blasius()
+        return x, np.sin(km * y), np.sin(self.omega * z * km * sm)
source
ChaoticDynamicalSystemsLibrary.Blasius — Function
   Blasius()
 
 A chaotic food web composed of interacting predators,, consumers, and vegetation.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.016657959617314
  • Lyapunov spectrum (estimated): Any[0.04712755400915782, 0.001803137034352918, -2.9362739640325555]
  • Maximum Lyapunov exponent (estimated): 0.12597339364128174
  • Multiscale Entropy: 0.5917316835585642
  • Non-autonomous: false
  • Period: 6.6038
  • Pesin entropy: 0.04893069104351076
Citation
Blasius, Huppert, Stone. Nature 1999
Original Python Code from the dysts package
class Blasius(DynSys):
     @staticjit
@@ -114,7 +114,7 @@
         xdot = a * x - alpha1 * x * y / (1 + k1 * x)
         ydot = -b * y + alpha1 * x * y / (1 + k1 * x) - alpha2 * y * z / (1 + k2 * y)
         zdot = -c * (z - zs) + alpha2 * y * z / (1 + k2 * y)
-        return xdot, ydot, zdot
source
Dysts.BlinkingRotlet — Function
   BlinkingRotlet()
The location of the mixer is chosen so that there is a stagnation point at its symmetric complement. Solution is given in polar coordinates (r, theta).
Stats
  • Embedding dimension: 3
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 2.910902586026695
  • Lyapunov spectrum (estimated): Any[5.609358269449125, 0.007963961496808246, -6.16801583110239]
  • Maximum Lyapunov exponent (estimated): 2.120338159738003
  • Multiscale Entropy: 1.2895030395332947
  • Non-autonomous: true
  • Period: 2.9520625
  • Pesin entropy: 5.617567417119514
Citation
Meleshko & Aref. A blinking rotlet model for chaotic advection. Physics of Fluids 1996. See also Tallapragada and Sudarsanam PRE 2019
Original Python Code from the dysts package
class BlinkingRotlet(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.BlinkingRotlet — Function
   BlinkingRotlet()
The location of the mixer is chosen so that there is a stagnation point at its symmetric complement. Solution is given in polar coordinates (r, theta).
Stats
  • Embedding dimension: 3
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 2.910902586026695
  • Lyapunov spectrum (estimated): Any[5.609358269449125, 0.007963961496808246, -6.16801583110239]
  • Maximum Lyapunov exponent (estimated): 2.120338159738003
  • Multiscale Entropy: 1.2895030395332947
  • Non-autonomous: true
  • Period: 2.9520625
  • Pesin entropy: 5.617567417119514
Citation
Meleshko & Aref. A blinking rotlet model for chaotic advection. Physics of Fluids 1996. See also Tallapragada and Sudarsanam PRE 2019
Original Python Code from the dysts package
class BlinkingRotlet(DynSys):
     @staticjit
     def _rotlet(r, theta, a, b, bc):
         """A rotlet velocity field"""
@@ -147,19 +147,19 @@
         return self.sigma * dr, self.sigma * dth, dtt
 
     def _postprocessing(self, r, th, tt):
-        return r * np.cos(th), r * np.sin(th), np.sin(2 * np.pi * tt / self.tau)
source
Dysts.Bouali2 — Function
   Bouali2()
A modified economic cycle model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 16.471858210544983
  • Lyapunov spectrum (estimated): Any[0.020389225004628773, 0.0033455417942589724, -0.009602768742817189]
  • Maximum Lyapunov exponent (estimated): 0.07681730213975516
  • Multiscale Entropy: 0.4520009826326611
  • Non-autonomous: false
  • Period: 2.6779
  • Pesin entropy: 0.02439941982137089
Citation
Bouali (1999). Feedback loop in extended Van der Pols equation applied to an economic model of cycles. International Journal of Bifurkation and Chaos, Vol. 9, No. 4
Original Python Code from the dysts package
class Bouali2(DynSys):
+        return r * np.cos(th), r * np.sin(th), np.sin(2 * np.pi * tt / self.tau)
source
ChaoticDynamicalSystemsLibrary.Bouali2 — Function
   Bouali2()
A modified economic cycle model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 16.471858210544983
  • Lyapunov spectrum (estimated): Any[0.020389225004628773, 0.0033455417942589724, -0.009602768742817189]
  • Maximum Lyapunov exponent (estimated): 0.07681730213975516
  • Multiscale Entropy: 0.4520009826326611
  • Non-autonomous: false
  • Period: 2.6779
  • Pesin entropy: 0.02439941982137089
Citation
Bouali (1999). Feedback loop in extended Van der Pols equation applied to an economic model of cycles. International Journal of Bifurkation and Chaos, Vol. 9, No. 4
Original Python Code from the dysts package
class Bouali2(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, bb, c, g, m, y0):
         xdot = a * y0 * x - a * x * y - b * z
         ydot = -g * y + g * y * x ** 2
         zdot = -1.5 * m * x + m * bb * x * z - c * z
-        return xdot, ydot, zdot
source
Dysts.BurkeShaw — Function
   BurkeShaw()
A scroll-like attractor with unique symmetry along the z axis.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2113447891991065
  • Lyapunov spectrum (estimated): Any[2.3453555864112117, 0.027056468824027846, -11.225316054517004]
  • Maximum Lyapunov exponent (estimated): 2.324596351534631
  • Multiscale Entropy: 0.906432166797223
  • Non-autonomous: false
  • Period: 0.78333
  • Pesin entropy: 2.37241205523524
Citation
Shaw (1981). Zeitschrift fur Naturforschung.
Original Python Code from the dysts package
class BurkeShaw(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.BurkeShaw — Function
   BurkeShaw()
A scroll-like attractor with unique symmetry along the z axis.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2113447891991065
  • Lyapunov spectrum (estimated): Any[2.3453555864112117, 0.027056468824027846, -11.225316054517004]
  • Maximum Lyapunov exponent (estimated): 2.324596351534631
  • Multiscale Entropy: 0.906432166797223
  • Non-autonomous: false
  • Period: 0.78333
  • Pesin entropy: 2.37241205523524
Citation
Shaw (1981). Zeitschrift fur Naturforschung.
Original Python Code from the dysts package
class BurkeShaw(DynSys):
     @staticjit
     def _rhs(x, y, z, t, e, n):
         xdot = -n * x - n * y
         ydot = y - n * x * z
         zdot = n * x * y + e
-        return xdot, ydot, zdot
source
Dysts.CaTwoPlus — Function
   CaTwoPlus()
Intracellular calcium ion oscillations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0647813731079956
  • Lyapunov spectrum (estimated): Any[0.5788123099350391, 0.022652249040266133, -9.284523260554844]
  • Maximum Lyapunov exponent (estimated): 0.5802865327706731
  • Multiscale Entropy: 0.5770296569860646
  • Non-autonomous: false
  • Period: 1.343
  • Pesin entropy: 0.6014645592138562
Citation
Houart, Dupont, Goldbeter. Bull Math Biol 1999.
Original Python Code from the dysts package
class CaTwoPlus(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.CaTwoPlus — Function
   CaTwoPlus()
Intracellular calcium ion oscillations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0647813731079956
  • Lyapunov spectrum (estimated): Any[0.5788123099350391, 0.022652249040266133, -9.284523260554844]
  • Maximum Lyapunov exponent (estimated): 0.5802865327706731
  • Multiscale Entropy: 0.5770296569860646
  • Non-autonomous: false
  • Period: 1.343
  • Pesin entropy: 0.6014645592138562
Citation
Houart, Dupont, Goldbeter. Bull Math Biol 1999.
Original Python Code from the dysts package
class CaTwoPlus(DynSys):
     def rhs(self, X, t):
         z, y, a = X
         Vin = self.V0 + self.V1 * self.beta
@@ -177,7 +177,7 @@
         zdot = Vin - V2 + V3 + self.kf * y - self.k * z
         ydot = V2 - V3 - self.kf * y
         adot = self.beta * self.V4 - V5 - self.eps * a
-        return (zdot, ydot, adot)
source
Dysts.CellCycle — Function
   CellCycle()
A simplified model of the cell cycle. The parameter Kim controls the bifurcations.
Stats
  • Embedding dimension: 6
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.020810681242375
  • Lyapunov spectrum (estimated): Any[0.00469985418122158, -0.0003567612310661562, -0.20915644240415007, -0.43394905365820846, -2.850536314608979, -6.767614340964144]
  • Maximum Lyapunov exponent (estimated): 0.010172128409635746
  • Multiscale Entropy: 1.021736994304444
  • Non-autonomous: false
  • Period: 204.71
  • Pesin entropy: 0.004765100518900255
Citation
Romond, Rustici, Gonze, Goldbeter. 1999.
Original Python Code from the dysts package
class CellCycle(DynSys):
+        return (zdot, ydot, adot)
source
ChaoticDynamicalSystemsLibrary.CellCycle — Function
   CellCycle()
A simplified model of the cell cycle. The parameter Kim controls the bifurcations.
Stats
  • Embedding dimension: 6
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.020810681242375
  • Lyapunov spectrum (estimated): Any[0.00469985418122158, -0.0003567612310661562, -0.20915644240415007, -0.43394905365820846, -2.850536314608979, -6.767614340964144]
  • Maximum Lyapunov exponent (estimated): 0.010172128409635746
  • Multiscale Entropy: 1.021736994304444
  • Non-autonomous: false
  • Period: 204.71
  • Pesin entropy: 0.004765100518900255
Citation
Romond, Rustici, Gonze, Goldbeter. 1999.
Original Python Code from the dysts package
class CellCycle(DynSys):
     def rhs(self, X, t):
         c1, m1, x1, c2, m2, x2 = X
         Vm1, Um1 = 2 * [self.Vm1]
@@ -202,7 +202,7 @@
         m2dot = U1 * (1 - m2) / (H1 + (1 - m2)) - U2 * m2 / (H2 + m2)
         x1dot = V3 * (1 - x1) / (K3 + (1 - x1)) - V4 * x1 / (K4 + x1)
         x2dot = U3 * (1 - x2) / (H3 + (1 - x2)) - U4 * x2 / (H4 + x2)
-        return c1dot, m1dot, x1dot, c2dot, m2dot, x2dot
source
Dysts.CellularNeuralNetwork — Function
   CellularNeuralNetwork()
Cellular neural network dynamics.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.167668286708633
  • Lyapunov spectrum (estimated): Any[0.09799656528996928, 0.0036354952626342415, -0.6061495728157424]
  • Maximum Lyapunov exponent (estimated): 2.3258468977096562
  • Multiscale Entropy: 1.2414841874061011
  • Non-autonomous: false
  • Period: 3.2
  • Pesin entropy: 0.1016320623247746
Citation
Arena, Caponetto, Fortuna, and Porto., Int J Bifurcat Chaos. 1998:  1527-1539.
Original Python Code from the dysts package
class CellularNeuralNetwork(DynSys):
+        return c1dot, m1dot, x1dot, c2dot, m2dot, x2dot
source
ChaoticDynamicalSystemsLibrary.CellularNeuralNetwork — Function
   CellularNeuralNetwork()
Cellular neural network dynamics.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.167668286708633
  • Lyapunov spectrum (estimated): Any[0.09799656528996928, 0.0036354952626342415, -0.6061495728157424]
  • Maximum Lyapunov exponent (estimated): 2.3258468977096562
  • Multiscale Entropy: 1.2414841874061011
  • Non-autonomous: false
  • Period: 3.2
  • Pesin entropy: 0.1016320623247746
Citation
Arena, Caponetto, Fortuna, and Porto., Int J Bifurcat Chaos. 1998:  1527-1539.
Original Python Code from the dysts package
class CellularNeuralNetwork(DynSys):
     @staticjit
     def f(x):
         return 0.5 * (np.abs(x + 1) - np.abs(x - 1))
@@ -212,26 +212,26 @@
         xdot = -x + self.d * self.f(x) - self.b * self.f(y) - self.b * self.f(z)
         ydot = -y - self.b * self.f(x) + self.c * self.f(y) - self.a * self.f(z)
         zdot = -z - self.b * self.f(x) + self.a * self.f(y) + self.f(z)
-        return (xdot, ydot, zdot)
source
Dysts.Chen — Function
   Chen()
A system based on feedback anti-control in engineering.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.168621926035506
  • Lyapunov spectrum (estimated): Any[2.0471208808332557, -0.05031354288886987, -11.841816202859391]
  • Maximum Lyapunov exponent (estimated): 2.005961349488129
  • Multiscale Entropy: 0.7604529380690989
  • Non-autonomous: false
  • Period: 0.58689
  • Pesin entropy: 2.047313967129477
Citation
Chen (1997). Proc. First Int. Conf. Control of Oscillations and Chaos.
Original Python Code from the dysts package
class Chen(DynSys):
+        return (xdot, ydot, zdot)
source
ChaoticDynamicalSystemsLibrary.Chen — Function
   Chen()
A system based on feedback anti-control in engineering.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.168621926035506
  • Lyapunov spectrum (estimated): Any[2.0471208808332557, -0.05031354288886987, -11.841816202859391]
  • Maximum Lyapunov exponent (estimated): 2.005961349488129
  • Multiscale Entropy: 0.7604529380690989
  • Non-autonomous: false
  • Period: 0.58689
  • Pesin entropy: 2.047313967129477
Citation
Chen (1997). Proc. First Int. Conf. Control of Oscillations and Chaos.
Original Python Code from the dysts package
class Chen(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = a * y - a * x
         ydot = (c - a) * x - x * z + c * y
         zdot = x * y - b * z
-        return xdot, ydot, zdot
source
Dysts.ChenLee — Function
   ChenLee()
A rigid body with feedback anti-control.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0300416878282137
  • Lyapunov spectrum (estimated): Any[0.16460631102343384, 0.00018225769802326938, -5.485329903201408]
  • Maximum Lyapunov exponent (estimated): 0.16363115430160305
  • Multiscale Entropy: 0.6224325755254609
  • Non-autonomous: false
  • Period: 2.8043
  • Pesin entropy: 0.16478856872146466
Citation
Chen HK, Lee CI (2004). Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals
Original Python Code from the dysts package
class ChenLee(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.ChenLee — Function
   ChenLee()
A rigid body with feedback anti-control.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0300416878282137
  • Lyapunov spectrum (estimated): Any[0.16460631102343384, 0.00018225769802326938, -5.485329903201408]
  • Maximum Lyapunov exponent (estimated): 0.16363115430160305
  • Multiscale Entropy: 0.6224325755254609
  • Non-autonomous: false
  • Period: 2.8043
  • Pesin entropy: 0.16478856872146466
Citation
Chen HK, Lee CI (2004). Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals
Original Python Code from the dysts package
class ChenLee(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = a * x - y * z
         ydot = b * y + x * z
         zdot = c * z + 0.3333333333333333333333333 * x * y
-        return xdot, ydot, zdot
source
Dysts.Chua — Function
   Chua()
An electronic circuit with a diode providing negative resistance.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.10727056025829
  • Lyapunov spectrum (estimated): Any[0.4605474829443073, 0.0022048281168492866, -4.319076735248446]
  • Maximum Lyapunov exponent (estimated): 1.2301163591725595
  • Multiscale Entropy: 1.2693527097629784
  • Non-autonomous: false
  • Period: 1.4235
  • Pesin entropy: 0.4645707969482971
Citation
Chua, L. O. (1969) Introduction to Nonlinear Network Theory, McGraw-Hill, New York.
Original Python Code from the dysts package
class Chua(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Chua — Function
   Chua()
An electronic circuit with a diode providing negative resistance.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.10727056025829
  • Lyapunov spectrum (estimated): Any[0.4605474829443073, 0.0022048281168492866, -4.319076735248446]
  • Maximum Lyapunov exponent (estimated): 1.2301163591725595
  • Multiscale Entropy: 1.2693527097629784
  • Non-autonomous: false
  • Period: 1.4235
  • Pesin entropy: 0.4645707969482971
Citation
Chua, L. O. (1969) Introduction to Nonlinear Network Theory, McGraw-Hill, New York.
Original Python Code from the dysts package
class Chua(DynSys):
     @staticjit
     def _rhs(x, y, z, t, alpha, beta, m0, m1):
         ramp_x = m1 * x + 0.5 * (m0 - m1) * (np.abs(x + 1) - np.abs(x - 1))
         xdot = alpha * (y - x - ramp_x)
         ydot = x - y + z
         zdot = -beta * y
-        return xdot, ydot, zdot
source
Dysts.CircadianRhythm — Function
   CircadianRhythm()
The Drosophila circadian rhythm under periodic light/dark forcing.
Stats
  • Embedding dimension: 5
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.640554412377753
  • Lyapunov spectrum (estimated): Any[0.09298369588063138, 0.00046501992305171574, -0.14586025062784352, -0.3598453092044599, -1.091393481159416]
  • Maximum Lyapunov exponent (estimated): 0.006263277393890506
  • Multiscale Entropy: 0.7054585345031309
  • Non-autonomous: true
  • Period: 27.0
  • Pesin entropy: 0.09372724379002938
Citation
Leloup, Gonze, Goldbeter. 1999. Gonze, Leloup, Goldbeter. 2000
Original Python Code from the dysts package
class CircadianRhythm(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.CircadianRhythm — Function
   CircadianRhythm()
The Drosophila circadian rhythm under periodic light/dark forcing.
Stats
  • Embedding dimension: 5
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.640554412377753
  • Lyapunov spectrum (estimated): Any[0.09298369588063138, 0.00046501992305171574, -0.14586025062784352, -0.3598453092044599, -1.091393481159416]
  • Maximum Lyapunov exponent (estimated): 0.006263277393890506
  • Multiscale Entropy: 0.7054585345031309
  • Non-autonomous: true
  • Period: 27.0
  • Pesin entropy: 0.09372724379002938
Citation
Leloup, Gonze, Goldbeter. 1999. Gonze, Leloup, Goldbeter. 2000
Original Python Code from the dysts package
class CircadianRhythm(DynSys):
     @staticjit
     def _rhs(
         m,
@@ -266,7 +266,7 @@
 
     @staticjit
     def _postprocessing(m, fc, fs, fn, th):
-        return m, fc, fs, fn, np.cos(th)
source
Dysts.CoevolvingPredatorPrey — Function
   CoevolvingPredatorPrey()
A system of predator-prey equations with co-evolving prey.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0026539580881706
  • Lyapunov spectrum (estimated): Any[0.0034879428860592264, 0.0017391810332012293, -1.9676998121970224]
  • Maximum Lyapunov exponent (estimated): 0.00732698039035783
  • Multiscale Entropy: 1.006450625395487
  • Non-autonomous: false
  • Period: 122.77
  • Pesin entropy: 0.005379136261058156
Citation
Gilpin & Feldman (2017). PLOS Comp Biol
Original Python Code from the dysts package
class CoevolvingPredatorPrey(DynSys):
+        return m, fc, fs, fn, np.cos(th)
source
ChaoticDynamicalSystemsLibrary.CoevolvingPredatorPrey — Function
   CoevolvingPredatorPrey()
A system of predator-prey equations with co-evolving prey.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0026539580881706
  • Lyapunov spectrum (estimated): Any[0.0034879428860592264, 0.0017391810332012293, -1.9676998121970224]
  • Maximum Lyapunov exponent (estimated): 0.00732698039035783
  • Multiscale Entropy: 1.006450625395487
  • Non-autonomous: false
  • Period: 122.77
  • Pesin entropy: 0.005379136261058156
Citation
Gilpin & Feldman (2017). PLOS Comp Biol
Original Python Code from the dysts package
class CoevolvingPredatorPrey(DynSys):
     @staticjit
     def _rhs(x, y, alpha, t, a1, a2, a3, b1, b2, d1, d2, delta, k1, k2, k4, vv):
         xdot = x * (
@@ -284,7 +284,7 @@
             -((a1 * k1 * x * alpha * delta) / (1 + b1 * alpha))
             - d1 * (-2 * k2 * alpha * delta ** 2 + 4 * k4 * alpha ** 3 * delta ** 4)
         )
-        return xdot, ydot, alphadot
source
Dysts.Colpitts — Function
   Colpitts()
An electrical circuit used as a signal generator.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.134959031654127
  • Lyapunov spectrum (estimated): Any[0.13750285290669104, -0.0021021241106901328, -1.0014048629576902]
  • Maximum Lyapunov exponent (estimated): 0.3736008151161584
  • Multiscale Entropy: 0.7266290517682168
  • Non-autonomous: false
  • Period: 4.2676
  • Pesin entropy: 0.1390751886208436
Citation
Kennedy (2007). IEEE Trans Circuits & Systems. 1994: 771-774. Li, Zhou, Yang. Chaos, Solitons, & Fractals.
Original Python Code from the dysts package
class Colpitts(DynSys):
+        return xdot, ydot, alphadot
source
ChaoticDynamicalSystemsLibrary.Colpitts — Function
   Colpitts()
An electrical circuit used as a signal generator.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.134959031654127
  • Lyapunov spectrum (estimated): Any[0.13750285290669104, -0.0021021241106901328, -1.0014048629576902]
  • Maximum Lyapunov exponent (estimated): 0.3736008151161584
  • Multiscale Entropy: 0.7266290517682168
  • Non-autonomous: false
  • Period: 4.2676
  • Pesin entropy: 0.1390751886208436
Citation
Kennedy (2007). IEEE Trans Circuits & Systems. 1994: 771-774. Li, Zhou, Yang. Chaos, Solitons, & Fractals.
Original Python Code from the dysts package
class Colpitts(DynSys):
     def rhs(self, X, t):
         x, y, z = X
         u = z - (self.e - 1)
@@ -292,25 +292,25 @@
         xdot = y - self.a * fz
         ydot = self.c - x - self.b * y - z
         zdot = y - self.d * z
-        return (xdot, ydot, zdot)
source
Dysts.Coullet — Function
   Coullet()
A variant of the Arneodo attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.252992250313846
  • Lyapunov spectrum (estimated): Any[0.14890693833273877, -6.504894715051528e-5, -0.5873658566804084]
  • Maximum Lyapunov exponent (estimated): 0.13718762024831516
  • Multiscale Entropy: 0.9055084400125268
  • Non-autonomous: false
  • Period: 5.7663
  • Pesin entropy: 0.14919849957322842
Citation
Arneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional volterra equations. Physics Letters A 79, 259–263 (1980)
Original Python Code from the dysts package
class Rucklidge(DynSys):
+        return (xdot, ydot, zdot)
source
ChaoticDynamicalSystemsLibrary.Coullet — Function
   Coullet()
A variant of the Arneodo attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.252992250313846
  • Lyapunov spectrum (estimated): Any[0.14890693833273877, -6.504894715051528e-5, -0.5873658566804084]
  • Maximum Lyapunov exponent (estimated): 0.13718762024831516
  • Multiscale Entropy: 0.9055084400125268
  • Non-autonomous: false
  • Period: 5.7663
  • Pesin entropy: 0.14919849957322842
Citation
Arneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional volterra equations. Physics Letters A 79, 259–263 (1980)
Original Python Code from the dysts package
class Rucklidge(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = -a * x + b * y - y * z
         ydot = x
         zdot = -z + y ** 2
-        return xdot, ydot, zdot
source
Dysts.Dadras — Function
   Dadras()
An electronic circuit capable of producing multiple attractors under bifurcations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0430103150081735
  • Lyapunov spectrum (estimated): Any[0.47344233096408, 0.0064562720284210215, -11.157725546142514]
  • Maximum Lyapunov exponent (estimated): 0.40591521049520063
  • Multiscale Entropy: 0.8537980405426894
  • Non-autonomous: false
  • Period: 3.2886
  • Pesin entropy: 0.4798988304624064
Citation
S Dadras, HR Momeni (2009). A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Physics Letters A.
Original Python Code from the dysts package
class Dadras(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Dadras — Function
   Dadras()
An electronic circuit capable of producing multiple attractors under bifurcations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0430103150081735
  • Lyapunov spectrum (estimated): Any[0.47344233096408, 0.0064562720284210215, -11.157725546142514]
  • Maximum Lyapunov exponent (estimated): 0.40591521049520063
  • Multiscale Entropy: 0.8537980405426894
  • Non-autonomous: false
  • Period: 3.2886
  • Pesin entropy: 0.4798988304624064
Citation
S Dadras, HR Momeni (2009). A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Physics Letters A.
Original Python Code from the dysts package
class Dadras(DynSys):
     @staticjit
     def _rhs(x, y, z, t, c, e, o, p, r):
         xdot = y - p * x + o * y * z
         ydot = r * y - x * z + z
         zdot = c * x * y - e * z
-        return xdot, ydot, zdot
source
Dysts.DequanLi — Function
   DequanLi()
Related to the Three Scroll unified attractor TSUCS-2
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.6848604732863537
  • Lyapunov spectrum (estimated): Any[1.8601963396884453, 0.09866428540326228, -2.867185439620747]
  • Maximum Lyapunov exponent (estimated): 0.739747976641268
  • Multiscale Entropy: 0.8319369667525882
  • Non-autonomous: false
  • Period: 0.083806
  • Pesin entropy: 1.9588606250917073
Citation
Li, Phys Lett A. 2008: 387-393.
Original Python Code from the dysts package
class DequanLi(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.DequanLi — Function
   DequanLi()
Related to the Three Scroll unified attractor TSUCS-2
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.6848604732863537
  • Lyapunov spectrum (estimated): Any[1.8601963396884453, 0.09866428540326228, -2.867185439620747]
  • Maximum Lyapunov exponent (estimated): 0.739747976641268
  • Multiscale Entropy: 0.8319369667525882
  • Non-autonomous: false
  • Period: 0.083806
  • Pesin entropy: 1.9588606250917073
Citation
Li, Phys Lett A. 2008: 387-393.
Original Python Code from the dysts package
class DequanLi(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, c, d, eps, f, k):
         xdot = a * y - a * x + d * x * z
         ydot = k * x + f * y - x * z
         zdot = c * z + x * y - eps * x ** 2
-        return xdot, ydot, zdot
source
Dysts.DoubleGyre — Function
   DoubleGyre()
A time-dependent fluid flow exhibiting Lagrangian coherent structures
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.881525629972418
  • Lyapunov spectrum (estimated): Any[0.06285936079346421, -0.0036702088073069326, -0.06633987533400756]
  • Maximum Lyapunov exponent (estimated): 0.12185254082886439
  • Multiscale Entropy: 0.66519273658672
  • Non-autonomous: true
  • Period: 13.953
  • Pesin entropy: 0.06285936079346421
Citation
Shadden, Lekien, Marsden (2005). Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D.
Original Python Code from the dysts package
class DoubleGyre(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.DoubleGyre — Function
   DoubleGyre()
A time-dependent fluid flow exhibiting Lagrangian coherent structures
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.881525629972418
  • Lyapunov spectrum (estimated): Any[0.06285936079346421, -0.0036702088073069326, -0.06633987533400756]
  • Maximum Lyapunov exponent (estimated): 0.12185254082886439
  • Multiscale Entropy: 0.66519273658672
  • Non-autonomous: true
  • Period: 13.953
  • Pesin entropy: 0.06285936079346421
Citation
Shadden, Lekien, Marsden (2005). Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D.
Original Python Code from the dysts package
class DoubleGyre(DynSys):
     @staticjit
     def _rhs(x, y, z, t, alpha, eps, omega):
         a = eps * np.sin(z)
@@ -323,7 +323,7 @@
 
     @staticjit
     def _postprocessing(x, y, z):
-        return x, y, np.sin(z)
source
Dysts.DoublePendulum — Function
   DoublePendulum()
Two coupled rigid pendula without damping.
Stats
  • Embedding dimension: 4
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 3.810221473882809
  • Lyapunov spectrum (estimated): Any[8.464443924343541, 0.46411672696182493, -0.17345525567611142, -10.805817517613212]
  • Maximum Lyapunov exponent (estimated): 3.985298067514186
  • Multiscale Entropy: 0.5591104791210428
  • Non-autonomous: false
  • Period: 4.0234
  • Pesin entropy: 8.928560651305366
Citation
See, for example: Marion (2013). Classical dynamics of particles and systems.
Original Python Code from the dysts package
class DoublePendulum(DynSys):
+        return x, y, np.sin(z)
source
ChaoticDynamicalSystemsLibrary.DoublePendulum — Function
   DoublePendulum()
Two coupled rigid pendula without damping.
Stats
  • Embedding dimension: 4
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 3.810221473882809
  • Lyapunov spectrum (estimated): Any[8.464443924343541, 0.46411672696182493, -0.17345525567611142, -10.805817517613212]
  • Maximum Lyapunov exponent (estimated): 3.985298067514186
  • Multiscale Entropy: 0.5591104791210428
  • Non-autonomous: false
  • Period: 4.0234
  • Pesin entropy: 8.928560651305366
Citation
See, for example: Marion (2013). Classical dynamics of particles and systems.
Original Python Code from the dysts package
class DoublePendulum(DynSys):
     @staticjit
     def _rhs(th1, th2, p1, p2, t, d, m):
         g = 9.82
@@ -341,7 +341,7 @@
             * (m * d ** 2)
             * (-th1_dot * th2_dot * np.sin(th1 - th2) + 3 * (g / d) * np.sin(th2))
         )
-        return th1_dot, th2_dot, p1_dot, p2_dot
source
Dysts.Duffing — Function
   Duffing()
A monochromatically-forced rigid pendulum, with the nonlinearity approximated as cubic.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.561219319140712
  • Lyapunov spectrum (estimated): Any[0.09952054532941002, 0.0003077550037828977, -0.17595301607629793]
  • Maximum Lyapunov exponent (estimated): 0.14026545752865122
  • Multiscale Entropy: 0.8686568999981737
  • Non-autonomous: true
  • Period: 7.4496
  • Pesin entropy: 0.10008657388524086
Citation
Duffing, G. (1918), Forced oscillations with variable natural frequency and their technical relevance, Heft 41/42, Braunschweig: Vieweg.
Original Python Code from the dysts package
class Duffing(DynSys):
+        return th1_dot, th2_dot, p1_dot, p2_dot
source
ChaoticDynamicalSystemsLibrary.Duffing — Function
   Duffing()
A monochromatically-forced rigid pendulum, with the nonlinearity approximated as cubic.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.561219319140712
  • Lyapunov spectrum (estimated): Any[0.09952054532941002, 0.0003077550037828977, -0.17595301607629793]
  • Maximum Lyapunov exponent (estimated): 0.14026545752865122
  • Multiscale Entropy: 0.8686568999981737
  • Non-autonomous: true
  • Period: 7.4496
  • Pesin entropy: 0.10008657388524086
Citation
Duffing, G. (1918), Forced oscillations with variable natural frequency and their technical relevance, Heft 41/42, Braunschweig: Vieweg.
Original Python Code from the dysts package
class Duffing(DynSys):
     @staticjit
     def _rhs(x, y, z, t, alpha, beta, delta, gamma, omega):
         xdot = y
@@ -351,7 +351,7 @@
 
     @staticjit
     def _postprocessing(x, y, z):
-        return x, y, np.cos(z)
source
Dysts.ExcitableCell — Function
   ExcitableCell()
A reduced-order variant of the Hodgkin-Huxley model. The parameter gkc controls the onset of chaos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1287842177704674
  • Lyapunov spectrum (estimated): Any[17.030315084501463, -0.3267053778962445, -129.70230355691595]
  • Maximum Lyapunov exponent (estimated): 0.7164303367057223
  • Multiscale Entropy: 0.7807524009298388
  • Non-autonomous: false
  • Period: 0.5528
  • Pesin entropy: 17.030315084501463
Citation
Teresa Chay. Chaos In A Three-variable Model Of An Excitable Cell. Physica D 1985
Original Python Code from the dysts package
class ExcitableCell(DynSys):
+        return x, y, np.cos(z)
source
ChaoticDynamicalSystemsLibrary.ExcitableCell — Function
   ExcitableCell()
A reduced-order variant of the Hodgkin-Huxley model. The parameter gkc controls the onset of chaos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1287842177704674
  • Lyapunov spectrum (estimated): Any[17.030315084501463, -0.3267053778962445, -129.70230355691595]
  • Maximum Lyapunov exponent (estimated): 0.7164303367057223
  • Multiscale Entropy: 0.7807524009298388
  • Non-autonomous: false
  • Period: 0.5528
  • Pesin entropy: 17.030315084501463
Citation
Teresa Chay. Chaos In A Three-variable Model Of An Excitable Cell. Physica D 1985
Original Python Code from the dysts package
class ExcitableCell(DynSys):
     def rhs(self, X, t):
         v, n, c = X
 
@@ -378,13 +378,13 @@
         )
         ndot = (ninf - n) / tau
         cdot = self.rho * (minf ** 3 * hinf * (self.vc - v) - self.kc * c)
-        return vdot, ndot, cdot
source
Dysts.Finance — Function
   Finance()
Stock fluctuations under varying investment demand and interest rates
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.172040321306389
  • Lyapunov spectrum (estimated): Any[0.10708598383274774, 0.0009835797228343356, -0.6279525877233353]
  • Maximum Lyapunov exponent (estimated): 0.09354106442670211
  • Multiscale Entropy: 0.7817861077722251
  • Non-autonomous: false
  • Period: 9.8932
  • Pesin entropy: 0.1084488028544319
Citation
Guoliang Cai, Juanjuan Huang. International Journal of Nonlinear Science, Vol. 3 (2007)
Original Python Code from the dysts package
class Finance(DynSys):
+        return vdot, ndot, cdot
source
ChaoticDynamicalSystemsLibrary.Finance — Function
   Finance()
Stock fluctuations under varying investment demand and interest rates
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.172040321306389
  • Lyapunov spectrum (estimated): Any[0.10708598383274774, 0.0009835797228343356, -0.6279525877233353]
  • Maximum Lyapunov exponent (estimated): 0.09354106442670211
  • Multiscale Entropy: 0.7817861077722251
  • Non-autonomous: false
  • Period: 9.8932
  • Pesin entropy: 0.1084488028544319
Citation
Guoliang Cai, Juanjuan Huang. International Journal of Nonlinear Science, Vol. 3 (2007)
Original Python Code from the dysts package
class Finance(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = (1 / b - a) * x + z + x * y
         ydot = -b * y - x ** 2
         zdot = -x - c * z
-        return xdot, ydot, zdot
source
Dysts.FluidTrampoline — Function
   FluidTrampoline()
A droplet bouncing on a horizontal soap film.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.8462636972682116
  • Lyapunov spectrum (estimated): Any[0.08081301489037329, -0.0011783910877478326, -0.09403052188090477]
  • Maximum Lyapunov exponent (estimated): 0.22311237411461174
  • Multiscale Entropy: 0.796312585554829
  • Non-autonomous: true
  • Period: 7.7351
  • Pesin entropy: 0.08094585172323163
Citation
Gilet, Bush. The fluid trampoline: droplets bouncing on a soap film. JFM 2009.
Original Python Code from the dysts package
class FluidTrampoline(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.FluidTrampoline — Function
   FluidTrampoline()
A droplet bouncing on a horizontal soap film.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.8462636972682116
  • Lyapunov spectrum (estimated): Any[0.08081301489037329, -0.0011783910877478326, -0.09403052188090477]
  • Maximum Lyapunov exponent (estimated): 0.22311237411461174
  • Multiscale Entropy: 0.796312585554829
  • Non-autonomous: true
  • Period: 7.7351
  • Pesin entropy: 0.08094585172323163
Citation
Gilet, Bush. The fluid trampoline: droplets bouncing on a soap film. JFM 2009.
Original Python Code from the dysts package
class FluidTrampoline(DynSys):
     @staticmethod
     def _rhs(x, y, th, t, gamma, psi, w):
         xdot = y
@@ -394,7 +394,7 @@
 
     @staticjit
     def _postprocessing(x, y, th):
-        return x, y, np.cos(th)
source
Dysts.ForcedBrusselator — Function
   ForcedBrusselator()
An autocatalytic chemical system.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.190019824057157
  • Lyapunov spectrum (estimated): Any[0.0028307087627546254, -0.014898681909916936, -0.23304448314240073]
  • Maximum Lyapunov exponent (estimated): 0.015326007974604663
  • Multiscale Entropy: 0.47299463051887286
  • Non-autonomous: true
  • Period: 20.411
  • Pesin entropy: 0.0028307087627546254
Citation
I. Prigogine, From Being to Becoming: Time and Complexity in the Physical Sciences, New York: W.H. Freeman and Company, 1980.
Original Python Code from the dysts package
class ForcedBrusselator(DynSys):
+        return x, y, np.cos(th)
source
ChaoticDynamicalSystemsLibrary.ForcedBrusselator — Function
   ForcedBrusselator()
An autocatalytic chemical system.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.190019824057157
  • Lyapunov spectrum (estimated): Any[0.0028307087627546254, -0.014898681909916936, -0.23304448314240073]
  • Maximum Lyapunov exponent (estimated): 0.015326007974604663
  • Multiscale Entropy: 0.47299463051887286
  • Non-autonomous: true
  • Period: 20.411
  • Pesin entropy: 0.0028307087627546254
Citation
I. Prigogine, From Being to Becoming: Time and Complexity in the Physical Sciences, New York: W.H. Freeman and Company, 1980.
Original Python Code from the dysts package
class ForcedBrusselator(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, f, w):
         xdot = a + x ** 2 * y - (b + 1) * x + f * np.cos(z)
@@ -404,7 +404,7 @@
 
     @staticjit
     def _postprocessing(x, y, z):
-        return x, y, np.sin(z)
source
Dysts.ForcedFitzHughNagumo — Function
   ForcedFitzHughNagumo()
A driven neuron model sustaining both quiesent and oscillating states.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.5705852319395741
  • Lyapunov spectrum (estimated): Any[0.0006271045837994497, -0.0017541532237019502, -0.9544351506941636]
  • Maximum Lyapunov exponent (estimated): 0.007561278655908403
  • Multiscale Entropy: 0.946734422406113
  • Non-autonomous: true
  • Period: 42.12235294117647
  • Pesin entropy: 0.000828176101772673
Citation
FitzHugh, Richard (1961). Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal. 1 (6).
Original Python Code from the dysts package
class ForcedFitzHughNagumo(DynSys):
+        return x, y, np.sin(z)
source
ChaoticDynamicalSystemsLibrary.ForcedFitzHughNagumo — Function
   ForcedFitzHughNagumo()
A driven neuron model sustaining both quiesent and oscillating states.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.5705852319395741
  • Lyapunov spectrum (estimated): Any[0.0006271045837994497, -0.0017541532237019502, -0.9544351506941636]
  • Maximum Lyapunov exponent (estimated): 0.007561278655908403
  • Multiscale Entropy: 0.946734422406113
  • Non-autonomous: true
  • Period: 42.12235294117647
  • Pesin entropy: 0.000828176101772673
Citation
FitzHugh, Richard (1961). Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal. 1 (6).
Original Python Code from the dysts package
class ForcedFitzHughNagumo(DynSys):
     @staticjit
     def _rhs(v, w, z, t, a, b, curr, f, gamma, omega):
         vdot = v - v ** 3 / 3 - w + curr + f * np.sin(z)
@@ -414,7 +414,7 @@
 
     @staticjit
     def _postprocessing(x, y, z):
-        return x, y, np.sin(z)
source
Dysts.ForcedVanDerPol — Function
   ForcedVanDerPol()
An electronic circuit containing a triode.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.012828649055009
  • Lyapunov spectrum (estimated): Any[0.2651824388815116, -0.0008450617218488057, -20.60632249066986]
  • Maximum Lyapunov exponent (estimated): 0.0035097622587646343
  • Multiscale Entropy: 0.39443058905146283
  • Non-autonomous: true
  • Period: 14.476875
  • Pesin entropy: 0.26518243888151155
Citation
B. van der Pol (1920). A theory of the amplitude of free and forced triode vibrations. Radio Review 1.
Original Python Code from the dysts package
class ForcedVanDerPol(DynSys):
+        return x, y, np.sin(z)
source
ChaoticDynamicalSystemsLibrary.ForcedVanDerPol — Function
   ForcedVanDerPol()
An electronic circuit containing a triode.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.012828649055009
  • Lyapunov spectrum (estimated): Any[0.2651824388815116, -0.0008450617218488057, -20.60632249066986]
  • Maximum Lyapunov exponent (estimated): 0.0035097622587646343
  • Multiscale Entropy: 0.39443058905146283
  • Non-autonomous: true
  • Period: 14.476875
  • Pesin entropy: 0.26518243888151155
Citation
B. van der Pol (1920). A theory of the amplitude of free and forced triode vibrations. Radio Review 1.
Original Python Code from the dysts package
class ForcedVanDerPol(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, mu, w):
         ydot = mu * (1 - x ** 2) * y - x + a * np.sin(z)
@@ -424,174 +424,174 @@
 
     @staticjit
     def _postprocessing(x, y, z):
-        return x, y, np.sin(z)
source
Dysts.GenesioTesi — Function
   GenesioTesi()
A nonlinear control system with feedback.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1892507433089325
  • Lyapunov spectrum (estimated): Any[0.09974381765927798, 0.00038146676582121423, -0.5282231255530121]
  • Maximum Lyapunov exponent (estimated): 0.10509061137519304
  • Multiscale Entropy: 0.7003576241153363
  • Non-autonomous: false
  • Period: 5.9431
  • Pesin entropy: 0.10027488648885732
Citation
Genesio, Tesi (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28.3 (1992): 531-548.
Original Python Code from the dysts package
class GenesioTesi(DynSys):
+        return x, y, np.sin(z)
source
ChaoticDynamicalSystemsLibrary.GenesioTesi — Function
   GenesioTesi()
A nonlinear control system with feedback.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1892507433089325
  • Lyapunov spectrum (estimated): Any[0.09974381765927798, 0.00038146676582121423, -0.5282231255530121]
  • Maximum Lyapunov exponent (estimated): 0.10509061137519304
  • Multiscale Entropy: 0.7003576241153363
  • Non-autonomous: false
  • Period: 5.9431
  • Pesin entropy: 0.10027488648885732
Citation
Genesio, Tesi (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28.3 (1992): 531-548.
Original Python Code from the dysts package
class GenesioTesi(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = y
         ydot = z
         zdot = -c * x - b * y - a * z + x ** 2
-        return xdot, ydot, zdot
source
Dysts.GuckenheimerHolmes — Function
   GuckenheimerHolmes()
A nonlinear oscillator.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.3663164666051353
  • Lyapunov spectrum (estimated): Any[0.34082003567895874, -0.0032191807339425173, -0.9216135618098852]
  • Maximum Lyapunov exponent (estimated): 0.7885729042404602
  • Multiscale Entropy: 0.5385101182847213
  • Non-autonomous: false
  • Period: 0.3076923076923077
  • Pesin entropy: 0.3408200356941137
Citation
Guckenheimer, John, and Philip Holmes (1983).  Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42. Springer Science & Business Media.
Original Python Code from the dysts package
class GuckenheimerHolmes(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.GuckenheimerHolmes — Function
   GuckenheimerHolmes()
A nonlinear oscillator.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.3663164666051353
  • Lyapunov spectrum (estimated): Any[0.34082003567895874, -0.0032191807339425173, -0.9216135618098852]
  • Maximum Lyapunov exponent (estimated): 0.7885729042404602
  • Multiscale Entropy: 0.5385101182847213
  • Non-autonomous: false
  • Period: 0.3076923076923077
  • Pesin entropy: 0.3408200356941137
Citation
Guckenheimer, John, and Philip Holmes (1983).  Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42. Springer Science & Business Media.
Original Python Code from the dysts package
class GuckenheimerHolmes(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d, e, f):
         xdot = a * x - b * y + c * z * x + d * z * x ** 2 + d * z * y ** 2
         ydot = a * y + b * x + c * z * y
         zdot = e - z ** 2 - f * x ** 2 - f * y ** 2 - a * z ** 3
-        return xdot, ydot, zdot
source
Dysts.Hadley — Function
   Hadley()
An atmospheric convective cell.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.7273838319442127
  • Lyapunov spectrum (estimated): Any[0.31166133927315953, -0.0010838827053467116, -0.42739317979015523]
  • Maximum Lyapunov exponent (estimated): 0.2386587703845738
  • Multiscale Entropy: 0.9351389987191853
  • Non-autonomous: false
  • Period: 1.4541
  • Pesin entropy: 0.3144504334934165
Citation
G. Hadley (1735). On the cause of the general trade winds. Philosophical Transactions of the Royal Society, vol. 34, pp. 58–62.
Original Python Code from the dysts package
class Hadley(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Hadley — Function
   Hadley()
An atmospheric convective cell.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.7273838319442127
  • Lyapunov spectrum (estimated): Any[0.31166133927315953, -0.0010838827053467116, -0.42739317979015523]
  • Maximum Lyapunov exponent (estimated): 0.2386587703845738
  • Multiscale Entropy: 0.9351389987191853
  • Non-autonomous: false
  • Period: 1.4541
  • Pesin entropy: 0.3144504334934165
Citation
G. Hadley (1735). On the cause of the general trade winds. Philosophical Transactions of the Royal Society, vol. 34, pp. 58–62.
Original Python Code from the dysts package
class Hadley(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, f, g):
         xdot = -y ** 2 - z ** 2 - a * x + a * f
         ydot = x * y - b * x * z - y + g
         zdot = b * x * y + x * z - z
-        return xdot, ydot, zdot
source
Dysts.Halvorsen — Function
   Halvorsen()
An algebraically-simple chaotic system with quadratic nonlinearity
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.118235268907861
  • Lyapunov spectrum (estimated): Any[0.586197150399321, -0.031953945256851884, -4.6876300977027565]
  • Maximum Lyapunov exponent (estimated): 0.6963711539102683
  • Multiscale Entropy: 0.7527831765757674
  • Non-autonomous: false
  • Period: 1.4889
  • Pesin entropy: 0.586197150399321
Citation
Sprott, Julien C (2010). Elegant chaos: algebraically simple chaotic flows. World Scientific, 2010.
Original Python Code from the dysts package
class Halvorsen(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Halvorsen — Function
   Halvorsen()
An algebraically-simple chaotic system with quadratic nonlinearity
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.118235268907861
  • Lyapunov spectrum (estimated): Any[0.586197150399321, -0.031953945256851884, -4.6876300977027565]
  • Maximum Lyapunov exponent (estimated): 0.6963711539102683
  • Multiscale Entropy: 0.7527831765757674
  • Non-autonomous: false
  • Period: 1.4889
  • Pesin entropy: 0.586197150399321
Citation
Sprott, Julien C (2010). Elegant chaos: algebraically simple chaotic flows. World Scientific, 2010.
Original Python Code from the dysts package
class Halvorsen(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = -a * x - b * y - b * z - y ** 2
         ydot = -a * y - b * z - b * x - z ** 2
         zdot = -a * z - b * x - b * y - x ** 2
-        return xdot, ydot, zdot
source
Dysts.HastingsPowell — Function
   HastingsPowell()
A three species food web.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.027757892627057
  • Lyapunov spectrum (estimated): Any[0.008585012825774563, 0.005765040490702786, -0.5169232002224432]
  • Maximum Lyapunov exponent (estimated): 0.012532796124563227
  • Multiscale Entropy: 0.47829796264231095
  • Non-autonomous: false
  • Period: 135.19
  • Pesin entropy: 0.014350053316477346
Citation
Hastings, Powell. Ecology 1991
Original Python Code from the dysts package
class HastingsPowell(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.HastingsPowell — Function
   HastingsPowell()
A three species food web.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.027757892627057
  • Lyapunov spectrum (estimated): Any[0.008585012825774563, 0.005765040490702786, -0.5169232002224432]
  • Maximum Lyapunov exponent (estimated): 0.012532796124563227
  • Multiscale Entropy: 0.47829796264231095
  • Non-autonomous: false
  • Period: 135.19
  • Pesin entropy: 0.014350053316477346
Citation
Hastings, Powell. Ecology 1991
Original Python Code from the dysts package
class HastingsPowell(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a1, a2, b1, b2, d1, d2):
         xdot = x * (1 - x) - y * a1 * x / (1 + b1 * x)
         ydot = y * a1 * x / (1 + b1 * x) - z * a2 * y / (1 + b2 * y) - d1 * y
         zdot = z * a2 * y / (1 + b2 * y) - d2 * z
-        return xdot, ydot, zdot
source
Dysts.HenonHeiles — Function
   HenonHeiles()
A star's motion around the galactic center.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 6.81141589838772
  • Lyapunov spectrum (estimated): Any[0.026747154244887203, 0.011500891097175434, -1.4057113933414076e-5, -0.011151041415888588]
  • Maximum Lyapunov exponent (estimated): 0.03551120315239528
  • Multiscale Entropy: 0.7137794258156454
  • Non-autonomous: false
  • Period: 6.37
  • Pesin entropy: 0.03870865064107525
Citation
Henon, M.; Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astronomical Journal. 69: 73.
Original Python Code from the dysts package
class HenonHeiles(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.HenonHeiles — Function
   HenonHeiles()
A star's motion around the galactic center.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 6.81141589838772
  • Lyapunov spectrum (estimated): Any[0.026747154244887203, 0.011500891097175434, -1.4057113933414076e-5, -0.011151041415888588]
  • Maximum Lyapunov exponent (estimated): 0.03551120315239528
  • Multiscale Entropy: 0.7137794258156454
  • Non-autonomous: false
  • Period: 6.37
  • Pesin entropy: 0.03870865064107525
Citation
Henon, M.; Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astronomical Journal. 69: 73.
Original Python Code from the dysts package
class HenonHeiles(DynSys):
     @staticjit
     def _rhs(x, y, px, py, t, lam):
         xdot = px
         ydot = py
         pxdot = -x - 2 * lam * x * y
         pydot = -y - lam * x ** 2 + lam * y ** 2
-        return xdot, ydot, pxdot, pydot
source
Dysts.HindmarshRose — Function
   HindmarshRose()
A neuron model exhibiting spiking and bursting.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0405369895957373
  • Lyapunov spectrum (estimated): Any[0.2621464274609406, 0.0017705383465481184, -6.510663065519547]
  • Maximum Lyapunov exponent (estimated): 0.6535393283242747
  • Multiscale Entropy: 0.8692605688019491
  • Non-autonomous: false
  • Period: 1.752
  • Pesin entropy: 0.2639169709155772
Citation
Marhl, Perc. Chaos, Solitons, Fractals 2005.
Original Python Code from the dysts package
class HindmarshRose(DynSys):
+        return xdot, ydot, pxdot, pydot
source
ChaoticDynamicalSystemsLibrary.HindmarshRose — Function
   HindmarshRose()
A neuron model exhibiting spiking and bursting.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0405369895957373
  • Lyapunov spectrum (estimated): Any[0.2621464274609406, 0.0017705383465481184, -6.510663065519547]
  • Maximum Lyapunov exponent (estimated): 0.6535393283242747
  • Multiscale Entropy: 0.8692605688019491
  • Non-autonomous: false
  • Period: 1.752
  • Pesin entropy: 0.2639169709155772
Citation
Marhl, Perc. Chaos, Solitons, Fractals 2005.
Original Python Code from the dysts package
class HindmarshRose(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d, s, tx, tz):
         xdot = -x + 1 / tx * y - a / tx * x ** 3 + b / tx * x ** 2 + 1 / tx * z
         ydot = -a * x ** 3 - (d - b) * x ** 2 + z
         zdot = -s / tz * x - 1 / tz * z + c / tz
-        return xdot, ydot, zdot
source
Dysts.Hopfield — Function
   Hopfield()
A neural network with frustrated connectivity
Stats
  • Embedding dimension: 6
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.264479110541058
  • Lyapunov spectrum (estimated): Any[0.07726437124258394, -0.00018152933777384922, -0.29083324580448194, -0.36906925337474944, -0.40604718386075256, -1.4425404449511432]
  • Maximum Lyapunov exponent (estimated): 0.5684266951234997
  • Multiscale Entropy: 0.9160145554679957
  • Non-autonomous: false
  • Period: 10.865
  • Pesin entropy: 0.07748903749076723
Citation
Lewis & Glass, Neur Comp (1992)
Original Python Code from the dysts package
class Hopfield(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Hopfield — Function
   Hopfield()
A neural network with frustrated connectivity
Stats
  • Embedding dimension: 6
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.264479110541058
  • Lyapunov spectrum (estimated): Any[0.07726437124258394, -0.00018152933777384922, -0.29083324580448194, -0.36906925337474944, -0.40604718386075256, -1.4425404449511432]
  • Maximum Lyapunov exponent (estimated): 0.5684266951234997
  • Multiscale Entropy: 0.9160145554679957
  • Non-autonomous: false
  • Period: 10.865
  • Pesin entropy: 0.07748903749076723
Citation
Lewis & Glass, Neur Comp (1992)
Original Python Code from the dysts package
class Hopfield(DynSys):
     def f(self, x):
         return (1 + np.tanh(x)) / 2
 
     def rhs(self, X, t):
         Xdot = -X / self.tau + self.f(self.eps * np.matmul(self.k, X)) - self.beta
-        return Xdot
source
Dysts.HyperBao — Function
   HyperBao()
Hyperchaos in the Lu system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0460429607933572
  • Lyapunov spectrum (estimated): Any[0.6959739496471207, 0.2076688444499756, 0.01646766869238271, -19.983737503321315]
  • Maximum Lyapunov exponent (estimated): 0.7941925747845208
  • Multiscale Entropy: 0.8910215561062699
  • Non-autonomous: false
  • Period: 0.5831505997603473
  • Pesin entropy: 0.9201105075982862
Citation
Bao, Liu  (2008). A hyperchaotic attractor coined from the chaotic Lu system. Chin. Physics Letters, Vol. 25, pp 2396-2399.
Original Python Code from the dysts package
class HyperBao(DynSys):
+        return Xdot
source
ChaoticDynamicalSystemsLibrary.HyperBao — Function
   HyperBao()
Hyperchaos in the Lu system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0460429607933572
  • Lyapunov spectrum (estimated): Any[0.6959739496471207, 0.2076688444499756, 0.01646766869238271, -19.983737503321315]
  • Maximum Lyapunov exponent (estimated): 0.7941925747845208
  • Multiscale Entropy: 0.8910215561062699
  • Non-autonomous: false
  • Period: 0.5831505997603473
  • Pesin entropy: 0.9201105075982862
Citation
Bao, Liu  (2008). A hyperchaotic attractor coined from the chaotic Lu system. Chin. Physics Letters, Vol. 25, pp 2396-2399.
Original Python Code from the dysts package
class HyperBao(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d, e):
         xdot = a * y - a * x + w
         ydot = c * y - x * z
         zdot = x * y - b * z
         wdot = e * x + d * y * z
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperCai — Function
   HyperCai()
A hyperchaotic variant of the Finance system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.138030987910704
  • Lyapunov spectrum (estimated): Any[1.634443623908957, 0.12845528254618424, -0.003328688360603957, -12.747646341795276]
  • Maximum Lyapunov exponent (estimated): 1.6777584899044127
  • Multiscale Entropy: 1.1155413734604414
  • Non-autonomous: false
  • Period: 0.6113936187028436
  • Pesin entropy: 1.7628989064551457
Citation
Guoliang, Huang (2007). A New Finance Chaotic Attractor. International Journal of Nonlinear Science.
Original Python Code from the dysts package
class HyperCai(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperCai — Function
   HyperCai()
A hyperchaotic variant of the Finance system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.138030987910704
  • Lyapunov spectrum (estimated): Any[1.634443623908957, 0.12845528254618424, -0.003328688360603957, -12.747646341795276]
  • Maximum Lyapunov exponent (estimated): 1.6777584899044127
  • Multiscale Entropy: 1.1155413734604414
  • Non-autonomous: false
  • Period: 0.6113936187028436
  • Pesin entropy: 1.7628989064551457
Citation
Guoliang, Huang (2007). A New Finance Chaotic Attractor. International Journal of Nonlinear Science.
Original Python Code from the dysts package
class HyperCai(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d, e):
         xdot = a * y - a * x
         ydot = b * x + c * y - x * z + w
         zdot = -d * z + y ** 2
         wdot = -e * x
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperJha — Function
   HyperJha()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.053181782412727
  • Lyapunov spectrum (estimated): Any[0.3949499860671474, 0.2920214751184925, 0.005568431034995312, -13.022125042186726]
  • Maximum Lyapunov exponent (estimated): 0.38695361636760767
  • Multiscale Entropy: 0.8934166449122942
  • Non-autonomous: false
  • Period: 0.8564467295817622
  • Pesin entropy: 0.6925398922206352
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperJha(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperJha — Function
   HyperJha()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.053181782412727
  • Lyapunov spectrum (estimated): Any[0.3949499860671474, 0.2920214751184925, 0.005568431034995312, -13.022125042186726]
  • Maximum Lyapunov exponent (estimated): 0.38695361636760767
  • Multiscale Entropy: 0.8934166449122942
  • Non-autonomous: false
  • Period: 0.8564467295817622
  • Pesin entropy: 0.6925398922206352
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperJha(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d):
         xdot = a * y - a * x + w
         ydot = -x * z + b * x - y
         zdot = x * y - c * z
         wdot = -x * z + d * w
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperLorenz — Function
   HyperLorenz()
A hyperchaotic variant of the Lorenz attractor.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.044709906668086
  • Lyapunov spectrum (estimated): Any[0.2925643599949083, 0.24691553258701357, 0.04704434899468292, -13.118440302968889]
  • Maximum Lyapunov exponent (estimated): 0.3288809596909799
  • Multiscale Entropy: 0.8026458791332332
  • Non-autonomous: false
  • Period: 1.277546730110058
  • Pesin entropy: 0.5865242415766048
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperLorenz(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperLorenz — Function
   HyperLorenz()
A hyperchaotic variant of the Lorenz attractor.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.044709906668086
  • Lyapunov spectrum (estimated): Any[0.2925643599949083, 0.24691553258701357, 0.04704434899468292, -13.118440302968889]
  • Maximum Lyapunov exponent (estimated): 0.3288809596909799
  • Multiscale Entropy: 0.8026458791332332
  • Non-autonomous: false
  • Period: 1.277546730110058
  • Pesin entropy: 0.5865242415766048
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperLorenz(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d):
         xdot = a * y - a * x + w
         ydot = -x * z + c * x - y
         zdot = -b * z + x * y
         wdot = d * w - x * z
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperLu — Function
   HyperLu()
A hyperchaotic variant of the Lu attractor.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0822889656900583
  • Lyapunov spectrum (estimated): Any[1.0855770127041702, 0.4753850931706396, 0.02849951886859947, -19.313823203887658]
  • Maximum Lyapunov exponent (estimated): 1.042961263138283
  • Multiscale Entropy: 1.0455788608813867
  • Non-autonomous: false
  • Period: 0.7162856981996777
  • Pesin entropy: 1.5894616248138722
Citation
Chen, A., Lu, J., Lü, J., & Yu, S. (2006). Generating hyperchaotic Lü attractor via state feedback control. Physica A: Statistical Mechanics and its Applications, 364, 103-110.
Original Python Code from the dysts package
class HyperLu(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperLu — Function
   HyperLu()
A hyperchaotic variant of the Lu attractor.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0822889656900583
  • Lyapunov spectrum (estimated): Any[1.0855770127041702, 0.4753850931706396, 0.02849951886859947, -19.313823203887658]
  • Maximum Lyapunov exponent (estimated): 1.042961263138283
  • Multiscale Entropy: 1.0455788608813867
  • Non-autonomous: false
  • Period: 0.7162856981996777
  • Pesin entropy: 1.5894616248138722
Citation
Chen, A., Lu, J., Lü, J., & Yu, S. (2006). Generating hyperchaotic Lü attractor via state feedback control. Physica A: Statistical Mechanics and its Applications, 364, 103-110.
Original Python Code from the dysts package
class HyperLu(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a=36, b=3, c=20, d=1.3):
         xdot = a * y - a * x + w
         ydot = -x * z + c * y
         zdot = x * y - b * z
         wdot = d * w + x * z
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperPang — Function
   HyperPang()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.08902125132107
  • Lyapunov spectrum (estimated): Any[1.586932998381499, 0.28444192185263556, -0.0011541322141891326, -21.007696151519628]
  • Maximum Lyapunov exponent (estimated): 1.4465315511406867
  • Multiscale Entropy: 0.9172158388570456
  • Non-autonomous: false
  • Period: 0.8439374360017028
  • Pesin entropy: 1.871913394885623
Citation
Pang, S., & Liu, Y. (2011). A new hyperchaotic system from the Lü system and its control. Journal of Computational and Applied Mathematics, 235(8), 2775-2789.
Original Python Code from the dysts package
class HyperPang(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperPang — Function
   HyperPang()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.08902125132107
  • Lyapunov spectrum (estimated): Any[1.586932998381499, 0.28444192185263556, -0.0011541322141891326, -21.007696151519628]
  • Maximum Lyapunov exponent (estimated): 1.4465315511406867
  • Multiscale Entropy: 0.9172158388570456
  • Non-autonomous: false
  • Period: 0.8439374360017028
  • Pesin entropy: 1.871913394885623
Citation
Pang, S., & Liu, Y. (2011). A new hyperchaotic system from the Lü system and its control. Journal of Computational and Applied Mathematics, 235(8), 2775-2789.
Original Python Code from the dysts package
class HyperPang(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a=36, b=3, c=20, d=2):
         xdot = a * y - a * x
         ydot = -x * z + c * y + w
         zdot = x * y - b * z
         wdot = -d * x - d * y
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperQi — Function
   HyperQi()
A hyperchaotic variant of the Qi system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.2816770286126777
  • Lyapunov spectrum (estimated): Any[13.908799876284396, 2.8519979827853357, -0.04957904444451209, -59.32758839771818]
  • Maximum Lyapunov exponent (estimated): 12.787249321134263
  • Multiscale Entropy: 0.9988440965886223
  • Non-autonomous: false
  • Period: 0.06186038560173437
  • Pesin entropy: 16.760797859069733
Citation
G. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.
Original Python Code from the dysts package
class HyperQi(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperQi — Function
   HyperQi()
A hyperchaotic variant of the Qi system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.2816770286126777
  • Lyapunov spectrum (estimated): Any[13.908799876284396, 2.8519979827853357, -0.04957904444451209, -59.32758839771818]
  • Maximum Lyapunov exponent (estimated): 12.787249321134263
  • Multiscale Entropy: 0.9988440965886223
  • Non-autonomous: false
  • Period: 0.06186038560173437
  • Pesin entropy: 16.760797859069733
Citation
G. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.
Original Python Code from the dysts package
class HyperQi(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d, e, f):
         xdot = a * y - a * x + y * z
         ydot = b * x + b * y - x * z
         zdot = -c * z - e * w + x * y
         wdot = -d * w + f * z + x * y
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperRossler — Function
   HyperRossler()
A hyperchaotic variant of the Rossler system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.916384112225743
  • Lyapunov spectrum (estimated): Any[0.27859233648384435, -7.749198930652314e-5, -0.3014660985555502, -70.26609894565237]
  • Maximum Lyapunov exponent (estimated): 0.1718849632568779
  • Multiscale Entropy: 0.9144754695112332
  • Non-autonomous: false
  • Period: 1.1157476694793766
  • Pesin entropy: 0.2789695684743801
Citation
Rossler, O. E. (1979). An equation for hyperchaos.
Original Python Code from the dysts package
class HyperRossler(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperRossler — Function
   HyperRossler()
A hyperchaotic variant of the Rossler system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.916384112225743
  • Lyapunov spectrum (estimated): Any[0.27859233648384435, -7.749198930652314e-5, -0.3014660985555502, -70.26609894565237]
  • Maximum Lyapunov exponent (estimated): 0.1718849632568779
  • Multiscale Entropy: 0.9144754695112332
  • Non-autonomous: false
  • Period: 1.1157476694793766
  • Pesin entropy: 0.2789695684743801
Citation
Rossler, O. E. (1979). An equation for hyperchaos.
Original Python Code from the dysts package
class HyperRossler(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d):
         xdot = -y - z
         ydot = x + a * y + w
         zdot = b + x * z
         wdot = -c * z + d * w
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperWang — Function
   HyperWang()
A hyperchaotic variant of the Wang system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.105321722354005
  • Lyapunov spectrum (estimated): Any[1.2224015619377895, 0.25827623672371736, -0.0075512136517132625, -13.986919236227278]
  • Maximum Lyapunov exponent (estimated): 1.2930140887300063
  • Multiscale Entropy: 0.9969261837212393
  • Non-autonomous: false
  • Period: 0.6481520567576332
  • Pesin entropy: 1.480677798661507
Citation
Wang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.
Original Python Code from the dysts package
class HyperWang(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperWang — Function
   HyperWang()
A hyperchaotic variant of the Wang system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.105321722354005
  • Lyapunov spectrum (estimated): Any[1.2224015619377895, 0.25827623672371736, -0.0075512136517132625, -13.986919236227278]
  • Maximum Lyapunov exponent (estimated): 1.2930140887300063
  • Multiscale Entropy: 0.9969261837212393
  • Non-autonomous: false
  • Period: 0.6481520567576332
  • Pesin entropy: 1.480677798661507
Citation
Wang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.
Original Python Code from the dysts package
class HyperWang(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a=10, b=40, c=2.5, d=10.6, e=4):
         xdot = a * y - a * x
         ydot = -x * z + b * x + w
         zdot = -c * z + e * x ** 2
         wdot = -d * x
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperXu — Function
   HyperXu()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0909471864561056
  • Lyapunov spectrum (estimated): Any[1.0420068843753962, 0.20717763316365526, -0.0033988115975040426, -13.697896294059921]
  • Maximum Lyapunov exponent (estimated): 0.9819830018599975
  • Multiscale Entropy: 0.9312323303231519
  • Non-autonomous: false
  • Period: 1.1328151255687646
  • Pesin entropy: 1.2491913312241498
Citation
Letellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.
Original Python Code from the dysts package
class HyperXu(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperXu — Function
   HyperXu()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0909471864561056
  • Lyapunov spectrum (estimated): Any[1.0420068843753962, 0.20717763316365526, -0.0033988115975040426, -13.697896294059921]
  • Maximum Lyapunov exponent (estimated): 0.9819830018599975
  • Multiscale Entropy: 0.9312323303231519
  • Non-autonomous: false
  • Period: 1.1328151255687646
  • Pesin entropy: 1.2491913312241498
Citation
Letellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.
Original Python Code from the dysts package
class HyperXu(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a=10, b=40, c=2.5, d=2, e=16):
         xdot = a * y - a * x + w
         ydot = b * x + e * x * z
         zdot = -c * z - x * y
         wdot = x * z - d * y
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperYan — Function
   HyperYan()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0711118082329567
  • Lyapunov spectrum (estimated): Any[0.9120352761173002, 0.15303345953001665, -14.977382134147613, -39.36365990052555]
  • Maximum Lyapunov exponent (estimated): 1.3945968455812914
  • Multiscale Entropy: 0.8166157518949907
  • Non-autonomous: false
  • Period: 0.9127028626966961
  • Pesin entropy: 1.0650687356473167
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperYan(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperYan — Function
   HyperYan()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0711118082329567
  • Lyapunov spectrum (estimated): Any[0.9120352761173002, 0.15303345953001665, -14.977382134147613, -39.36365990052555]
  • Maximum Lyapunov exponent (estimated): 1.3945968455812914
  • Multiscale Entropy: 0.8166157518949907
  • Non-autonomous: false
  • Period: 0.9127028626966961
  • Pesin entropy: 1.0650687356473167
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperYan(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a=37, b=3, c=26, d=38):
         xdot = a * y - a * x
         ydot = (c - a) * x - x * z + c * y
         zdot = -b * z + x * y - y * z + x * z - w
         wdot = -d * w + y * z - x * z
-        return xdot, ydot, zdot, wdot
source
Dysts.HyperYangChen — Function
   HyperYangChen()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.026605264361547
  • Lyapunov spectrum (estimated): Any[0.5742688696462735, 0.327365255173604, 0.023510192276255986, -34.77284163099938]
  • Maximum Lyapunov exponent (estimated): 0.42042732521941667
  • Multiscale Entropy: 0.9730322217427024
  • Non-autonomous: false
  • Period: 0.9127028626966961
  • Pesin entropy: 0.9255171125652686
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperYangChen(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.HyperYangChen — Function
   HyperYangChen()
A hyperchaotic system.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.026605264361547
  • Lyapunov spectrum (estimated): Any[0.5742688696462735, 0.327365255173604, 0.023510192276255986, -34.77284163099938]
  • Maximum Lyapunov exponent (estimated): 0.42042732521941667
  • Multiscale Entropy: 0.9730322217427024
  • Non-autonomous: false
  • Period: 0.9127028626966961
  • Pesin entropy: 0.9255171125652686
Citation
Jürgen Meier (2003). Presentation of Attractors with Cinema.
Original Python Code from the dysts package
class HyperYangChen(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a=30, b=3, c=35, d=8):
         xdot = a * y - a * x
         ydot = c * x - x * z + w
         zdot = -b * z + x * y
         wdot = -d * x
-        return xdot, ydot, zdot, wdot
source
Dysts.IkedaDelay — Function
   IkedaDelay()
A passive optical resonator system. A standard delay dimension of three is used.
Stats
  • Embedding dimension: 10
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 8.284295212220359
  • Lyapunov spectrum (estimated): Any[0.7139744431963619, 0.03258370277261943, -0.06766062243103049, -0.07744565521894307, -0.09054735418410836, -0.10900317234818346, -0.13696057562417352, -0.18439894250590672, -0.2833034824174493, -0.6392438255039592]
  • Maximum Lyapunov exponent (estimated): 0.448298365567425
  • Non-autonomous: false
  • Period: 1.486528
  • Pesin entropy: 0.7465581459689814
Citation
K. Ikeda and K. Matsumoto (1987). High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29 (1-2), 223-235
Original Python Code from the dysts package
class IkedaDelay(DynSysDelay):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.IkedaDelay — Function
   IkedaDelay()
A passive optical resonator system. A standard delay dimension of three is used.
Stats
  • Embedding dimension: 10
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 8.284295212220359
  • Lyapunov spectrum (estimated): Any[0.7139744431963619, 0.03258370277261943, -0.06766062243103049, -0.07744565521894307, -0.09054735418410836, -0.10900317234818346, -0.13696057562417352, -0.18439894250590672, -0.2833034824174493, -0.6392438255039592]
  • Maximum Lyapunov exponent (estimated): 0.448298365567425
  • Non-autonomous: false
  • Period: 1.486528
  • Pesin entropy: 0.7465581459689814
Citation
K. Ikeda and K. Matsumoto (1987). High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29 (1-2), 223-235
Original Python Code from the dysts package
class IkedaDelay(DynSysDelay):
     @staticjit
     def _rhs(x, xt, t, c, mu, tau, x0):
         xdot = mu * np.sin(xt - x0) - c * x
-        return xdot
source
Dysts.IsothermalChemical — Function
   IsothermalChemical()
An isothermal chemical system with mixed-mode oscillations
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0385823810364654
  • Lyapunov spectrum (estimated): Any[0.3365871343421436, -0.05136527222093221, -7.392541736904205]
  • Maximum Lyapunov exponent (estimated): 0.450090505359164
  • Multiscale Entropy: 0.6589422003608746
  • Non-autonomous: false
  • Period: 0.7302799999999999
  • Pesin entropy: 0.3365871343421436
Citation
Petrov, Scott, Showalter. Mixed-mode oscillations in chemical systems. J Chem Phys 1992
Original Python Code from the dysts package
class IsothermalChemical(DynSys):
+        return xdot
source
ChaoticDynamicalSystemsLibrary.IsothermalChemical — Function
   IsothermalChemical()
An isothermal chemical system with mixed-mode oscillations
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0385823810364654
  • Lyapunov spectrum (estimated): Any[0.3365871343421436, -0.05136527222093221, -7.392541736904205]
  • Maximum Lyapunov exponent (estimated): 0.450090505359164
  • Multiscale Entropy: 0.6589422003608746
  • Non-autonomous: false
  • Period: 0.7302799999999999
  • Pesin entropy: 0.3365871343421436
Citation
Petrov, Scott, Showalter. Mixed-mode oscillations in chemical systems. J Chem Phys 1992
Original Python Code from the dysts package
class IsothermalChemical(DynSys):
     @staticmethod
     def _rhs(alpha, beta, gamma, t, delta, kappa, mu, sigma):
         alphadot = mu * (kappa + gamma) - alpha * beta ** 2 - alpha
         betadot = (alpha * beta ** 2 + alpha - beta) / sigma
         gammadot = (beta - gamma) / delta
-        return alphadot, betadot, gammadot
source
Dysts.ItikBanksTumor — Function
   ItikBanksTumor()
A model of cancer cell populations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0152772613355485
  • Lyapunov spectrum (estimated): Any[0.022156938955395417, -0.0135394920593605, -0.5610833889492404]
  • Maximum Lyapunov exponent (estimated): 0.03621509684991421
  • Multiscale Entropy: 0.5883298730506914
  • Non-autonomous: false
  • Period: 49.74
  • Pesin entropy: 0.022156938955395417
Citation
Itik, Banks. Int J Bifurcat Chaos 2010
Original Python Code from the dysts package
class ItikBanksTumor(DynSys):
+        return alphadot, betadot, gammadot
source
ChaoticDynamicalSystemsLibrary.ItikBanksTumor — Function
   ItikBanksTumor()
A model of cancer cell populations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0152772613355485
  • Lyapunov spectrum (estimated): Any[0.022156938955395417, -0.0135394920593605, -0.5610833889492404]
  • Maximum Lyapunov exponent (estimated): 0.03621509684991421
  • Multiscale Entropy: 0.5883298730506914
  • Non-autonomous: false
  • Period: 49.74
  • Pesin entropy: 0.022156938955395417
Citation
Itik, Banks. Int J Bifurcat Chaos 2010
Original Python Code from the dysts package
class ItikBanksTumor(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a12, a13, a21, a31, d3, k3, r2, r3):
         xdot = x * (1 - x) - a12 * x * y - a13 * x * z
         ydot = r2 * y * (1 - y) - a21 * x * y
         zdot = r3 * x * z / (x + k3) - a31 * x * z - d3 * z
-        return xdot, ydot, zdot
source
Dysts.JerkCircuit — Function
   JerkCircuit()
An electronic circuit with nonlinearity provided by a diode.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.5082428902531144
  • Lyapunov spectrum (estimated): Any[0.394872929553187, 0.1849026306096344, -1.1407351759248092]
  • Maximum Lyapunov exponent (estimated): 0.2922722771543928
  • Multiscale Entropy: 0.38139916676685187
  • Non-autonomous: false
  • Period: 19.4546
  • Pesin entropy: 0.5797755601628214
Citation
Sprott (2011). A new chaotic jerk circuit. IEEE Circ. Sys. 2011.
Original Python Code from the dysts package
class JerkCircuit(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.JerkCircuit — Function
   JerkCircuit()
An electronic circuit with nonlinearity provided by a diode.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.5082428902531144
  • Lyapunov spectrum (estimated): Any[0.394872929553187, 0.1849026306096344, -1.1407351759248092]
  • Maximum Lyapunov exponent (estimated): 0.2922722771543928
  • Multiscale Entropy: 0.38139916676685187
  • Non-autonomous: false
  • Period: 19.4546
  • Pesin entropy: 0.5797755601628214
Citation
Sprott (2011). A new chaotic jerk circuit. IEEE Circ. Sys. 2011.
Original Python Code from the dysts package
class JerkCircuit(DynSys):
     @staticjit
     def _rhs(x, y, z, t, eps, y0):
         xdot = y
         ydot = z
         zdot = -z - x - eps * (np.exp(y / y0) - 1)
-        return xdot, ydot, zdot
source
Dysts.KawczynskiStrizhak — Function
   KawczynskiStrizhak()
A chemical oscillator model describing mixed-modes in the BZ equations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.4235221965773541
  • Lyapunov spectrum (estimated): Any[0.025607519929323416, -0.06145426210744069, -0.31683393262481707]
  • Maximum Lyapunov exponent (estimated): 0.0757060839542753
  • Multiscale Entropy: 0.5935098791328474
  • Non-autonomous: false
  • Period: 7.991666666666667
  • Pesin entropy: 0.02560751992932341
Citation
P. E. Strizhak and A. L. Kawczynski, J. Phys. Chem. 99, 10830 (1995).
Original Python Code from the dysts package
class KawczynskiStrizhak(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.KawczynskiStrizhak — Function
   KawczynskiStrizhak()
A chemical oscillator model describing mixed-modes in the BZ equations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.4235221965773541
  • Lyapunov spectrum (estimated): Any[0.025607519929323416, -0.06145426210744069, -0.31683393262481707]
  • Maximum Lyapunov exponent (estimated): 0.0757060839542753
  • Multiscale Entropy: 0.5935098791328474
  • Non-autonomous: false
  • Period: 7.991666666666667
  • Pesin entropy: 0.02560751992932341
Citation
P. E. Strizhak and A. L. Kawczynski, J. Phys. Chem. 99, 10830 (1995).
Original Python Code from the dysts package
class KawczynskiStrizhak(DynSys):
     @staticjit
     def _rhs(x, y, z, t, beta, gamma, kappa, mu):
         xdot = gamma * y - gamma * x ** 3 + 3 * mu * gamma * x
         ydot = -2 * mu * x - y - z + beta
         zdot = kappa * x - kappa * z
-        return xdot, ydot, zdot
source
Dysts.Laser — Function
   Laser()
A semiconductor laser model
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.049321615812801
  • Lyapunov spectrum (estimated): Any[0.7637435195473311, 0.020943250302999646, -15.909545615344538]
  • Maximum Lyapunov exponent (estimated): 0.6080602180196161
  • Multiscale Entropy: 1.004437967414151
  • Non-autonomous: false
  • Period: 0.94667
  • Pesin entropy: 0.7848314215142518
Citation
Abooee, Yaghini-Bonabi, Jahed-Motlagh (2013). Analysis and circuitry realization of a novel three-dimensional chaotic system. Comm Nonline Sci 2013
Original Python Code from the dysts package
class Laser(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Laser — Function
   Laser()
A semiconductor laser model
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.049321615812801
  • Lyapunov spectrum (estimated): Any[0.7637435195473311, 0.020943250302999646, -15.909545615344538]
  • Maximum Lyapunov exponent (estimated): 0.6080602180196161
  • Multiscale Entropy: 1.004437967414151
  • Non-autonomous: false
  • Period: 0.94667
  • Pesin entropy: 0.7848314215142518
Citation
Abooee, Yaghini-Bonabi, Jahed-Motlagh (2013). Analysis and circuitry realization of a novel three-dimensional chaotic system. Comm Nonline Sci 2013
Original Python Code from the dysts package
class Laser(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d, h, k):
         xdot = a * y - a * x + b * y * z ** 2
         ydot = c * x + d * x * z ** 2
         zdot = h * z + k * x ** 2
-        return xdot, ydot, zdot
source
Dysts.LidDrivenCavityFlow — Function
   LidDrivenCavityFlow()
A two-dimensional flow alternately driven by translation of the left and right walls.
Stats
  • Embedding dimension: 3
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 2.9856669109882814
  • Lyapunov spectrum (estimated): Any[1.00757312, -0.00276829, -1.01941623]
  • Maximum Lyapunov exponent (estimated): 2.172601412109087
  • Multiscale Entropy: 0.7815824521381879
  • Non-autonomous: true
  • Period: 2.7333134945588955
  • Pesin entropy: 1.007573134065856
Citation
Grover et al. Chaos (2012). Topological chaos, braiding and bifurcation of almost-cyclic sets.
Original Python Code from the dysts package
class LidDrivenCavityFlow(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.LidDrivenCavityFlow — Function
   LidDrivenCavityFlow()
A two-dimensional flow alternately driven by translation of the left and right walls.
Stats
  • Embedding dimension: 3
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 2.9856669109882814
  • Lyapunov spectrum (estimated): Any[1.00757312, -0.00276829, -1.01941623]
  • Maximum Lyapunov exponent (estimated): 2.172601412109087
  • Multiscale Entropy: 0.7815824521381879
  • Non-autonomous: true
  • Period: 2.7333134945588955
  • Pesin entropy: 1.007573134065856
Citation
Grover et al. Chaos (2012). Topological chaos, braiding and bifurcation of almost-cyclic sets.
Original Python Code from the dysts package
class LidDrivenCavityFlow(DynSys):
     @staticjit
     def _lid(x, y, a, b, tau, u1, u2):
         """The velocity field when the left domain drives"""
@@ -645,13 +645,13 @@
         return dx, dy, dtt
 
     def _postprocessing(self, x, y, tt):
-        return x, y, np.sin(2 * np.pi * tt / self.tau)
source
Dysts.LiuChen — Function
   LiuChen()
Derived from Sakarya.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.2188137323261288
  • Lyapunov spectrum (estimated): Any[0.21332142584135133, -0.9748996261500176, -18.807059276226646]
  • Maximum Lyapunov exponent (estimated): 0.2457084451352168
  • Multiscale Entropy: 0.7641473604711979
  • Non-autonomous: false
  • Period: 8.0164
  • Pesin entropy: 0.2133214258413513
Citation
Liu, Chen. Int J Bifurcat Chaos. 2004: 1395-1403.
Original Python Code from the dysts package
class RayleighBenard(DynSys):
+        return x, y, np.sin(2 * np.pi * tt / self.tau)
source
ChaoticDynamicalSystemsLibrary.LiuChen — Function
   LiuChen()
Derived from Sakarya.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.2188137323261288
  • Lyapunov spectrum (estimated): Any[0.21332142584135133, -0.9748996261500176, -18.807059276226646]
  • Maximum Lyapunov exponent (estimated): 0.2457084451352168
  • Multiscale Entropy: 0.7641473604711979
  • Non-autonomous: false
  • Period: 8.0164
  • Pesin entropy: 0.2133214258413513
Citation
Liu, Chen. Int J Bifurcat Chaos. 2004: 1395-1403.
Original Python Code from the dysts package
class RayleighBenard(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, r):
         xdot = a * y - a * x
         ydot = r * y - x * z
         zdot = x * y - b * z
-        return xdot, ydot, zdot
source
Dysts.Lorenz — Method
   Lorenz()
A minimal weather model based on atmospheric convection.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.075158758095728
  • Lyapunov spectrum (estimated): Any[1.0910931847726466, 0.02994120961308413, -14.915552395875103]
  • Maximum Lyapunov exponent (estimated): 0.8917098035724058
  • Multiscale Entropy: 1.1541457906835575
  • Non-autonomous: false
  • Period: 1.5008
  • Pesin entropy: 1.121034394385731
Citation
Lorenz, Edward N (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.
Original Python Code from the dysts package
class Lorenz(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Lorenz — Method
   Lorenz()
A minimal weather model based on atmospheric convection.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.075158758095728
  • Lyapunov spectrum (estimated): Any[1.0910931847726466, 0.02994120961308413, -14.915552395875103]
  • Maximum Lyapunov exponent (estimated): 0.8917098035724058
  • Multiscale Entropy: 1.1541457906835575
  • Non-autonomous: false
  • Period: 1.5008
  • Pesin entropy: 1.121034394385731
Citation
Lorenz, Edward N (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.
Original Python Code from the dysts package
class Lorenz(DynSys):
     @staticjit
     def _rhs(x, y, z, t, beta, rho, sigma):
         xdot = sigma * y - sigma * x
@@ -663,26 +663,26 @@
         row1 = [-sigma, sigma, 0]
         row2 = [rho - z, -1, -x]
         row3 = [y, x, -beta]
-        return [row1, row2, row3]
source
Dysts.Lorenz84 — Function
   Lorenz84()
Atmospheric circulation analogous to Hadley convection.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.4449761695022096
  • Lyapunov spectrum (estimated): Any[1.0474807193950644, -0.045398035352994866, -2.251992293230755]
  • Maximum Lyapunov exponent (estimated): 0.46146683857936616
  • Multiscale Entropy: 0.6685802317710752
  • Non-autonomous: false
  • Period: 6.2586
  • Pesin entropy: 1.0474807193950644
Citation
E. Lorenz (1984). Irregularity: a fundamental property of the atmosphere. Tellus A, vol. 36, no. 2, pp. 98–110, 1984.
Original Python Code from the dysts package
class Lorenz84(DynSys):
+        return [row1, row2, row3]
source
ChaoticDynamicalSystemsLibrary.Lorenz84 — Function
   Lorenz84()
Atmospheric circulation analogous to Hadley convection.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.4449761695022096
  • Lyapunov spectrum (estimated): Any[1.0474807193950644, -0.045398035352994866, -2.251992293230755]
  • Maximum Lyapunov exponent (estimated): 0.46146683857936616
  • Multiscale Entropy: 0.6685802317710752
  • Non-autonomous: false
  • Period: 6.2586
  • Pesin entropy: 1.0474807193950644
Citation
E. Lorenz (1984). Irregularity: a fundamental property of the atmosphere. Tellus A, vol. 36, no. 2, pp. 98–110, 1984.
Original Python Code from the dysts package
class Lorenz84(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, f, g):
         xdot = -a * x - y ** 2 - z ** 2 + a * f
         ydot = -y + x * y - b * x * z + g
         zdot = -z + b * x * y + x * z
-        return xdot, ydot, zdot
source
Dysts.Lorenz96 — Function
   Lorenz96()
A climate model containing fluid-like advective nonlinearities, and multiscale coupling of fast and slow variables.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.1698163415908196
  • Lyapunov spectrum (estimated): Any[1.6247012663091516, 0.17185232259745578, -1.1936282999960617, -3.550455057872212]
  • Maximum Lyapunov exponent (estimated): 1.3361667787286362
  • Multiscale Entropy: 0.8046726640855828
  • Non-autonomous: false
  • Period: 2.1969
  • Pesin entropy: 1.7965535889066073
Citation
Lorenz, Edward (1996). Predictability: A problem partly solved. Seminar on Predictability, Vol. I, ECMWF.
Original Python Code from the dysts package
class Lorenz96(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Lorenz96 — Function
   Lorenz96()
A climate model containing fluid-like advective nonlinearities, and multiscale coupling of fast and slow variables.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.1698163415908196
  • Lyapunov spectrum (estimated): Any[1.6247012663091516, 0.17185232259745578, -1.1936282999960617, -3.550455057872212]
  • Maximum Lyapunov exponent (estimated): 1.3361667787286362
  • Multiscale Entropy: 0.8046726640855828
  • Non-autonomous: false
  • Period: 2.1969
  • Pesin entropy: 1.7965535889066073
Citation
Lorenz, Edward (1996). Predictability: A problem partly solved. Seminar on Predictability, Vol. I, ECMWF.
Original Python Code from the dysts package
class Lorenz96(DynSys):
     def rhs(self, X, t):
         Xdot = np.zeros_like(X)
         Xdot[0] = (X[1] - X[-2]) * X[-1] - X[0] + self.f
         Xdot[1] = (X[2] - X[-1]) * X[0] - X[1] + self.f
         Xdot[-1] = (X[0] - X[-3]) * X[-2] - X[-1] + self.f
         Xdot[2:-1] = (X[3:] - X[:-3]) * X[1:-2] - X[2:-1] + self.f
-        return Xdot
source
Dysts.LorenzBounded — Method
   LorenzBounded()
The Lorenz attractor in the presence of a confining potential.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0704391073439155
  • Lyapunov spectrum (estimated): Any[0.8497533717540197, 0.001413707285345669, -12.120634680260636]
  • Maximum Lyapunov exponent (estimated): 0.7124135270827626
  • Multiscale Entropy: 1.1983193915488475
  • Non-autonomous: false
  • Period: 1.3716
  • Pesin entropy: 0.8513198508911037
Citation
Sprott & Xiong (2015). Chaos.
Original Python Code from the dysts package
class LorenzBounded(DynSys):
+        return Xdot
source
ChaoticDynamicalSystemsLibrary.LorenzBounded — Method
   LorenzBounded()
The Lorenz attractor in the presence of a confining potential.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0704391073439155
  • Lyapunov spectrum (estimated): Any[0.8497533717540197, 0.001413707285345669, -12.120634680260636]
  • Maximum Lyapunov exponent (estimated): 0.7124135270827626
  • Multiscale Entropy: 1.1983193915488475
  • Non-autonomous: false
  • Period: 1.3716
  • Pesin entropy: 0.8513198508911037
Citation
Sprott & Xiong (2015). Chaos.
Original Python Code from the dysts package
class LorenzBounded(DynSys):
     @staticjit
     def _rhs(x, y, z, t, beta, r, rho, sigma):
         xdot = sigma * y - sigma * x - sigma/r**2 * y * x ** 2 - sigma/r**2 * y ** 3 - sigma/r**2 * y * z ** 2 + sigma/r**2 * x ** 3 + sigma/r**2 * x * y ** 2 + sigma/r**2 * x * z ** 2
         ydot = rho * x - x * z - y - rho/r**2 * x ** 3 - rho/r**2 * x * y ** 2 - rho/r**2 * x * z ** 2 + 1/r**2 * z * x ** 3 + 1/r**2 * x * z * y ** 2 + 1/r**2 * x * z ** 3 + 1/r**2 * y * x ** 2 + 1/r**2 * y ** 3 + 1/r**2 * y * z ** 2
         zdot = x * y - beta * z - 1/r**2 * y * x ** 3 - 1/r**2 * x * y ** 3 - 1/r**2 * x * y * z ** 2 + beta/r**2 * z * x ** 2 + beta/r**2 * z * y ** 2 + beta/r**2 * z ** 3
-        return xdot, ydot, zdot
source
Dysts.LorenzCoupled — Method
   LorenzCoupled()
Two coupled Lorenz attractors.
Stats
  • Embedding dimension: 6
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0408069075490425
  • Lyapunov spectrum (estimated): Any[1.2218762961515068, 0.1222254715421154, -1.265209618630438, -1.9333037412103384, -13.618111101381306, -15.073328947851422]
  • Maximum Lyapunov exponent (estimated): 1.0284398414904556
  • Multiscale Entropy: 1.1717493753138832
  • Non-autonomous: false
  • Period: 1.6207
  • Pesin entropy: 1.3441017676936224
Citation
Lorenz, Edward N. Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.
Original Python Code from the dysts package
class LorenzCoupled(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.LorenzCoupled — Method
   LorenzCoupled()
Two coupled Lorenz attractors.
Stats
  • Embedding dimension: 6
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.0408069075490425
  • Lyapunov spectrum (estimated): Any[1.2218762961515068, 0.1222254715421154, -1.265209618630438, -1.9333037412103384, -13.618111101381306, -15.073328947851422]
  • Maximum Lyapunov exponent (estimated): 1.0284398414904556
  • Multiscale Entropy: 1.1717493753138832
  • Non-autonomous: false
  • Period: 1.6207
  • Pesin entropy: 1.3441017676936224
Citation
Lorenz, Edward N. Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.
Original Python Code from the dysts package
class LorenzCoupled(DynSys):
     @staticjit
     def _rhs(x1, y1, z1, x2, y2, z2, t, beta, eps, rho, rho1, rho2, sigma):
         x1dot = sigma * y1 - sigma * x1
@@ -691,26 +691,26 @@
         x2dot = sigma * y2 - sigma * x2 + eps * x1 - eps * x2
         y2dot = rho2 * x2 - x2 * z2 - y2
         z2dot = x2 * y2 - beta * z2
-        return x1dot, y1dot, z1dot, x2dot, y2dot, z2dot
source
Dysts.LorenzStenflo — Function
   LorenzStenflo()
Atmospheric acoustic-gravity waves.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1738121741011174
  • Lyapunov spectrum (estimated): Any[0.44944459925783387, 0.026077542739630446, -2.7358399678801906, -3.4213273148340293]
  • Maximum Lyapunov exponent (estimated): 0.3807742646421621
  • Multiscale Entropy: 1.0299236916146275
  • Non-autonomous: false
  • Period: 3.8877142857142855
  • Pesin entropy: 0.47552214199746434
Citation
Letellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.
Original Python Code from the dysts package
class LorenzStenflo(DynSys):
+        return x1dot, y1dot, z1dot, x2dot, y2dot, z2dot
source
ChaoticDynamicalSystemsLibrary.LorenzStenflo — Function
   LorenzStenflo()
Atmospheric acoustic-gravity waves.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1738121741011174
  • Lyapunov spectrum (estimated): Any[0.44944459925783387, 0.026077542739630446, -2.7358399678801906, -3.4213273148340293]
  • Maximum Lyapunov exponent (estimated): 0.3807742646421621
  • Multiscale Entropy: 1.0299236916146275
  • Non-autonomous: false
  • Period: 3.8877142857142855
  • Pesin entropy: 0.47552214199746434
Citation
Letellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.
Original Python Code from the dysts package
class LorenzStenflo(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d):
         xdot = a * y - a * x + d * w
         ydot = c * x - x * z - y
         zdot = x * y - b * z
         wdot = -x - a * w
-        return xdot, ydot, zdot, wdot
source
Dysts.LuChen — Function
   LuChen()
A system that switches shapes between the Lorenz and Chen attractors as the parameter c is varied from 12.7 to 28.5. Also called the Chen Celikovsky attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0358980309317456
  • Lyapunov spectrum (estimated): Any[0.6651845739025707, 0.1450338199952451, -22.56999542505058]
  • Maximum Lyapunov exponent (estimated): 0.3266868574498627
  • Multiscale Entropy: 0.4483903807918106
  • Non-autonomous: false
  • Period: 2.3385
  • Pesin entropy: 0.8102183938978158
Citation
Lu, Chen. Int J Bifurcat Chaos. 2002: 659-661.
Original Python Code from the dysts package
class LuChen(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.LuChen — Function
   LuChen()
A system that switches shapes between the Lorenz and Chen attractors as the parameter c is varied from 12.7 to 28.5. Also called the Chen Celikovsky attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0358980309317456
  • Lyapunov spectrum (estimated): Any[0.6651845739025707, 0.1450338199952451, -22.56999542505058]
  • Maximum Lyapunov exponent (estimated): 0.3266868574498627
  • Multiscale Entropy: 0.4483903807918106
  • Non-autonomous: false
  • Period: 2.3385
  • Pesin entropy: 0.8102183938978158
Citation
Lu, Chen. Int J Bifurcat Chaos. 2002: 659-661.
Original Python Code from the dysts package
class LuChen(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = a * y - a * x
         ydot = -x * z + c * y
         zdot = x * y - b * z
-        return xdot, ydot, zdot
source
Dysts.LuChenCheng — Function
   LuChenCheng()
A four scroll attractor that reduces to Lorenz-like dynamics under bifurcations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0293036172873244
  • Lyapunov spectrum (estimated): Any[0.35106210160478324, -0.012414362698175424, -11.556516575635818]
  • Maximum Lyapunov exponent (estimated): 0.2612775880231121
  • Multiscale Entropy: 0.6029763775293518
  • Non-autonomous: false
  • Period: 0.92333
  • Pesin entropy: 0.3510621518701211
Citation
Lu, Chen, Cheng. Int J Bifurcat Chaos. 2004: 1507–1537.
Original Python Code from the dysts package
class LuChenCheng(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.LuChenCheng — Function
   LuChenCheng()
A four scroll attractor that reduces to Lorenz-like dynamics under bifurcations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0293036172873244
  • Lyapunov spectrum (estimated): Any[0.35106210160478324, -0.012414362698175424, -11.556516575635818]
  • Maximum Lyapunov exponent (estimated): 0.2612775880231121
  • Multiscale Entropy: 0.6029763775293518
  • Non-autonomous: false
  • Period: 0.92333
  • Pesin entropy: 0.3510621518701211
Citation
Lu, Chen, Cheng. Int J Bifurcat Chaos. 2004: 1507–1537.
Original Python Code from the dysts package
class LuChenCheng(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = -(a * b) / (a + b) * x - y * z + c
         ydot = a * y + x * z
         zdot = b * z + x * y
-        return xdot, ydot, zdot
source
Dysts.MacArthur — Function
   MacArthur()
Population abundances in a plankton community, based on MacArthur's consumer resource model. Growth rates are determined using Liebig's law of the maximum
Stats
  • Embedding dimension: 10
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.8961444775216614
  • Lyapunov spectrum (estimated): Any[0.1934617869736995, 0.004866584152339167, -0.0431132227766039, -0.17665092099562724, -0.30357752906614904, -0.30392441898183636, -0.30474608910196055, -0.3051840587615213, -0.3058908399877519, -17.37453227551654]
  • Maximum Lyapunov exponent (estimated): 0.2494638510848618
  • Multiscale Entropy: 0.7844902548776685
  • Non-autonomous: false
  • Period: 43.406
  • Pesin entropy: 0.19832837112603868
Citation
MacArthur, R. 1969. Species packing, and what competition minimizes. PNAS 64:1369. Parameter values taken from a model of phytoplankton in Huisman & Weissing. Nature 1999.
Original Python Code from the dysts package
class MacArthur(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.MacArthur — Function
   MacArthur()
Population abundances in a plankton community, based on MacArthur's consumer resource model. Growth rates are determined using Liebig's law of the maximum
Stats
  • Embedding dimension: 10
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 3.8961444775216614
  • Lyapunov spectrum (estimated): Any[0.1934617869736995, 0.004866584152339167, -0.0431132227766039, -0.17665092099562724, -0.30357752906614904, -0.30392441898183636, -0.30474608910196055, -0.3051840587615213, -0.3058908399877519, -17.37453227551654]
  • Maximum Lyapunov exponent (estimated): 0.2494638510848618
  • Multiscale Entropy: 0.7844902548776685
  • Non-autonomous: false
  • Period: 43.406
  • Pesin entropy: 0.19832837112603868
Citation
MacArthur, R. 1969. Species packing, and what competition minimizes. PNAS 64:1369. Parameter values taken from a model of phytoplankton in Huisman & Weissing. Nature 1999.
Original Python Code from the dysts package
class MacArthur(DynSys):
     def growth_rate(self, rr):
         u0 = rr / (self.k.T + rr)
         u = self.r * u0.T
@@ -721,17 +721,17 @@
         mu = self.growth_rate(rr)
         nndot = nn * (mu - self.m)
         rrdot = self.d * (self.s - rr) - np.matmul(self.c, (mu * nn))
-        return np.hstack([nndot, rrdot])
source
Dysts.MackeyGlass — Function
   MackeyGlass()
A physiological circuit with time-delayed feedback. A standard delay dimension of three is used.
Stats
  • Embedding dimension: 10
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 8.012917999650943
  • Lyapunov spectrum (estimated): Any[0.21081023801544096, 0.16332731438425996, -0.03580777756738295, -0.041336844485590406, -0.04889078450243534, -0.059836905162829496, -0.07714200039885702, -0.10872244658769878, -0.18584871959889662, -2.2412562118189827]
  • Maximum Lyapunov exponent (estimated): 0.07287527454565443
  • Non-autonomous: false
  • Period: 5.9612230962292045
  • Pesin entropy: 0.3741375523997009
Citation
Glass, L. and Mackey, M. C. (1979). Pathological physiological conditions resulting from instabilities in physiological control systems. Ann. NY. Acad. Sci.
Original Python Code from the dysts package
class MackeyGlass(DynSysDelay):
+        return np.hstack([nndot, rrdot])
source
ChaoticDynamicalSystemsLibrary.MackeyGlass — Function
   MackeyGlass()
A physiological circuit with time-delayed feedback. A standard delay dimension of three is used.
Stats
  • Embedding dimension: 10
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 8.012917999650943
  • Lyapunov spectrum (estimated): Any[0.21081023801544096, 0.16332731438425996, -0.03580777756738295, -0.041336844485590406, -0.04889078450243534, -0.059836905162829496, -0.07714200039885702, -0.10872244658769878, -0.18584871959889662, -2.2412562118189827]
  • Maximum Lyapunov exponent (estimated): 0.07287527454565443
  • Non-autonomous: false
  • Period: 5.9612230962292045
  • Pesin entropy: 0.3741375523997009
Citation
Glass, L. and Mackey, M. C. (1979). Pathological physiological conditions resulting from instabilities in physiological control systems. Ann. NY. Acad. Sci.
Original Python Code from the dysts package
class MackeyGlass(DynSysDelay):
     @staticjit
     def _rhs(x, xt, t, beta, gamma, n, tau):
         xdot = beta * (xt / (1 + xt ** n)) - gamma * x
-        return xdot
source
Dysts.MooreSpiegel — Function
   MooreSpiegel()
A thermo-mechanical oscillator.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.3217713120052443
  • Lyapunov spectrum (estimated): Any[0.4074353399833356, -0.022079955878999143, -1.1975751283606624]
  • Maximum Lyapunov exponent (estimated): 0.6976623003008144
  • Multiscale Entropy: 0.5401952179209213
  • Non-autonomous: false
  • Period: 1.2887
  • Pesin entropy: 0.4076334003995097
Citation
Moore, Spiegel. A Thermally Excited Nonlinear Oscillator, Ap. J., 143, 871 (1966)
Original Python Code from the dysts package
class MooreSpiegel(DynSys):
+        return xdot
source
ChaoticDynamicalSystemsLibrary.MooreSpiegel — Function
   MooreSpiegel()
A thermo-mechanical oscillator.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.3217713120052443
  • Lyapunov spectrum (estimated): Any[0.4074353399833356, -0.022079955878999143, -1.1975751283606624]
  • Maximum Lyapunov exponent (estimated): 0.6976623003008144
  • Multiscale Entropy: 0.5401952179209213
  • Non-autonomous: false
  • Period: 1.2887
  • Pesin entropy: 0.4076334003995097
Citation
Moore, Spiegel. A Thermally Excited Nonlinear Oscillator, Ap. J., 143, 871 (1966)
Original Python Code from the dysts package
class MooreSpiegel(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, eps):
         xdot = y
         ydot = a * z
         zdot = -z + eps * y - y * x ** 2 - b * x
-        return xdot, ydot, zdot
source
Dysts.MultiChua — Function
   MultiChua()
Multiple interacting Chua electronic circuits.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0710214570482988
  • Lyapunov spectrum (estimated): Any[0.2210571234274344, -0.00140279701621115, -3.10264254580399]
  • Maximum Lyapunov exponent (estimated): 1.462134784614238
  • Multiscale Entropy: 0.8118881305702679
  • Non-autonomous: false
  • Period: 2.2586
  • Pesin entropy: 0.22151080996231726
Citation
Mufcstak E. Yalcin, Johan A. K. Suykens, Joos Vandewalle. Cellular Neural Networks, Multi-Scroll Chaos And Synchronization. 
Original Python Code from the dysts package
class MultiChua(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.MultiChua — Function
   MultiChua()
Multiple interacting Chua electronic circuits.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0710214570482988
  • Lyapunov spectrum (estimated): Any[0.2210571234274344, -0.00140279701621115, -3.10264254580399]
  • Maximum Lyapunov exponent (estimated): 1.462134784614238
  • Multiscale Entropy: 0.8118881305702679
  • Non-autonomous: false
  • Period: 2.2586
  • Pesin entropy: 0.22151080996231726
Citation
Mufcstak E. Yalcin, Johan A. K. Suykens, Joos Vandewalle. Cellular Neural Networks, Multi-Scroll Chaos And Synchronization. 
Original Python Code from the dysts package
class MultiChua(DynSys):
     def diode(self, x):
         m, c = self.m, self.c
         total = m[-1] * x
@@ -744,26 +744,26 @@
         xdot = self.a * (y - self.diode(x))
         ydot = x - y + z
         zdot = -self.b * y
-        return (xdot, ydot, zdot)
source
Dysts.NewtonLiepnik — Function
   NewtonLiepnik()
Euler's equations for a rigid body, augmented with a linear feedback term.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2206092778719664
  • Lyapunov spectrum (estimated): Any[0.1723564207468104, -0.00036976661148194504, -0.7795789311467557]
  • Maximum Lyapunov exponent (estimated): 0.14945341784639976
  • Multiscale Entropy: 0.6322918700854281
  • Non-autonomous: false
  • Period: 8.5095
  • Pesin entropy: 0.17242608410350868
Citation
Leipnik, R. B., and T. A. Newton (1981). Double strange attractors in rigid body motion with linear feedback control. Physics Letters A 86.2 (1981): 63-67.
Original Python Code from the dysts package
class NewtonLiepnik(DynSys):
+        return (xdot, ydot, zdot)
source
ChaoticDynamicalSystemsLibrary.NewtonLiepnik — Function
   NewtonLiepnik()
Euler's equations for a rigid body, augmented with a linear feedback term.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2206092778719664
  • Lyapunov spectrum (estimated): Any[0.1723564207468104, -0.00036976661148194504, -0.7795789311467557]
  • Maximum Lyapunov exponent (estimated): 0.14945341784639976
  • Multiscale Entropy: 0.6322918700854281
  • Non-autonomous: false
  • Period: 8.5095
  • Pesin entropy: 0.17242608410350868
Citation
Leipnik, R. B., and T. A. Newton (1981). Double strange attractors in rigid body motion with linear feedback control. Physics Letters A 86.2 (1981): 63-67.
Original Python Code from the dysts package
class NewtonLiepnik(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = -a * x + y + 10 * y * z
         ydot = -x - 0.4 * y + 5 * x * z
         zdot = b * z - 5 * x * y
-        return xdot, ydot, zdot
source
Dysts.NoseHoover — Function
   NoseHoover()
Fixed temperature molecular dynamics for a strained flow under a reduced order model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 6.213204677134776
  • Lyapunov spectrum (estimated): Any[0.04757010871740569, 0.0013844486958287268, -0.011653762286922777]
  • Maximum Lyapunov exponent (estimated): 0.010464383779400952
  • Multiscale Entropy: 0.5656074689683536
  • Non-autonomous: false
  • Period: 5.8673
  • Pesin entropy: 0.04895455741323442
Citation
Nose, S (1985). A unified formulation of the constant temperature molecular-dynamics methods. Journal of Chemical Physics (1984). Hoover, William G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A. Hoover, William G., Anthony JC Ladd, and Bill Moran (1982). High-strain-rate plastic flow studied via nonequilibrium molecular dynamics. Physical Review Letters.
Original Python Code from the dysts package
class NoseHoover(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.NoseHoover — Function
   NoseHoover()
Fixed temperature molecular dynamics for a strained flow under a reduced order model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 6.213204677134776
  • Lyapunov spectrum (estimated): Any[0.04757010871740569, 0.0013844486958287268, -0.011653762286922777]
  • Maximum Lyapunov exponent (estimated): 0.010464383779400952
  • Multiscale Entropy: 0.5656074689683536
  • Non-autonomous: false
  • Period: 5.8673
  • Pesin entropy: 0.04895455741323442
Citation
Nose, S (1985). A unified formulation of the constant temperature molecular-dynamics methods. Journal of Chemical Physics (1984). Hoover, William G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A. Hoover, William G., Anthony JC Ladd, and Bill Moran (1982). High-strain-rate plastic flow studied via nonequilibrium molecular dynamics. Physical Review Letters.
Original Python Code from the dysts package
class NoseHoover(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = y
         ydot = -x + y * z
         zdot = a - y ** 2
-        return xdot, ydot, zdot
source
Dysts.NuclearQuadrupole — Function
   NuclearQuadrupole()
A quadrupole boson Hamiltonian that produces chaos via a resonance overlap mechanism
Stats
  • Embedding dimension: 4
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 4.280996083113027
  • Lyapunov spectrum (estimated): Any[0.27858365099437626, 0.06629535725881477, -0.00449888101008936, -0.2658004759134054]
  • Maximum Lyapunov exponent (estimated): 0.29554197767663787
  • Multiscale Entropy: 0.8209529097052416
  • Non-autonomous: false
  • Period: 6.023076923076923
  • Pesin entropy: 0.3448827969293456
Citation
Baran V. and Raduta A. A. (1998), International Journal of Modern Physics E
Original Python Code from the dysts package
class NuclearQuadrupole(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.NuclearQuadrupole — Function
   NuclearQuadrupole()
A quadrupole boson Hamiltonian that produces chaos via a resonance overlap mechanism
Stats
  • Embedding dimension: 4
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 4.280996083113027
  • Lyapunov spectrum (estimated): Any[0.27858365099437626, 0.06629535725881477, -0.00449888101008936, -0.2658004759134054]
  • Maximum Lyapunov exponent (estimated): 0.29554197767663787
  • Multiscale Entropy: 0.8209529097052416
  • Non-autonomous: false
  • Period: 6.023076923076923
  • Pesin entropy: 0.3448827969293456
Citation
Baran V. and Raduta A. A. (1998), International Journal of Modern Physics E
Original Python Code from the dysts package
class NuclearQuadrupole(DynSys):
     @staticjit
     def _rhs(q1, q2, p1, p2, t, a, b, d):
         q1dot = a * p1
         q2dot = a * p2
         p1dot = - a * q1 + 3 / np.sqrt(2) * b * q1 ** 2 - 3 / np.sqrt(2) * b * q2 ** 2 - d * q1 ** 3 - d * q1 * q2 ** 2
         p2dot = -a * q2 - 3 * np.sqrt(2) * b * q1 * q2 - d * q2 * q1 ** 2 - d * q2 ** 3
-        return q1dot, q2dot, p1dot, p2dot
source
Dysts.OscillatingFlow — Function
   OscillatingFlow()
A model fluid flow that produces KAM tori. Original parameters have been rescaled by k
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.962629189215532
  • Lyapunov spectrum (estimated): Any[0.4274159768365289, -0.005029615144529367, -0.43888500460322705]
  • Maximum Lyapunov exponent (estimated): 0.13470179997834794
  • Multiscale Entropy: 0.6177682255070645
  • Non-autonomous: true
  • Period: 23.857
  • Pesin entropy: 0.4274159768365288
Citation
T. H. Solomon and J. P. Gollub, Phys. Rev. A 38, 6280 (1988).
Original Python Code from the dysts package
class OscillatingFlow(DynSys):
+        return q1dot, q2dot, p1dot, p2dot
source
ChaoticDynamicalSystemsLibrary.OscillatingFlow — Function
   OscillatingFlow()
A model fluid flow that produces KAM tori. Original parameters have been rescaled by k
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.962629189215532
  • Lyapunov spectrum (estimated): Any[0.4274159768365289, -0.005029615144529367, -0.43888500460322705]
  • Maximum Lyapunov exponent (estimated): 0.13470179997834794
  • Multiscale Entropy: 0.6177682255070645
  • Non-autonomous: true
  • Period: 23.857
  • Pesin entropy: 0.4274159768365288
Citation
T. H. Solomon and J. P. Gollub, Phys. Rev. A 38, 6280 (1988).
Original Python Code from the dysts package
class OscillatingFlow(DynSys):
     @staticjit
     def _rhs(x, y, z, t, b, k, omega, u):
         f = x + b * np.sin(z)
@@ -773,200 +773,200 @@
         return dx, dy, dz
 
     def _postprocessing(self, x, y, z):
-        return np.cos(self.k * x), y, np.sin(z)
source
Dysts.PehlivanWei — Function
   PehlivanWei()
A system with quadratic nonlinearity, which undergoes Hopf bifurcations of codimension 1 and 2.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1563635563177295
  • Lyapunov spectrum (estimated): Any[0.09512327528959069, 0.0015323304198873254, -0.6168618108533457]
  • Maximum Lyapunov exponent (estimated): 0.10415076085878888
  • Multiscale Entropy: 0.6547493000137818
  • Non-autonomous: false
  • Period: 3.7487330316742087
  • Pesin entropy: 0.09665562724246173
Citation
Pehlivan, Ihsan, and Wei Zhouchao (2012). Analysis, nonlinear control, and chaos generator circuit of another strange chaotic system.
Original Python Code from the dysts package
class PehlivanWei(DynSys):
+        return np.cos(self.k * x), y, np.sin(z)
source
ChaoticDynamicalSystemsLibrary.PehlivanWei — Function
   PehlivanWei()
A system with quadratic nonlinearity, which undergoes Hopf bifurcations of codimension 1 and 2.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1563635563177295
  • Lyapunov spectrum (estimated): Any[0.09512327528959069, 0.0015323304198873254, -0.6168618108533457]
  • Maximum Lyapunov exponent (estimated): 0.10415076085878888
  • Multiscale Entropy: 0.6547493000137818
  • Non-autonomous: false
  • Period: 3.7487330316742087
  • Pesin entropy: 0.09665562724246173
Citation
Pehlivan, Ihsan, and Wei Zhouchao (2012). Analysis, nonlinear control, and chaos generator circuit of another strange chaotic system.
Original Python Code from the dysts package
class PehlivanWei(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = y - y * z
         ydot = y + y * z - 2 * x
         zdot = 2 - x * y - y ** 2
-        return xdot, ydot, zdot
source
Dysts.Qi — Function
   Qi()
A hyperchaotic system with a wide power spectrum.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.498645226832379
  • Lyapunov spectrum (estimated): Any[2.0360429519545677, 0.1320125824983059, -4.347891883424509, -44.356041824978305]
  • Maximum Lyapunov exponent (estimated): 2.1916060472989556
  • Multiscale Entropy: 0.8855652233700573
  • Non-autonomous: false
  • Period: 0.40820999999999996
  • Pesin entropy: 2.168055534452874
Citation
G. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.
Original Python Code from the dysts package
class Qi(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Qi — Function
   Qi()
A hyperchaotic system with a wide power spectrum.
Stats
  • Embedding dimension: 4
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.498645226832379
  • Lyapunov spectrum (estimated): Any[2.0360429519545677, 0.1320125824983059, -4.347891883424509, -44.356041824978305]
  • Maximum Lyapunov exponent (estimated): 2.1916060472989556
  • Multiscale Entropy: 0.8855652233700573
  • Non-autonomous: false
  • Period: 0.40820999999999996
  • Pesin entropy: 2.168055534452874
Citation
G. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.
Original Python Code from the dysts package
class Qi(DynSys):
     @staticjit
     def _rhs(x, y, z, w, t, a, b, c, d):
         xdot = a * y - a * x + y * z * w
         ydot = b * x + b * y - x * z * w
         zdot = -c * z + x * y * w
         wdot = -d * w + x * y * z
-        return xdot, ydot, zdot, wdot
source
Dysts.QiChen — Function
   QiChen()
A double-wing chaotic attractor that arises from two bistable attractors. The system has quadratic nonlinearity.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0917466081236316
  • Lyapunov spectrum (estimated): Any[3.8880075026301206, 0.13379585631170307, -43.8359895934497]
  • Maximum Lyapunov exponent (estimated): 4.0275703123027435
  • Multiscale Entropy: 0.846453868214798
  • Non-autonomous: false
  • Period: 0.39179
  • Pesin entropy: 4.021803358941824
Citation
Qi et al. Chaos, Solitons & Fractals 2008.
Original Python Code from the dysts package
class QiChen(DynSys):
+        return xdot, ydot, zdot, wdot
source
ChaoticDynamicalSystemsLibrary.QiChen — Function
   QiChen()
A double-wing chaotic attractor that arises from two bistable attractors. The system has quadratic nonlinearity.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0917466081236316
  • Lyapunov spectrum (estimated): Any[3.8880075026301206, 0.13379585631170307, -43.8359895934497]
  • Maximum Lyapunov exponent (estimated): 4.0275703123027435
  • Multiscale Entropy: 0.846453868214798
  • Non-autonomous: false
  • Period: 0.39179
  • Pesin entropy: 4.021803358941824
Citation
Qi et al. Chaos, Solitons & Fractals 2008.
Original Python Code from the dysts package
class QiChen(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = a * y - a * x + y * z
         ydot = c * x + y - x * z
         zdot = x * y - b * z
-        return xdot, ydot, zdot
source
Dysts.RabinovichFabrikant — Function
   RabinovichFabrikant()
A reduced-order model of propagating waves in monequilibrium media.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.36076141745315
  • Lyapunov spectrum (estimated): Any[0.24233580922074083, 1.7068504600119996e-5, -0.6717710576786404]
  • Maximum Lyapunov exponent (estimated): 0.2541443669572464
  • Multiscale Entropy: 0.9189339042839536
  • Non-autonomous: false
  • Period: 2.3356
  • Pesin entropy: 0.24239107527154555
Citation
Rabinovich, Mikhail I.; Fabrikant, A. L. (1979). Stochastic Self-Modulation of Waves in Nonequilibrium Media. Sov. Phys. JETP.
Original Python Code from the dysts package
class RabinovichFabrikant(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.RabinovichFabrikant — Function
   RabinovichFabrikant()
A reduced-order model of propagating waves in monequilibrium media.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.36076141745315
  • Lyapunov spectrum (estimated): Any[0.24233580922074083, 1.7068504600119996e-5, -0.6717710576786404]
  • Maximum Lyapunov exponent (estimated): 0.2541443669572464
  • Multiscale Entropy: 0.9189339042839536
  • Non-autonomous: false
  • Period: 2.3356
  • Pesin entropy: 0.24239107527154555
Citation
Rabinovich, Mikhail I.; Fabrikant, A. L. (1979). Stochastic Self-Modulation of Waves in Nonequilibrium Media. Sov. Phys. JETP.
Original Python Code from the dysts package
class RabinovichFabrikant(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, g):
         xdot = y * z - y + y * x ** 2 + g * x
         ydot = 3 * x * z + x - x ** 3 + g * y
         zdot = -2 * a * z  - 2 * x * y * z
-        return (xdot, ydot, zdot)
source
Dysts.RikitakeDynamo — Function
   RikitakeDynamo()
Electric current and magnetic field of two coupled disk dynamos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0672834652602137
  • Lyapunov spectrum (estimated): Any[0.14571192786308473, 0.0007222098218782572, -2.1743990404663296]
  • Maximum Lyapunov exponent (estimated): 0.1318127385277225
  • Multiscale Entropy: 0.9664471212975273
  • Non-autonomous: false
  • Period: 5.8975
  • Pesin entropy: 0.14655639292019562
Citation
Rikitake, T., Oscillations of a system of disk dynamos, Mathematical Proceedings of the Cambridge Philosophical Society, 1958.
Original Python Code from the dysts package
class RikitakeDynamo(DynSys):
+        return (xdot, ydot, zdot)
source
ChaoticDynamicalSystemsLibrary.RikitakeDynamo — Function
   RikitakeDynamo()
Electric current and magnetic field of two coupled disk dynamos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0672834652602137
  • Lyapunov spectrum (estimated): Any[0.14571192786308473, 0.0007222098218782572, -2.1743990404663296]
  • Maximum Lyapunov exponent (estimated): 0.1318127385277225
  • Multiscale Entropy: 0.9664471212975273
  • Non-autonomous: false
  • Period: 5.8975
  • Pesin entropy: 0.14655639292019562
Citation
Rikitake, T., Oscillations of a system of disk dynamos, Mathematical Proceedings of the Cambridge Philosophical Society, 1958.
Original Python Code from the dysts package
class RikitakeDynamo(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, mu):
         xdot = -mu * x + y * z
         ydot = -mu * y - a * x + x * z
         zdot = 1 - x * y
-        return xdot, ydot, zdot
source
Dysts.Rossler — Function
   Rossler()
Spiral-type chaos in a simple oscillator model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0146095059018845
  • Lyapunov spectrum (estimated): Any[0.0763301393780693, 0.007522476799147231, -5.741024338716747]
  • Maximum Lyapunov exponent (estimated): 0.15059688939547888
  • Multiscale Entropy: 0.605878349292794
  • Non-autonomous: false
  • Period: 5.9086
  • Pesin entropy: 0.08385261617721652
Citation
Rossler, O. E. (1976), An Equation for Continuous Chaos, Physics Letters, 57A (5): 3972013398,
Original Python Code from the dysts package
class Rossler(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Rossler — Function
   Rossler()
Spiral-type chaos in a simple oscillator model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0146095059018845
  • Lyapunov spectrum (estimated): Any[0.0763301393780693, 0.007522476799147231, -5.741024338716747]
  • Maximum Lyapunov exponent (estimated): 0.15059688939547888
  • Multiscale Entropy: 0.605878349292794
  • Non-autonomous: false
  • Period: 5.9086
  • Pesin entropy: 0.08385261617721652
Citation
Rossler, O. E. (1976), An Equation for Continuous Chaos, Physics Letters, 57A (5): 3972013398,
Original Python Code from the dysts package
class Rossler(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c):
         xdot = -y - z
         ydot = x + a * y
         zdot = b + z * x - c * z
-        return xdot, ydot, zdot
source
Dysts.Sakarya — Function
   Sakarya()
An attractor that arises due to merging of two disjoint bistable attractors.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.245095314885476
  • Lyapunov spectrum (estimated): Any[0.3080921246264039, 0.00035292037656675234, -1.2584697568737606]
  • Maximum Lyapunov exponent (estimated): 0.26501036485491514
  • Multiscale Entropy: 0.6082767822064958
  • Non-autonomous: false
  • Period: 4.9844
  • Pesin entropy: 0.3084450450029706
Citation
Li, Chunbiao, et al (2015). A novel four-wing strange attractor born in bistability. IEICE Electronics Express 12.4.
Original Python Code from the dysts package
class Sakarya(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Sakarya — Function
   Sakarya()
An attractor that arises due to merging of two disjoint bistable attractors.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.245095314885476
  • Lyapunov spectrum (estimated): Any[0.3080921246264039, 0.00035292037656675234, -1.2584697568737606]
  • Maximum Lyapunov exponent (estimated): 0.26501036485491514
  • Multiscale Entropy: 0.6082767822064958
  • Non-autonomous: false
  • Period: 4.9844
  • Pesin entropy: 0.3084450450029706
Citation
Li, Chunbiao, et al (2015). A novel four-wing strange attractor born in bistability. IEICE Electronics Express 12.4.
Original Python Code from the dysts package
class Sakarya(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, h, p, q, r, s):
         xdot = a * x + h * y + s * y * z
         ydot = -b * y - p * x + q * x * z
         zdot = c * z - r * x * y
-        return xdot, ydot, zdot
source
Dysts.SaltonSea — Function
   SaltonSea()
An eco-epidemiological model of bird and fish abundances in the Salton sea of southern California.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0843843542474496
  • Lyapunov spectrum (estimated): Any[0.3914445015077359, -0.0037578848522063194, -4.592867667006148]
  • Maximum Lyapunov exponent (estimated): 1.0702552017260383
  • Multiscale Entropy: 0.9335269725171643
  • Non-autonomous: false
  • Period: 0.96738
  • Pesin entropy: 0.39160860195935765
Citation
Upadhyay, Bairagi, Kundu, Chattopadhyay (2007).Chaos in eco-epidemiological problem of the Salton Sea and its possible control. Applied mathematics and computation.
Original Python Code from the dysts package
class SaltonSea(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SaltonSea — Function
   SaltonSea()
An eco-epidemiological model of bird and fish abundances in the Salton sea of southern California.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0843843542474496
  • Lyapunov spectrum (estimated): Any[0.3914445015077359, -0.0037578848522063194, -4.592867667006148]
  • Maximum Lyapunov exponent (estimated): 1.0702552017260383
  • Multiscale Entropy: 0.9335269725171643
  • Non-autonomous: false
  • Period: 0.96738
  • Pesin entropy: 0.39160860195935765
Citation
Upadhyay, Bairagi, Kundu, Chattopadhyay (2007).Chaos in eco-epidemiological problem of the Salton Sea and its possible control. Applied mathematics and computation.
Original Python Code from the dysts package
class SaltonSea(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, d, k, lam, m, mu, r, th):
         xdot = r * x * (1 - (x + y) / k) - lam * x * y
         ydot = lam * x * y - m * y * z / (y + a) - mu * y
         zdot = th * y * z / (y + a) - d * z
-        return xdot, ydot, zdot
source
Dysts.SanUmSrisuchinwong — Function
   SanUmSrisuchinwong()
A two-scroll attractor arising from dynamical equations with piecewise nonlinearities. Equivalent to the Wimol-Banlue system
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.194219372064799
  • Lyapunov spectrum (estimated): Any[0.23913890446639047, -0.0002916165416126154, -1.2296759041605134]
  • Maximum Lyapunov exponent (estimated): 0.2606135207962152
  • Multiscale Entropy: 0.9947542017597873
  • Non-autonomous: false
  • Period: 7.4654
  • Pesin entropy: 0.23946267672516214
Citation
San-Um, Srisuchinwong. J. Comp 2012
Original Python Code from the dysts package
class SanUmSrisuchinwong(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SanUmSrisuchinwong — Function
   SanUmSrisuchinwong()
A two-scroll attractor arising from dynamical equations with piecewise nonlinearities. Equivalent to the Wimol-Banlue system
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.194219372064799
  • Lyapunov spectrum (estimated): Any[0.23913890446639047, -0.0002916165416126154, -1.2296759041605134]
  • Maximum Lyapunov exponent (estimated): 0.2606135207962152
  • Multiscale Entropy: 0.9947542017597873
  • Non-autonomous: false
  • Period: 7.4654
  • Pesin entropy: 0.23946267672516214
Citation
San-Um, Srisuchinwong. J. Comp 2012
Original Python Code from the dysts package
class SanUmSrisuchinwong(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = y - x
         ydot = -z * np.tanh(x)
         zdot = -a + x * y + np.abs(y)
-        return xdot, ydot, zdot
source
Dysts.ShimizuMorioka — Function
   ShimizuMorioka()
A system that bifurcates from a symmetric limit cycle to an asymmetric one, similar to the Lorenz attractor.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0312187919639926
  • Lyapunov spectrum (estimated): Any[0.044032234819090736, -1.3942528300804223e-5, -1.409991553164768]
  • Maximum Lyapunov exponent (estimated): 0.03577425334463406
  • Multiscale Entropy: 0.7156563118943259
  • Non-autonomous: false
  • Period: 12.038
  • Pesin entropy: 0.044239420179671494
Citation
Shimizu, Morioka. Phys Lett A. 1980: 201-204
Original Python Code from the dysts package
class ShimizuMorioka(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.ShimizuMorioka — Function
   ShimizuMorioka()
A system that bifurcates from a symmetric limit cycle to an asymmetric one, similar to the Lorenz attractor.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0312187919639926
  • Lyapunov spectrum (estimated): Any[0.044032234819090736, -1.3942528300804223e-5, -1.409991553164768]
  • Maximum Lyapunov exponent (estimated): 0.03577425334463406
  • Multiscale Entropy: 0.7156563118943259
  • Non-autonomous: false
  • Period: 12.038
  • Pesin entropy: 0.044239420179671494
Citation
Shimizu, Morioka. Phys Lett A. 1980: 201-204
Original Python Code from the dysts package
class ShimizuMorioka(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = y
         ydot = x - a * y - x * z
         zdot = -b * z + x ** 2
-        return xdot, ydot, zdot
source
Dysts.SprottA — Function
   SprottA()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 60.66528307527488
  • Lyapunov spectrum (estimated): Any[0.029259920155059096, 0.004990159457687409, -0.0015561317158397701]
  • Maximum Lyapunov exponent (estimated): 0.012993052652429771
  • Multiscale Entropy: 0.5748267225365974
  • Non-autonomous: false
  • Period: 6.3728461538461545
  • Pesin entropy: 0.0343008088722448
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottA(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottA — Function
   SprottA()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 60.66528307527488
  • Lyapunov spectrum (estimated): Any[0.029259920155059096, 0.004990159457687409, -0.0015561317158397701]
  • Maximum Lyapunov exponent (estimated): 0.012993052652429771
  • Multiscale Entropy: 0.5748267225365974
  • Non-autonomous: false
  • Period: 6.3728461538461545
  • Pesin entropy: 0.0343008088722448
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottA(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = y
         ydot = -x + y * z
         zdot = 1 - y ** 2
-        return xdot, ydot, zdot
source
Dysts.SprottB — Function
   SprottB()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1852994894688953
  • Lyapunov spectrum (estimated): Any[0.21987503866971636, 0.005187437926969091, -1.2145826242115807]
  • Maximum Lyapunov exponent (estimated): 0.20674275256006883
  • Multiscale Entropy: 0.8826029903839796
  • Non-autonomous: false
  • Period: 11.050142857142857
  • Pesin entropy: 0.2250624765966855
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottB(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottB — Function
   SprottB()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1852994894688953
  • Lyapunov spectrum (estimated): Any[0.21987503866971636, 0.005187437926969091, -1.2145826242115807]
  • Maximum Lyapunov exponent (estimated): 0.20674275256006883
  • Multiscale Entropy: 0.8826029903839796
  • Non-autonomous: false
  • Period: 11.050142857142857
  • Pesin entropy: 0.2250624765966855
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottB(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = y * z
         ydot = x - y
         zdot = 1 - x * y
-        return xdot, ydot, zdot
source
Dysts.SprottC — Function
   SprottC()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1529236029587073
  • Lyapunov spectrum (estimated): Any[0.17898394736414464, -0.0007304674803429964, -1.1644550651924115]
  • Maximum Lyapunov exponent (estimated): 0.15806924766944033
  • Multiscale Entropy: 1.14839462464542
  • Non-autonomous: false
  • Period: 8.8852
  • Pesin entropy: 0.17933456833596748
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottC(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottC — Function
   SprottC()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1529236029587073
  • Lyapunov spectrum (estimated): Any[0.17898394736414464, -0.0007304674803429964, -1.1644550651924115]
  • Maximum Lyapunov exponent (estimated): 0.15806924766944033
  • Multiscale Entropy: 1.14839462464542
  • Non-autonomous: false
  • Period: 8.8852
  • Pesin entropy: 0.17933456833596748
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottC(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = y * z
         ydot = x - y
         zdot = 1 - x ** 2
-        return xdot, ydot, zdot
source
Dysts.SprottD — Function
   SprottD()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0807565309303526
  • Lyapunov spectrum (estimated): Any[0.11415315682442645, 0.001787998513315278, -1.4601870646976078]
  • Maximum Lyapunov exponent (estimated): 0.10170941886932346
  • Multiscale Entropy: 1.0030259093693934
  • Non-autonomous: false
  • Period: 4.8333
  • Pesin entropy: 0.11594526886491094
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottD(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottD — Function
   SprottD()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0807565309303526
  • Lyapunov spectrum (estimated): Any[0.11415315682442645, 0.001787998513315278, -1.4601870646976078]
  • Maximum Lyapunov exponent (estimated): 0.10170941886932346
  • Multiscale Entropy: 1.0030259093693934
  • Non-autonomous: false
  • Period: 4.8333
  • Pesin entropy: 0.11594526886491094
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottD(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = -y
         ydot = x + z
         zdot = x * z + 3 * y ** 2
-        return xdot, ydot, zdot
source
Dysts.SprottE — Function
   SprottE()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0894640654250147
  • Lyapunov spectrum (estimated): Any[0.09834470642927143, -0.00031069798260046597, -1.0953799295090543]
  • Maximum Lyapunov exponent (estimated): 0.09016067590084897
  • Multiscale Entropy: 0.8387888716672988
  • Non-autonomous: false
  • Period: 7.6275
  • Pesin entropy: 0.09866310814686535
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottE(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottE — Function
   SprottE()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0894640654250147
  • Lyapunov spectrum (estimated): Any[0.09834470642927143, -0.00031069798260046597, -1.0953799295090543]
  • Maximum Lyapunov exponent (estimated): 0.09016067590084897
  • Multiscale Entropy: 0.8387888716672988
  • Non-autonomous: false
  • Period: 7.6275
  • Pesin entropy: 0.09866310814686535
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottE(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = y * z
         ydot = x ** 2 - y
         zdot = 1 - 4 * x
-        return xdot, ydot, zdot
source
Dysts.SprottF — Function
   SprottF()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2516153966506316
  • Lyapunov spectrum (estimated): Any[0.13246006583273318, 0.0192716855937824, -0.603011686855317]
  • Maximum Lyapunov exponent (estimated): 0.1183090101484132
  • Multiscale Entropy: 0.7674306147320477
  • Non-autonomous: false
  • Period: 19.244
  • Pesin entropy: 0.1517317514265156
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottF(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottF — Function
   SprottF()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2516153966506316
  • Lyapunov spectrum (estimated): Any[0.13246006583273318, 0.0192716855937824, -0.603011686855317]
  • Maximum Lyapunov exponent (estimated): 0.1183090101484132
  • Multiscale Entropy: 0.7674306147320477
  • Non-autonomous: false
  • Period: 19.244
  • Pesin entropy: 0.1517317514265156
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottF(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = y + z
         ydot = -x + a * y
         zdot = x ** 2 - z
-        return xdot, ydot, zdot
source
Dysts.SprottG — Function
   SprottG()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0519020031307744
  • Lyapunov spectrum (estimated): Any[0.03349131570257798, -0.0012749369106980868, -0.6182204265060449]
  • Maximum Lyapunov exponent (estimated): 0.09067832253388744
  • Multiscale Entropy: 0.6185493944257671
  • Non-autonomous: false
  • Period: 8.7097
  • Pesin entropy: 0.03403484153064641
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottG(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottG — Function
   SprottG()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0519020031307744
  • Lyapunov spectrum (estimated): Any[0.03349131570257798, -0.0012749369106980868, -0.6182204265060449]
  • Maximum Lyapunov exponent (estimated): 0.09067832253388744
  • Multiscale Entropy: 0.6185493944257671
  • Non-autonomous: false
  • Period: 8.7097
  • Pesin entropy: 0.03403484153064641
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottG(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = a * x + z
         ydot = x * z - y
         zdot = -x + y
-        return xdot, ydot, zdot
source
Dysts.SprottH — Function
   SprottH()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2414973487697853
  • Lyapunov spectrum (estimated): Any[0.13871794302385684, 0.003501431527456669, -0.5889065665901029]
  • Maximum Lyapunov exponent (estimated): 0.13100182470690253
  • Multiscale Entropy: 0.6325364503922819
  • Non-autonomous: false
  • Period: 20.229
  • Pesin entropy: 0.14221937455131348
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottH(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottH — Function
   SprottH()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2414973487697853
  • Lyapunov spectrum (estimated): Any[0.13871794302385684, 0.003501431527456669, -0.5889065665901029]
  • Maximum Lyapunov exponent (estimated): 0.13100182470690253
  • Multiscale Entropy: 0.6325364503922819
  • Non-autonomous: false
  • Period: 20.229
  • Pesin entropy: 0.14221937455131348
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottH(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = -y + z ** 2
         ydot = x + a * y
         zdot = x - z
-        return xdot, ydot, zdot
source
Dysts.SprottI — Function
   SprottI()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.019844528960972
  • Lyapunov spectrum (estimated): Any[0.020459651934364873, 9.478591964272237e-6, -1.031514712617076]
  • Maximum Lyapunov exponent (estimated): 0.023218704376506836
  • Multiscale Entropy: 0.5893805046352655
  • Non-autonomous: false
  • Period: 10.93
  • Pesin entropy: 0.02077411702709478
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottI(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottI — Function
   SprottI()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.019844528960972
  • Lyapunov spectrum (estimated): Any[0.020459651934364873, 9.478591964272237e-6, -1.031514712617076]
  • Maximum Lyapunov exponent (estimated): 0.023218704376506836
  • Multiscale Entropy: 0.5893805046352655
  • Non-autonomous: false
  • Period: 10.93
  • Pesin entropy: 0.02077411702709478
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottI(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = -a * y
         ydot = x + z
         zdot = x + y ** 2 - z
-        return xdot, ydot, zdot
source
Dysts.SprottJ — Function
   SprottJ()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0221468017281006
  • Lyapunov spectrum (estimated): Any[0.04451808076432957, 0.0010621228852282804, -2.055842238064512]
  • Maximum Lyapunov exponent (estimated): 0.12290789099642041
  • Multiscale Entropy: 0.6352974298685767
  • Non-autonomous: false
  • Period: 4.836
  • Pesin entropy: 0.04588958821017871
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottJ(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottJ — Function
   SprottJ()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0221468017281006
  • Lyapunov spectrum (estimated): Any[0.04451808076432957, 0.0010621228852282804, -2.055842238064512]
  • Maximum Lyapunov exponent (estimated): 0.12290789099642041
  • Multiscale Entropy: 0.6352974298685767
  • Non-autonomous: false
  • Period: 4.836
  • Pesin entropy: 0.04588958821017871
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottJ(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = 2 * z
         ydot = -2 * y + z
         zdot = -x + y + y ** 2
-        return (xdot, ydot, zdot)
source
Dysts.SprottJerk — Function
   SprottJerk()
An algebraidally simple flow depending on a third time derivative. Numerical integration of this attractor is extremely sensitive to the initial conditions
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.031190248069004
  • Lyapunov spectrum (estimated): Any[0.06738258231535933, -0.0007831911114099333, -2.1347047232043663]
  • Maximum Lyapunov exponent (estimated): 0.08785190282162628
  • Multiscale Entropy: 0.5314887786603959
  • Non-autonomous: false
  • Period: 11.921
  • Pesin entropy: 0.06776939078007471
Citation
Sprott, J. C. Simplest dissipative chaotic flow. Physics Letters A (1997).
Original Python Code from the dysts package
class SprottJerk(DynSys):
+        return (xdot, ydot, zdot)
source
ChaoticDynamicalSystemsLibrary.SprottJerk — Function
   SprottJerk()
An algebraidally simple flow depending on a third time derivative. Numerical integration of this attractor is extremely sensitive to the initial conditions
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.031190248069004
  • Lyapunov spectrum (estimated): Any[0.06738258231535933, -0.0007831911114099333, -2.1347047232043663]
  • Maximum Lyapunov exponent (estimated): 0.08785190282162628
  • Multiscale Entropy: 0.5314887786603959
  • Non-autonomous: false
  • Period: 11.921
  • Pesin entropy: 0.06776939078007471
Citation
Sprott, J. C. Simplest dissipative chaotic flow. Physics Letters A (1997).
Original Python Code from the dysts package
class SprottJerk(DynSys):
     @staticjit
     def _rhs(x, y, z, t, mu):
         xdot = y
         ydot = z
         zdot = -x + y ** 2 - mu * z
-        return xdot, ydot, zdot
source
Dysts.SprottK — Function
   SprottK()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0417036451446204
  • Lyapunov spectrum (estimated): Any[0.03778259119323125, -0.0005109461395399151, -0.8934722146671759]
  • Maximum Lyapunov exponent (estimated): 0.05962593786799159
  • Multiscale Entropy: 0.6562812206790845
  • Non-autonomous: false
  • Period: 7.006
  • Pesin entropy: 0.03788160511007809
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottK(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottK — Function
   SprottK()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0417036451446204
  • Lyapunov spectrum (estimated): Any[0.03778259119323125, -0.0005109461395399151, -0.8934722146671759]
  • Maximum Lyapunov exponent (estimated): 0.05962593786799159
  • Multiscale Entropy: 0.6562812206790845
  • Non-autonomous: false
  • Period: 7.006
  • Pesin entropy: 0.03788160511007809
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottK(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = x * y - z
         ydot = x - y
         zdot = x + a * z
-        return xdot, ydot, zdot
source
Dysts.SprottL — Function
   SprottL()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0329815956000377
  • Lyapunov spectrum (estimated): Any[0.03407149258791976, -0.000391471796353918, -1.012962157697122]
  • Maximum Lyapunov exponent (estimated): 0.09216197583296963
  • Multiscale Entropy: 0.5605870039438788
  • Non-autonomous: false
  • Period: 6.7262
  • Pesin entropy: 0.0343958205328805
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottL(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottL — Function
   SprottL()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0329815956000377
  • Lyapunov spectrum (estimated): Any[0.03407149258791976, -0.000391471796353918, -1.012962157697122]
  • Maximum Lyapunov exponent (estimated): 0.09216197583296963
  • Multiscale Entropy: 0.5605870039438788
  • Non-autonomous: false
  • Period: 6.7262
  • Pesin entropy: 0.0343958205328805
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottL(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = y + b * z
         ydot = a * x ** 2 - y
         zdot = 1 - x
-        return xdot, ydot, zdot
source
Dysts.SprottM — Function
   SprottM()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.04996202872584
  • Lyapunov spectrum (estimated): Any[0.05230117379497565, -0.00033063791832836577, -1.0391447706177515]
  • Maximum Lyapunov exponent (estimated): 0.08422043130700084
  • Multiscale Entropy: 0.7481178461456197
  • Non-autonomous: false
  • Period: 5.3644
  • Pesin entropy: 0.05234518101225785
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottM(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottM — Function
   SprottM()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.04996202872584
  • Lyapunov spectrum (estimated): Any[0.05230117379497565, -0.00033063791832836577, -1.0391447706177515]
  • Maximum Lyapunov exponent (estimated): 0.08422043130700084
  • Multiscale Entropy: 0.7481178461456197
  • Non-autonomous: false
  • Period: 5.3644
  • Pesin entropy: 0.05234518101225785
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottM(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = -z
         ydot = -x ** 2 - y
         zdot = a + a * x + y
-        return xdot, ydot, zdot
source
Dysts.SprottMore — Function
   SprottMore()
A multifractal system with a nearly 3D attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.999952495039482
  • Lyapunov spectrum (estimated): Any[0.32499834596046606, -0.0001641583999746131, -0.31918544080126926]
  • Maximum Lyapunov exponent (estimated): 1.0946909441433958
  • Multiscale Entropy: 0.6370154977546665
  • Non-autonomous: false
  • Period: 6.62095890410959
  • Pesin entropy: 0.32515558132375566
Citation
Sprott, J. C. (2020). Do We Need More Chaos Examples? Chaos Theory and Applications 2(2),1-3, 2020.
Original Python Code from the dysts package
class SprottMore(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottMore — Function
   SprottMore()
A multifractal system with a nearly 3D attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.999952495039482
  • Lyapunov spectrum (estimated): Any[0.32499834596046606, -0.0001641583999746131, -0.31918544080126926]
  • Maximum Lyapunov exponent (estimated): 1.0946909441433958
  • Multiscale Entropy: 0.6370154977546665
  • Non-autonomous: false
  • Period: 6.62095890410959
  • Pesin entropy: 0.32515558132375566
Citation
Sprott, J. C. (2020). Do We Need More Chaos Examples? Chaos Theory and Applications 2(2),1-3, 2020.
Original Python Code from the dysts package
class SprottMore(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = y
         ydot = -x - np.sign(z) * y
         zdot = y ** 2 - np.exp(-(x ** 2))
-        return xdot, ydot, zdot
source
Dysts.SprottN — Function
   SprottN()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.028951405806632
  • Lyapunov spectrum (estimated): Any[0.06095861449194353, 6.560209716837452e-5, -2.101648044064128]
  • Maximum Lyapunov exponent (estimated): 0.11869612209728901
  • Multiscale Entropy: 0.6385218945868028
  • Non-autonomous: false
  • Period: 4.8135
  • Pesin entropy: 0.061478272616693266
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottN(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottN — Function
   SprottN()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.028951405806632
  • Lyapunov spectrum (estimated): Any[0.06095861449194353, 6.560209716837452e-5, -2.101648044064128]
  • Maximum Lyapunov exponent (estimated): 0.11869612209728901
  • Multiscale Entropy: 0.6385218945868028
  • Non-autonomous: false
  • Period: 4.8135
  • Pesin entropy: 0.061478272616693266
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottN(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = -2 * y
         ydot = x + z ** 2
         zdot = 1 + y - 2 * z
-        return xdot, ydot, zdot
source
Dysts.SprottO — Function
   SprottO()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2474645346215256
  • Lyapunov spectrum (estimated): Any[0.06685684927644238, 0.011533284505120292, -0.3165440349833235]
  • Maximum Lyapunov exponent (estimated): 0.09141077854590315
  • Multiscale Entropy: 0.4161154943097029
  • Non-autonomous: false
  • Period: 9.755
  • Pesin entropy: 0.07839013378156268
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottO(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottO — Function
   SprottO()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2474645346215256
  • Lyapunov spectrum (estimated): Any[0.06685684927644238, 0.011533284505120292, -0.3165440349833235]
  • Maximum Lyapunov exponent (estimated): 0.09141077854590315
  • Multiscale Entropy: 0.4161154943097029
  • Non-autonomous: false
  • Period: 9.755
  • Pesin entropy: 0.07839013378156268
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottO(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = y
         ydot = x - z
         zdot = x + x * z + a * y
-        return xdot, ydot, zdot
source
Dysts.SprottP — Function
   SprottP()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2122245136720187
  • Lyapunov spectrum (estimated): Any[0.1026165705645786, 0.00025136328640969387, -0.4843551814513015]
  • Maximum Lyapunov exponent (estimated): 0.1064676723437159
  • Multiscale Entropy: 0.8486396469370624
  • Non-autonomous: false
  • Period: 5.3414
  • Pesin entropy: 0.10329419803150507
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottP(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottP — Function
   SprottP()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2122245136720187
  • Lyapunov spectrum (estimated): Any[0.1026165705645786, 0.00025136328640969387, -0.4843551814513015]
  • Maximum Lyapunov exponent (estimated): 0.1064676723437159
  • Multiscale Entropy: 0.8486396469370624
  • Non-autonomous: false
  • Period: 5.3414
  • Pesin entropy: 0.10329419803150507
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottP(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a):
         xdot = a * y + z
         ydot = -x + y ** 2
         zdot = x + y
-        return xdot, ydot, zdot
source
Dysts.SprottQ — Function
   SprottQ()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2246852931286063
  • Lyapunov spectrum (estimated): Any[0.13698250147166127, 0.0010386716745898255, -0.6140516336902575]
  • Maximum Lyapunov exponent (estimated): 0.17169784338063596
  • Multiscale Entropy: 0.8105158145249107
  • Non-autonomous: false
  • Period: 4.8443
  • Pesin entropy: 0.13852213114060488
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottQ(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottQ — Function
   SprottQ()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.2246852931286063
  • Lyapunov spectrum (estimated): Any[0.13698250147166127, 0.0010386716745898255, -0.6140516336902575]
  • Maximum Lyapunov exponent (estimated): 0.17169784338063596
  • Multiscale Entropy: 0.8105158145249107
  • Non-autonomous: false
  • Period: 4.8443
  • Pesin entropy: 0.13852213114060488
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottQ(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = -z
         ydot = x - y
         zdot = a * x + y ** 2 + b * z
-        return (xdot, ydot, zdot)
source
Dysts.SprottR — Function
   SprottR()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.071744469374985
  • Lyapunov spectrum (estimated): Any[0.07608481335025485, 0.0013697096816499383, -1.0784000269941107]
  • Maximum Lyapunov exponent (estimated): 0.09661415277799693
  • Multiscale Entropy: 0.5877166245477188
  • Non-autonomous: false
  • Period: 7.0133
  • Pesin entropy: 0.07840404228136309
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottR(DynSys):
+        return (xdot, ydot, zdot)
source
ChaoticDynamicalSystemsLibrary.SprottR — Function
   SprottR()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.071744469374985
  • Lyapunov spectrum (estimated): Any[0.07608481335025485, 0.0013697096816499383, -1.0784000269941107]
  • Maximum Lyapunov exponent (estimated): 0.09661415277799693
  • Multiscale Entropy: 0.5877166245477188
  • Non-autonomous: false
  • Period: 7.0133
  • Pesin entropy: 0.07840404228136309
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottR(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = a - y
         ydot = b + z
         zdot = x * y - z
-        return xdot, ydot, zdot
source
Dysts.SprottS — Function
   SprottS()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.182390107009358
  • Lyapunov spectrum (estimated): Any[0.2149312025337423, 0.0030194370306177213, -1.1945344436340624]
  • Maximum Lyapunov exponent (estimated): 0.2737629235389973
  • Multiscale Entropy: 0.6690400524585065
  • Non-autonomous: false
  • Period: 3.3505
  • Pesin entropy: 0.21876911249150574
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottS(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottS — Function
   SprottS()
A member of the Sprott family of algebraically-simple chaotic sytems.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.182390107009358
  • Lyapunov spectrum (estimated): Any[0.2149312025337423, 0.0030194370306177213, -1.1945344436340624]
  • Maximum Lyapunov exponent (estimated): 0.2737629235389973
  • Multiscale Entropy: 0.6690400524585065
  • Non-autonomous: false
  • Period: 3.3505
  • Pesin entropy: 0.21876911249150574
Citation
Sprott (1994). Some simple chaotic flows. Physical Review E (1994).
Original Python Code from the dysts package
class SprottS(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = -x - 4 * y
         ydot = x + z ** 2
         zdot = 1 + x
-        return xdot, ydot, zdot
source
Dysts.SprottTorus — Function
   SprottTorus()
A multiattractor system that goes to a torus or a complex attractor depending on the initial conditions.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.759693065535405
  • Lyapunov spectrum (estimated): Any[0.29656066211909027, 0.0005468818242604863, -0.3910853761095065]
  • Maximum Lyapunov exponent (estimated): 0.1056964344650892
  • Multiscale Entropy: 0.9562901894463037
  • Non-autonomous: false
  • Period: 13.607
  • Pesin entropy: 0.29710937347161515
Citation
Sprott Physics Letters A 2014
Original Python Code from the dysts package
class SprottTorus(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.SprottTorus — Function
   SprottTorus()
A multiattractor system that goes to a torus or a complex attractor depending on the initial conditions.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.759693065535405
  • Lyapunov spectrum (estimated): Any[0.29656066211909027, 0.0005468818242604863, -0.3910853761095065]
  • Maximum Lyapunov exponent (estimated): 0.1056964344650892
  • Multiscale Entropy: 0.9562901894463037
  • Non-autonomous: false
  • Period: 13.607
  • Pesin entropy: 0.29710937347161515
Citation
Sprott Physics Letters A 2014
Original Python Code from the dysts package
class SprottTorus(DynSys):
     @staticjit
     def _rhs(x, y, z, t):
         xdot = y + 2 * x * y + x * z
         ydot = 1 - 2 * x ** 2 + y * z
         zdot = x - x ** 2 - y ** 2
-        return xdot, ydot, zdot
source
Dysts.StickSlipOscillator — Function
   StickSlipOscillator()
A weakly forced (quasiautonomous) oscillator with dynamics similar to stick-slip friction. The parameter gamma controls the transition to chaos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 5.682155291949877
  • Lyapunov spectrum (estimated): Any[0.07073329226101453, 3.356854711815694e-5, -0.02131392884593247]
  • Maximum Lyapunov exponent (estimated): 0.06406131604734494
  • Multiscale Entropy: 0.9213029740441252
  • Non-autonomous: true
  • Period: 1.6129
  • Pesin entropy: 0.07108101456222432
Citation
Awrejcewicz, Jan, and M. M. Holicke (1999). Int J of Bifurcation and Chaos.
Original Python Code from the dysts package
class StickSlipOscillator(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.StickSlipOscillator — Function
   StickSlipOscillator()
A weakly forced (quasiautonomous) oscillator with dynamics similar to stick-slip friction. The parameter gamma controls the transition to chaos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 5.682155291949877
  • Lyapunov spectrum (estimated): Any[0.07073329226101453, 3.356854711815694e-5, -0.02131392884593247]
  • Maximum Lyapunov exponent (estimated): 0.06406131604734494
  • Multiscale Entropy: 0.9213029740441252
  • Non-autonomous: true
  • Period: 1.6129
  • Pesin entropy: 0.07108101456222432
Citation
Awrejcewicz, Jan, and M. M. Holicke (1999). Int J of Bifurcation and Chaos.
Original Python Code from the dysts package
class StickSlipOscillator(DynSys):
     def _t(self, v):
         return self.t0 * np.sign(v) - self.alpha * v + self.beta * v ** 3
 
@@ -980,7 +980,7 @@
 
     @staticjit
     def _postprocessing(x, v, th):
-        return x, v, np.cos(th)
source
Dysts.SwingingAtwood — Function
   SwingingAtwood()
A mechanical system consisting of two swinging weights connected by ropes and pulleys. This is only chaotic when m2 is sufficiently larger than m1, and there are nonzero initial momenta
Stats
  • Embedding dimension: 4
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 4.0198847313975445
  • Lyapunov spectrum (estimated): Any[7.735006558021491, 3.0380875684439133, -3.002075505122432, -7.664246463289054]
  • Maximum Lyapunov exponent (estimated): 1.0111284961954359
  • Non-autonomous: false
  • Period: 0.3417647058823529
  • Pesin entropy: 10.773094126465404
Citation
Tufillaro, Nicholas B.; Abbott, Tyler A.; Griffiths, David J. (1984). Swinging Atwood's Machine. American Journal of Physics. 52 (10): 895–903.
Original Python Code from the dysts package
class SwingingAtwood(DynSys):
+        return x, v, np.cos(th)
source
ChaoticDynamicalSystemsLibrary.SwingingAtwood — Function
   SwingingAtwood()
A mechanical system consisting of two swinging weights connected by ropes and pulleys. This is only chaotic when m2 is sufficiently larger than m1, and there are nonzero initial momenta
Stats
  • Embedding dimension: 4
  • Hamiltonian: true
  • Kaplan-Yorke dimension: 4.0198847313975445
  • Lyapunov spectrum (estimated): Any[7.735006558021491, 3.0380875684439133, -3.002075505122432, -7.664246463289054]
  • Maximum Lyapunov exponent (estimated): 1.0111284961954359
  • Non-autonomous: false
  • Period: 0.3417647058823529
  • Pesin entropy: 10.773094126465404
Citation
Tufillaro, Nicholas B.; Abbott, Tyler A.; Griffiths, David J. (1984). Swinging Atwood's Machine. American Journal of Physics. 52 (10): 895–903.
Original Python Code from the dysts package
class SwingingAtwood(DynSys):
     @staticjit
     def _rhs(r, th, pr, pth, t, m1, m2):
         g = 9.82
@@ -992,13 +992,13 @@
 
     @staticjit
     def _postprocessing(r, th, pr, pth):
-        return r, np.sin(th), pr, pth
source
Dysts.Thomas — Function
   Thomas()
A cyclically-symmetric attractor correspondng to a frictionally-damped particle traversing a 3D lattice of forces
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1271136075295507
  • Lyapunov spectrum (estimated): Any[0.7589685797630259, 0.057355308262307496, -6.421950974462397]
  • Maximum Lyapunov exponent (estimated): 0.6324054000339867
  • Multiscale Entropy: 0.9921951854111759
  • Non-autonomous: false
  • Period: 4.969
  • Pesin entropy: 0.8163238880253334
Citation
Thomas, Rene (1999). Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, labyrinth chaos. Int. J. Bifurc. Chaos.
Original Python Code from the dysts package
class Thomas(DynSys):
+        return r, np.sin(th), pr, pth
source
ChaoticDynamicalSystemsLibrary.Thomas — Function
   Thomas()
A cyclically-symmetric attractor correspondng to a frictionally-damped particle traversing a 3D lattice of forces
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1271136075295507
  • Lyapunov spectrum (estimated): Any[0.7589685797630259, 0.057355308262307496, -6.421950974462397]
  • Maximum Lyapunov exponent (estimated): 0.6324054000339867
  • Multiscale Entropy: 0.9921951854111759
  • Non-autonomous: false
  • Period: 4.969
  • Pesin entropy: 0.8163238880253334
Citation
Thomas, Rene (1999). Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, labyrinth chaos. Int. J. Bifurc. Chaos.
Original Python Code from the dysts package
class Thomas(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b):
         xdot = -a * x + b * np.sin(y)
         ydot = -a * y + b * np.sin(z)
         zdot = -a * z + b * np.sin(x)
-        return xdot, ydot, zdot
source
Dysts.Torus — Function
   Torus()
A minimal quasiperiodic flow on a torus. All lyapunov exponents and related quantities are zero
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0
  • Lyapunov spectrum (estimated): Any[0.0, 0.0, 0.0]
  • Maximum Lyapunov exponent (estimated): 0.00817284128815307
  • Multiscale Entropy: 0.47330913201518593
  • Non-autonomous: false
  • Period: 6.36
  • Pesin entropy: 0.0
Citation
See, for example, Strogatz (1994). Nonlinear Dynamics and Chaos.
Original Python Code from the dysts package
class Torus(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.Torus — Function
   Torus()
A minimal quasiperiodic flow on a torus. All lyapunov exponents and related quantities are zero
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0
  • Lyapunov spectrum (estimated): Any[0.0, 0.0, 0.0]
  • Maximum Lyapunov exponent (estimated): 0.00817284128815307
  • Multiscale Entropy: 0.47330913201518593
  • Non-autonomous: false
  • Period: 6.36
  • Pesin entropy: 0.0
Citation
See, for example, Strogatz (1994). Nonlinear Dynamics and Chaos.
Original Python Code from the dysts package
class Torus(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, n, r):
         xdot = (-a * n * np.sin(n * t)) * np.cos(t) - (r + a * np.cos(n * t)) * np.sin(
@@ -1008,7 +1008,7 @@
             t
         )
         zdot = a * n * np.cos(n * t)
-        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.TurchinHanski — Function
   TurchinHanski()
 
 A chaotic three species food web. The species growth rate has been increased from the default parameters in the original paper, in order to ensure that the system exhibits sustained chaotic oscillations.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 1.2053069221786739
  • Lyapunov spectrum (estimated): Any[0.03719830596656144, -0.181183885924572, -2.522759887828014]
  • Maximum Lyapunov exponent (estimated): 0.12906862565697674
  • Multiscale Entropy: 0.4989636261373046
  • Non-autonomous: true
  • Period: 4.40529411764706
  • Pesin entropy: 0.03719830596656144
Citation
Turchin, Hanski. The American Naturalist 1997. Turchin, Hanski. Ecology 2000
Original Python Code from the dysts package
class TurchinHanski(DynSys):
     @staticjit
@@ -1025,19 +1025,19 @@
 
     @staticjit
     def _postprocessing(x, y, z):
-        return x, y, np.sin(z)
source
Dysts.VallisElNino — Function
   VallisElNino()
Atmospheric temperature fluctuations with annual forcing.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1205065379188333
  • Lyapunov spectrum (estimated): Any[0.6561566154431996, 0.022689184568283626, -5.6332694618503885]
  • Maximum Lyapunov exponent (estimated): 0.5477586276153029
  • Multiscale Entropy: 0.9058707360850027
  • Non-autonomous: false
  • Period: 2.2087
  • Pesin entropy: 0.6788458000114831
Citation
Vallis GK. Conceptual models of El Nio and the southern oscillation. J Geophys Res 1988
Original Python Code from the dysts package
class VallisElNino(DynSys):
+        return x, y, np.sin(z)
source
ChaoticDynamicalSystemsLibrary.VallisElNino — Function
   VallisElNino()
Atmospheric temperature fluctuations with annual forcing.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1205065379188333
  • Lyapunov spectrum (estimated): Any[0.6561566154431996, 0.022689184568283626, -5.6332694618503885]
  • Maximum Lyapunov exponent (estimated): 0.5477586276153029
  • Multiscale Entropy: 0.9058707360850027
  • Non-autonomous: false
  • Period: 2.2087
  • Pesin entropy: 0.6788458000114831
Citation
Vallis GK. Conceptual models of El Nio and the southern oscillation. J Geophys Res 1988
Original Python Code from the dysts package
class VallisElNino(DynSys):
     @staticmethod
     def _rhs(x, y, z, t, b, c, p):
         xdot = b * y - c * x - c * p
         ydot = -y + x * z
         zdot = -z - x * y + 1
-        return xdot, ydot, zdot
source
Dysts.WangSun — Function
   WangSun()
A four-scroll attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0595560098996604
  • Lyapunov spectrum (estimated): Any[0.07990874520682271, -0.001043918775422804, -1.3240808432405]
  • Maximum Lyapunov exponent (estimated): 0.06735463002480017
  • Multiscale Entropy: 0.9045422097501858
  • Non-autonomous: false
  • Period: 26.958
  • Pesin entropy: 0.07994326076367073
Citation
Wang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.
Original Python Code from the dysts package
class WangSun(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.WangSun — Function
   WangSun()
A four-scroll attractor
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0595560098996604
  • Lyapunov spectrum (estimated): Any[0.07990874520682271, -0.001043918775422804, -1.3240808432405]
  • Maximum Lyapunov exponent (estimated): 0.06735463002480017
  • Multiscale Entropy: 0.9045422097501858
  • Non-autonomous: false
  • Period: 26.958
  • Pesin entropy: 0.07994326076367073
Citation
Wang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.
Original Python Code from the dysts package
class WangSun(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, d, e, f, q):
         xdot = a * x + q * y * z
         ydot = b * x + d * y - x * z
         zdot = e * z + f * x * y
-        return xdot, ydot, zdot
source
Dysts.WindmiReduced — Function
   WindmiReduced()
Energy transfer into the ionosphere and magnetosphere by the solar wind. The parameter vsw controls the onset of chaos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0947817139683895
  • Lyapunov spectrum (estimated): Any[0.15595238514031914, 9.695760899839663e-5, -1.6450480911820564]
  • Maximum Lyapunov exponent (estimated): 0.403521000622694
  • Multiscale Entropy: 0.5896070173085494
  • Non-autonomous: false
  • Period: 4.6733
  • Pesin entropy: 0.15635149449218705
Citation
Smith, Thiffeault, Horton. J Geophys Res. 2000. Horton, Weigel, Sprott. Chaos 2001.
Original Python Code from the dysts package
class WindmiReduced(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.WindmiReduced — Function
   WindmiReduced()
Energy transfer into the ionosphere and magnetosphere by the solar wind. The parameter vsw controls the onset of chaos.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0947817139683895
  • Lyapunov spectrum (estimated): Any[0.15595238514031914, 9.695760899839663e-5, -1.6450480911820564]
  • Maximum Lyapunov exponent (estimated): 0.403521000622694
  • Multiscale Entropy: 0.5896070173085494
  • Non-autonomous: false
  • Period: 4.6733
  • Pesin entropy: 0.15635149449218705
Citation
Smith, Thiffeault, Horton. J Geophys Res. 2000. Horton, Weigel, Sprott. Chaos 2001.
Original Python Code from the dysts package
class WindmiReduced(DynSys):
     @staticjit
     def _rhs(i, v, p, t, a1, b1, b2, b3, d1, vsw):
         idot = a1 * (vsw - v)
@@ -1045,22 +1045,22 @@
         pdot = (
             vsw ** 2 - p ** (5 / 4) * vsw ** (1 / 2) * (1 + np.tanh(d1 * (i - 1))) / 2
         )
-        return idot, vdot, pdot
source
Dysts.YuWang — Function
   YuWang()
A temperature-compensation circuit with an operational amplifier
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1703721122420614
  • Lyapunov spectrum (estimated): Any[2.459880656234634, -0.07297917814211805, -14.00993065520751]
  • Maximum Lyapunov exponent (estimated): 1.8525516537708415
  • Multiscale Entropy: 0.6340382196367295
  • Non-autonomous: false
  • Period: 1.3366
  • Pesin entropy: 2.459880656234634
Citation
Yu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.
Original Python Code from the dysts package
class YuWang(DynSys):
+        return idot, vdot, pdot
source
ChaoticDynamicalSystemsLibrary.YuWang — Function
   YuWang()
A temperature-compensation circuit with an operational amplifier
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.1703721122420614
  • Lyapunov spectrum (estimated): Any[2.459880656234634, -0.07297917814211805, -14.00993065520751]
  • Maximum Lyapunov exponent (estimated): 1.8525516537708415
  • Multiscale Entropy: 0.6340382196367295
  • Non-autonomous: false
  • Period: 1.3366
  • Pesin entropy: 2.459880656234634
Citation
Yu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.
Original Python Code from the dysts package
class YuWang(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d):
         xdot = a * (y - x)
         ydot = b * x - c * x * z
         zdot = np.exp(x * y) - d * z
-        return xdot, ydot, zdot
source
Dysts.YuWang2 — Function
   YuWang2()
An alternative temperature-compensation circuit with an operational amplifier
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0619881661890873
  • Lyapunov spectrum (estimated): Any[0.802715215563475, 0.007966964664720393, -13.078014115524866]
  • Maximum Lyapunov exponent (estimated): 1.1576066800869989
  • Multiscale Entropy: 0.9335570471768684
  • Non-autonomous: false
  • Period: 0.39265
  • Pesin entropy: 0.8106827804626555
Citation
Yu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.
Original Python Code from the dysts package
class YuWang2(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.YuWang2 — Function
   YuWang2()
An alternative temperature-compensation circuit with an operational amplifier
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.0619881661890873
  • Lyapunov spectrum (estimated): Any[0.802715215563475, 0.007966964664720393, -13.078014115524866]
  • Maximum Lyapunov exponent (estimated): 1.1576066800869989
  • Multiscale Entropy: 0.9335570471768684
  • Non-autonomous: false
  • Period: 0.39265
  • Pesin entropy: 0.8106827804626555
Citation
Yu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.
Original Python Code from the dysts package
class YuWang2(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d):
         xdot = a * (y - x)
         ydot = b * x - c * x * z
         zdot = np.cosh(x * y) - d * z
-        return xdot, ydot, zdot
source
Dysts.ZhouChen — Function
   ZhouChen()
A feedback circuit model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.045920730005392
  • Lyapunov spectrum (estimated): Any[0.5917259125455694, 0.005203946388490041, -12.999237688229242]
  • Maximum Lyapunov exponent (estimated): 0.6828000288089435
  • Multiscale Entropy: 0.7337896982114851
  • Non-autonomous: false
  • Period: 0.87518
  • Pesin entropy: 0.5969734164748293
Citation
Zhou, Chen (2004). A simple smooth chaotic system with a 3-layer attractor. Int J Bifurcat Chaos, 2004
Original Python Code from the dysts package
class ZhouChen(DynSys):
+        return xdot, ydot, zdot
source
ChaoticDynamicalSystemsLibrary.ZhouChen — Function
   ZhouChen()
A feedback circuit model.
Stats
  • Embedding dimension: 3
  • Hamiltonian: false
  • Kaplan-Yorke dimension: 2.045920730005392
  • Lyapunov spectrum (estimated): Any[0.5917259125455694, 0.005203946388490041, -12.999237688229242]
  • Maximum Lyapunov exponent (estimated): 0.6828000288089435
  • Multiscale Entropy: 0.7337896982114851
  • Non-autonomous: false
  • Period: 0.87518
  • Pesin entropy: 0.5969734164748293
Citation
Zhou, Chen (2004). A simple smooth chaotic system with a 3-layer attractor. Int J Bifurcat Chaos, 2004
Original Python Code from the dysts package
class ZhouChen(DynSys):
     @staticjit
     def _rhs(x, y, z, t, a, b, c, d, e):
         xdot = a * x + b * y + y * z
         ydot = c * y - x * z + d * y * z
         zdot = e * z - x * y
-        return xdot, ydot, zdot
source

+        return xdot, ydot, zdot
source
diff --git a/dev/search_index.js b/dev/search_index.js index 945156b..3cf9267 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"","page":"Home","title":"Home","text":"CurrentModule = Dysts","category":"page"},{"location":"#Dysts.jl","page":"Home","title":"Dysts.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Full list of all dynamical systems:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Modules = [Dysts]\nPublic = false\nPages = [\"chaotic_attractors.jl\"]","category":"page"},{"location":"#Dysts.Aizawa","page":"Home","title":"Dysts.Aizawa","text":" Aizawa()\n\nA torus-like attractor related to the forced Lorenz system.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.3501297618370773\nLyapunov spectrum (estimated): Any[0.08947878317195473, 0.020305496211675024, -0.3090729926541944]\nMaximum Lyapunov exponent (estimated): 0.13489555530106362\nMultiscale Entropy: 0.514512246123007\nNon-autonomous: false\nPeriod: 2.5837\nPesin entropy: 0.10978427938362978\n\nCitation\n\nAizawa, Yoji, and Tatsuya Uezu (1982). Topological Aspects in Chaos and in 2 k-Period Doubling Cascade. Progress of Theoretical Physics 67.3 (1982): 982-985. See also Langford (1984). Numerical studies of torus bifurcations. Numerical Methods for Bifurcation Problems. Birkhauser, Basel, 1984. 285-295.\n\nOriginal Python Code from the dysts package\n\nclass Aizawa(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, e, f):\n xdot = x * z - b * x - d * y\n ydot = d * x + y * z - b * y\n zdot = c + a * z - 0.333333333333333333 * z ** 3 - x ** 2 - y ** 2 - e * z * x ** 2 - e * z * y ** 2 + f * z * x ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.AnishchenkoAstakhov","page":"Home","title":"Dysts.AnishchenkoAstakhov","text":" AnishchenkoAstakhov()\n\nStochastic resonance in forced oscillators.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0916403758729327\nLyapunov spectrum (estimated): Any[0.039777554890210017, 0.0008785544495802295, -0.44025308728606666]\nMaximum Lyapunov exponent (estimated): 0.10107726791527546\nMultiscale Entropy: 0.7612343892027325\nNon-autonomous: false\nPeriod: 6.814\nPesin entropy: 0.040715925817307534\n\nCitation\n\nAnishchenko, et al. Nonlinear dynamics of chaotic and stochastic systems: tutorial and modern developments. 2007.\n\nOriginal Python Code from the dysts package\n\nclass AnishchenkoAstakhov(DynSys):\n def rhs(self, X, t):\n x, y, z = X\n mu, eta = self.mu, self.eta\n xdot = mu * x + y - x * z\n ydot = -x\n zdot = -eta * z + eta * np.heaviside(x, 0) * x ** 2\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Arneodo","page":"Home","title":"Dysts.Arneodo","text":" Arneodo()\n\nA modified Lotka-Volterra ecosystem, also known as the ACT attractor.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.198607392233255\nLyapunov spectrum (estimated): Any[0.24285661379003065, -0.0006168721448237418, -1.2169777901647385]\nMaximum Lyapunov exponent (estimated): 0.23737468667051287\nMultiscale Entropy: 0.7388404221540498\nNon-autonomous: false\nPeriod: 3.1641\nPesin entropy: 0.2428749592514372\n\nCitation\n\nArneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional Volterra equations. Physics Letters A 79, 259–263 (1980)\n\nOriginal Python Code from the dysts package\n\nclass Arneodo(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d):\n xdot = y\n ydot = z\n zdot = -a * x - b * y - c * z + d * x ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ArnoldBeltramiChildress","page":"Home","title":"Dysts.ArnoldBeltramiChildress","text":" ArnoldBeltramiChildress()\n\nAn exact solution of Euler's equation for inviscid flow in 3D. Often referred to as the ABC flow.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.612281260580053\nLyapunov spectrum (estimated): Any[1.47222087191395, -0.01824039630029517, -2.378562986862767]\nMaximum Lyapunov exponent (estimated): 0.3667300080060703\nMultiscale Entropy: 1.2759526834024162\nNon-autonomous: false\nPeriod: 2.8508\nPesin entropy: 1.47222087191395\n\nCitation\n\nV. I. Arnold, Journal of Applied Mathematics and Mechanics 30, 223 (1966).\n\nOriginal Python Code from the dysts package\n\nclass ArnoldBeltramiChildress(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n dx = a * np.sin(z) + c * np.cos(y)\n dy = b * np.sin(x) + a * np.cos(z)\n dz = c * np.sin(y) + b * np.cos(x)\n return dx, dy, dz\n\n @staticjit\n def _postprocessing(x, y, z):\n return np.sin(x), np.cos(y), np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ArnoldWeb","page":"Home","title":"Dysts.ArnoldWeb","text":" ArnoldWeb()\n\nA quasi-integrable system that transitions to Chirikov diffusive dynamics when perturbed.\n\nStats\n\nEmbedding dimension: 5\nHamiltonian: true\nKaplan-Yorke dimension: 22.554733912627043\nLyapunov spectrum (estimated): Any[0.06266308160064184, 0.0032177165561558833, 0.0011281526899086728, -0.00233619507966634, -0.0039196281630406]\nMaximum Lyapunov exponent (estimated): 0.06514413995171335\nMultiscale Entropy: 0.620621428966386\nNon-autonomous: true\nPeriod: 67.25\nPesin entropy: 0.06706808893310089\n\nCitation\n\nFroeschle, C., Guzzo, M. & Legga, E (2000). Graphical evolution of the Arnold web: From order to chaos. Science 289.\n\nOriginal Python Code from the dysts package\n\nclass ArnoldWeb(DynSys):\n @staticjit\n def _rhs(p1, p2, x1, x2, z, t, mu, w):\n denom = 4 + np.cos(z) + np.cos(x1) + np.cos(x2)\n p1dot = -mu * np.sin(x1) / denom ** 2\n p2dot = -mu * np.sin(x2) / denom ** 2\n x1dot = p1\n x2dot = p2\n zdot = w\n return p1dot, p2dot, x1dot, x2dot, zdot\n\n @staticjit\n def _postprocessing(p1, p2, x1, x2, z):\n return p1, p2, np.sin(x1), np.sin(x2), np.cos(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.AtmosphericRegime","page":"Home","title":"Dysts.AtmosphericRegime","text":" AtmosphericRegime()\n\nA model of regime transition in atmospheric flows. Contains widely-separated timescales due a quadratic term, which itself arises from a conservation law.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1859596374283137\nLyapunov spectrum (estimated): nothing\nMaximum Lyapunov exponent (estimated): 0.010543274932274862\nMultiscale Entropy: 0.4278776303819699\nNon-autonomous: false\nPeriod: 89.59496756697331\nPesin entropy: 0.3055574134566322\n\nCitation\n\nTuwankotta (2006). Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity\n\nOriginal Python Code from the dysts package\n\nclass AtmosphericRegime(DynSys):\n @staticjit\n def _rhs(\n x, y, z, t, alpha, beta, mu1, mu2, omega, sigma\n ):\n xdot = mu1 * x + sigma * x * y\n ydot = mu2 * y + omega * z + alpha * y * z + beta * z ** 2 - sigma * x ** 2\n zdot = mu2 * z - omega * y - alpha * y ** 2 - beta * y * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.BeerRNN","page":"Home","title":"Dysts.BeerRNN","text":" BeerRNN()\n\nA two-neuron minimal model nervous system.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0303797735862443\nLyapunov spectrum (estimated): Any[0.016274571192515633, -0.00011803510272572572, -0.5318869010933206]\nMaximum Lyapunov exponent (estimated): 0.038767800131815223\nMultiscale Entropy: 0.5244807630539362\nNon-autonomous: false\nPeriod: 181.58\nPesin entropy: 0.01647451962951782\n\nCitation\n\nBeer, R. D. (1995). On the dynamics of small continuous-time recurrent neural networks. Adapt. Behav., 3(4), 4692013509.\n\nOriginal Python Code from the dysts package\n\nclass BeerRNN(DynSys):\n @staticjit\n def _sig(x):\n return 1.0 / (1.0 + np.exp(-x))\n\n def rhs(self, X, t):\n Xdot = (-X + np.matmul(self.w, self._sig(X + self.theta))) / self.tau\n return Xdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.BelousovZhabotinsky","page":"Home","title":"Dysts.BelousovZhabotinsky","text":" BelousovZhabotinsky()\n\nA reduced-order model of the BZ reaction that exhibits period doubling. The bifurcation parameter for controlling the onset of chaos is kf. The system undergoes regular cycling when kf=3e-4, and chaotic oscillations when kf=3.5e-4\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0206505961796535\nLyapunov spectrum (estimated): Any[114.64513768787073, -0.4669861532822692, -5529.048679334775]\nMaximum Lyapunov exponent (estimated): 132.08766434104578\nMultiscale Entropy: 0.8764191281600929\nNon-autonomous: false\nPeriod: 0.018257\nPesin entropy: 114.64513768787073\n\nCitation\n\nGyorgyi and Field (1992). A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction. Nature.\n\nOriginal Python Code from the dysts package\n\nclass BelousovZhabotinsky(DynSys):\n @staticjit\n def _rhs(\n x,\n z,\n v,\n t,\n c1,\n c10,\n c11,\n c12,\n c13,\n c2,\n c3,\n c4,\n c5,\n c6,\n c7,\n c8,\n c9,\n ci,\n kf,\n t0,\n y0,\n yb1,\n yb2,\n yb3,\n z0,\n ):\n ybar = (1 / y0) * yb1 * z * v / (yb2 * x + yb3 + kf)\n if x < 0.0:\n x = 0\n rf = (ci - z0 * z) * np.sqrt(x)\n xdot = c1 * x * ybar + c2 * ybar + c3 * x ** 2 + c4 * rf + c5 * x * z - kf * x\n zdot = (c6 / z0) * rf + c7 * x * z + c8 * z * v + c9 * z - kf * z\n vdot = c10 * x * ybar + c11 * ybar + c12 * x ** 2 + c13 * z * v - kf * v\n return xdot * t0, zdot * t0, vdot * t0\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.BickleyJet","page":"Home","title":"Dysts.BickleyJet","text":" BickleyJet()\n\nA zonal jet passing between two counter rotating vortices. A steady Hamiltonian with a time-dependent perturbation.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: true\nKaplan-Yorke dimension: 4.066710025133773\nLyapunov spectrum (estimated): Any[1.9651794377408363e-7, 2.5409880958463764e-9, -9.631983584415862e-8]\nMaximum Lyapunov exponent (estimated): 3.0318672544949706e-6\nMultiscale Entropy: 0.733566708397195\nNon-autonomous: false\nPeriod: 87826.57\nPesin entropy: 1.99059498764899e-7\n\nCitation\n\nHadjighasem, Karrasch, Teramoto, Haller (2016). A Spectral Clustering Approach to Lagrangian Vortex Detection Phys Rev E.\n\nOriginal Python Code from the dysts package\n\nclass BickleyJet(DynSys):\n @staticjit\n def _rhs(y, x, z, t, ell, eps, k, omega, sigma, u):\n sechy = 1 / np.cosh(y / ell)\n inds = np.arange(3)\n un = k[inds] * (x - z * sigma[inds])\n dx = u * sechy ** 2 * (-1 - 2 * np.dot(np.cos(un), eps) * np.tanh(y / ell))\n dy = ell * u * sechy ** 2 * np.dot(eps * k, np.sin(un))\n dz = omega\n return dy, dx, dz\n\n def _postprocessing(self, x, y, z):\n km = np.min(self.k)\n sm = np.min(self.sigma)\n return x, np.sin(km * y), np.sin(self.omega * z * km * sm)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Blasius","page":"Home","title":"Dysts.Blasius","text":" Blasius()\n\nA chaotic food web composed of interacting predators,, consumers, and vegetation.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.016657959617314\nLyapunov spectrum (estimated): Any[0.04712755400915782, 0.001803137034352918, -2.9362739640325555]\nMaximum Lyapunov exponent (estimated): 0.12597339364128174\nMultiscale Entropy: 0.5917316835585642\nNon-autonomous: false\nPeriod: 6.6038\nPesin entropy: 0.04893069104351076\n\nCitation\n\nBlasius, Huppert, Stone. Nature 1999\n\nOriginal Python Code from the dysts package\n\nclass Blasius(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, alpha1, alpha2, b, c, k1, k2, zs):\n xdot = a * x - alpha1 * x * y / (1 + k1 * x)\n ydot = -b * y + alpha1 * x * y / (1 + k1 * x) - alpha2 * y * z / (1 + k2 * y)\n zdot = -c * (z - zs) + alpha2 * y * z / (1 + k2 * y)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.BlinkingRotlet","page":"Home","title":"Dysts.BlinkingRotlet","text":" BlinkingRotlet()\n\nThe location of the mixer is chosen so that there is a stagnation point at its symmetric complement. Solution is given in polar coordinates (r, theta).\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: true\nKaplan-Yorke dimension: 2.910902586026695\nLyapunov spectrum (estimated): Any[5.609358269449125, 0.007963961496808246, -6.16801583110239]\nMaximum Lyapunov exponent (estimated): 2.120338159738003\nMultiscale Entropy: 1.2895030395332947\nNon-autonomous: true\nPeriod: 2.9520625\nPesin entropy: 5.617567417119514\n\nCitation\n\nMeleshko & Aref. A blinking rotlet model for chaotic advection. Physics of Fluids 1996. See also Tallapragada and Sudarsanam PRE 2019\n\nOriginal Python Code from the dysts package\n\nclass BlinkingRotlet(DynSys):\n @staticjit\n def _rotlet(r, theta, a, b, bc):\n \"\"\"A rotlet velocity field\"\"\"\n kappa = a ** 2 + (b ** 2 * r ** 2) / a ** 2 - 2 * b * r * np.cos(theta)\n gamma = (1 - r ** 2 / a ** 2) * (a ** 2 - (b ** 2 * r ** 2) / a ** 2)\n iota = (b ** 2 * r) / a ** 2 - b * np.cos(theta)\n zeta = b ** 2 + r ** 2 - 2 * b * r * np.cos(theta)\n nu = a ** 2 + b ** 2 - (2 * b ** 2 * r ** 2) / a ** 2\n vr = b * np.sin(theta) * (-bc * (gamma / kappa ** 2) - 1 / kappa + 1 / zeta)\n vth = (\n bc * (gamma * iota) / kappa ** 2\n + bc * r * nu / (a ** 2 * kappa)\n + iota / kappa\n - (r - b * np.cos(theta)) / zeta\n )\n return vr, vth\n\n @staticjit\n def _protocol(t, tau, stiffness=20):\n return 0.5 + 0.5 * np.tanh(tau * stiffness * np.sin(2 * np.pi * t / tau))\n\n def rhs(self, X, t):\n r, theta, tt = X\n weight = self._protocol(tt, self.tau)\n dr1, dth1 = self._rotlet(r, theta, self.a, self.b, self.bc)\n dr2, dth2 = self._rotlet(r, theta, self.a, -self.b, self.bc)\n dr = weight * dr1 + (1 - weight) * dr2\n dth = (weight * dth1 + (1 - weight) * dth2) / r\n dtt = 1\n return self.sigma * dr, self.sigma * dth, dtt\n\n def _postprocessing(self, r, th, tt):\n return r * np.cos(th), r * np.sin(th), np.sin(2 * np.pi * tt / self.tau)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Bouali2","page":"Home","title":"Dysts.Bouali2","text":" Bouali2()\n\nA modified economic cycle model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 16.471858210544983\nLyapunov spectrum (estimated): Any[0.020389225004628773, 0.0033455417942589724, -0.009602768742817189]\nMaximum Lyapunov exponent (estimated): 0.07681730213975516\nMultiscale Entropy: 0.4520009826326611\nNon-autonomous: false\nPeriod: 2.6779\nPesin entropy: 0.02439941982137089\n\nCitation\n\nBouali (1999). Feedback loop in extended Van der Pols equation applied to an economic model of cycles. International Journal of Bifurkation and Chaos, Vol. 9, No. 4\n\nOriginal Python Code from the dysts package\n\nclass Bouali2(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, bb, c, g, m, y0):\n xdot = a * y0 * x - a * x * y - b * z\n ydot = -g * y + g * y * x ** 2\n zdot = -1.5 * m * x + m * bb * x * z - c * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.BurkeShaw","page":"Home","title":"Dysts.BurkeShaw","text":" BurkeShaw()\n\nA scroll-like attractor with unique symmetry along the z axis.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2113447891991065\nLyapunov spectrum (estimated): Any[2.3453555864112117, 0.027056468824027846, -11.225316054517004]\nMaximum Lyapunov exponent (estimated): 2.324596351534631\nMultiscale Entropy: 0.906432166797223\nNon-autonomous: false\nPeriod: 0.78333\nPesin entropy: 2.37241205523524\n\nCitation\n\nShaw (1981). Zeitschrift fur Naturforschung.\n\nOriginal Python Code from the dysts package\n\nclass BurkeShaw(DynSys):\n @staticjit\n def _rhs(x, y, z, t, e, n):\n xdot = -n * x - n * y\n ydot = y - n * x * z\n zdot = n * x * y + e\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.CaTwoPlus","page":"Home","title":"Dysts.CaTwoPlus","text":" CaTwoPlus()\n\nIntracellular calcium ion oscillations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0647813731079956\nLyapunov spectrum (estimated): Any[0.5788123099350391, 0.022652249040266133, -9.284523260554844]\nMaximum Lyapunov exponent (estimated): 0.5802865327706731\nMultiscale Entropy: 0.5770296569860646\nNon-autonomous: false\nPeriod: 1.343\nPesin entropy: 0.6014645592138562\n\nCitation\n\nHouart, Dupont, Goldbeter. Bull Math Biol 1999.\n\nOriginal Python Code from the dysts package\n\nclass CaTwoPlus(DynSys):\n def rhs(self, X, t):\n z, y, a = X\n Vin = self.V0 + self.V1 * self.beta\n V2 = self.Vm2 * (z ** 2) / (self.K2 ** 2 + z ** 2)\n V3 = (\n (self.Vm3 * (z ** self.m) / (self.Kz ** self.m + z ** self.m))\n * (y ** 2 / (self.Ky ** 2 + y ** 2))\n * (a ** 4 / (self.Ka ** 4 + a ** 4))\n )\n V5 = (\n self.Vm5\n * (a ** self.p / (self.K5 ** self.p + a ** self.p))\n * (z ** self.n / (self.Kd ** self.n + z ** self.n))\n )\n zdot = Vin - V2 + V3 + self.kf * y - self.k * z\n ydot = V2 - V3 - self.kf * y\n adot = self.beta * self.V4 - V5 - self.eps * a\n return (zdot, ydot, adot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.CellCycle","page":"Home","title":"Dysts.CellCycle","text":" CellCycle()\n\nA simplified model of the cell cycle. The parameter Kim controls the bifurcations.\n\nStats\n\nEmbedding dimension: 6\nHamiltonian: false\nKaplan-Yorke dimension: 2.020810681242375\nLyapunov spectrum (estimated): Any[0.00469985418122158, -0.0003567612310661562, -0.20915644240415007, -0.43394905365820846, -2.850536314608979, -6.767614340964144]\nMaximum Lyapunov exponent (estimated): 0.010172128409635746\nMultiscale Entropy: 1.021736994304444\nNon-autonomous: false\nPeriod: 204.71\nPesin entropy: 0.004765100518900255\n\nCitation\n\nRomond, Rustici, Gonze, Goldbeter. 1999.\n\nOriginal Python Code from the dysts package\n\nclass CellCycle(DynSys):\n def rhs(self, X, t):\n c1, m1, x1, c2, m2, x2 = X\n Vm1, Um1 = 2 * [self.Vm1]\n vi1, vi2 = 2 * [self.vi]\n H1, H2, H3, H4 = 4 * [self.K]\n K1, K2, K3, K4 = 4 * [self.K]\n V2, U2 = 2 * [self.V2]\n Vm3, Um3 = 2 * [self.Vm3]\n V4, U4 = 2 * [self.V4]\n Kc1, Kc2 = 2 * [self.Kc]\n vd1, vd2 = 2 * [self.vd]\n Kd1, Kd2 = 2 * [self.Kd1]\n kd1, kd2 = 2 * [self.kd1]\n Kim1, Kim2 = 2 * [self.Kim]\n V1 = Vm1 * c1 / (Kc1 + c1)\n U1 = Um1 * c2 / (Kc2 + c2)\n V3 = m1 * Vm3\n U3 = m2 * Um3\n c1dot = vi1 * Kim1 / (Kim1 + m2) - vd1 * x1 * c1 / (Kd1 + c1) - kd1 * c1\n c2dot = vi2 * Kim2 / (Kim2 + m1) - vd2 * x2 * c2 / (Kd2 + c2) - kd2 * c2\n m1dot = V1 * (1 - m1) / (K1 + (1 - m1)) - V2 * m1 / (K2 + m1)\n m2dot = U1 * (1 - m2) / (H1 + (1 - m2)) - U2 * m2 / (H2 + m2)\n x1dot = V3 * (1 - x1) / (K3 + (1 - x1)) - V4 * x1 / (K4 + x1)\n x2dot = U3 * (1 - x2) / (H3 + (1 - x2)) - U4 * x2 / (H4 + x2)\n return c1dot, m1dot, x1dot, c2dot, m2dot, x2dot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.CellularNeuralNetwork","page":"Home","title":"Dysts.CellularNeuralNetwork","text":" CellularNeuralNetwork()\n\nCellular neural network dynamics.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.167668286708633\nLyapunov spectrum (estimated): Any[0.09799656528996928, 0.0036354952626342415, -0.6061495728157424]\nMaximum Lyapunov exponent (estimated): 2.3258468977096562\nMultiscale Entropy: 1.2414841874061011\nNon-autonomous: false\nPeriod: 3.2\nPesin entropy: 0.1016320623247746\n\nCitation\n\nArena, Caponetto, Fortuna, and Porto., Int J Bifurcat Chaos. 1998: 1527-1539.\n\nOriginal Python Code from the dysts package\n\nclass CellularNeuralNetwork(DynSys):\n @staticjit\n def f(x):\n return 0.5 * (np.abs(x + 1) - np.abs(x - 1))\n\n def rhs(self, X, t):\n x, y, z = X\n xdot = -x + self.d * self.f(x) - self.b * self.f(y) - self.b * self.f(z)\n ydot = -y - self.b * self.f(x) + self.c * self.f(y) - self.a * self.f(z)\n zdot = -z - self.b * self.f(x) + self.a * self.f(y) + self.f(z)\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Chen","page":"Home","title":"Dysts.Chen","text":" Chen()\n\nA system based on feedback anti-control in engineering.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.168621926035506\nLyapunov spectrum (estimated): Any[2.0471208808332557, -0.05031354288886987, -11.841816202859391]\nMaximum Lyapunov exponent (estimated): 2.005961349488129\nMultiscale Entropy: 0.7604529380690989\nNon-autonomous: false\nPeriod: 0.58689\nPesin entropy: 2.047313967129477\n\nCitation\n\nChen (1997). Proc. First Int. Conf. Control of Oscillations and Chaos.\n\nOriginal Python Code from the dysts package\n\nclass Chen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * y - a * x\n ydot = (c - a) * x - x * z + c * y\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ChenLee","page":"Home","title":"Dysts.ChenLee","text":" ChenLee()\n\nA rigid body with feedback anti-control.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0300416878282137\nLyapunov spectrum (estimated): Any[0.16460631102343384, 0.00018225769802326938, -5.485329903201408]\nMaximum Lyapunov exponent (estimated): 0.16363115430160305\nMultiscale Entropy: 0.6224325755254609\nNon-autonomous: false\nPeriod: 2.8043\nPesin entropy: 0.16478856872146466\n\nCitation\n\nChen HK, Lee CI (2004). Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals\n\nOriginal Python Code from the dysts package\n\nclass ChenLee(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * x - y * z\n ydot = b * y + x * z\n zdot = c * z + 0.3333333333333333333333333 * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Chua","page":"Home","title":"Dysts.Chua","text":" Chua()\n\nAn electronic circuit with a diode providing negative resistance.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.10727056025829\nLyapunov spectrum (estimated): Any[0.4605474829443073, 0.0022048281168492866, -4.319076735248446]\nMaximum Lyapunov exponent (estimated): 1.2301163591725595\nMultiscale Entropy: 1.2693527097629784\nNon-autonomous: false\nPeriod: 1.4235\nPesin entropy: 0.4645707969482971\n\nCitation\n\nChua, L. O. (1969) Introduction to Nonlinear Network Theory, McGraw-Hill, New York.\n\nOriginal Python Code from the dysts package\n\nclass Chua(DynSys):\n @staticjit\n def _rhs(x, y, z, t, alpha, beta, m0, m1):\n ramp_x = m1 * x + 0.5 * (m0 - m1) * (np.abs(x + 1) - np.abs(x - 1))\n xdot = alpha * (y - x - ramp_x)\n ydot = x - y + z\n zdot = -beta * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.CircadianRhythm","page":"Home","title":"Dysts.CircadianRhythm","text":" CircadianRhythm()\n\nThe Drosophila circadian rhythm under periodic light/dark forcing.\n\nStats\n\nEmbedding dimension: 5\nHamiltonian: false\nKaplan-Yorke dimension: 2.640554412377753\nLyapunov spectrum (estimated): Any[0.09298369588063138, 0.00046501992305171574, -0.14586025062784352, -0.3598453092044599, -1.091393481159416]\nMaximum Lyapunov exponent (estimated): 0.006263277393890506\nMultiscale Entropy: 0.7054585345031309\nNon-autonomous: true\nPeriod: 27.0\nPesin entropy: 0.09372724379002938\n\nCitation\n\nLeloup, Gonze, Goldbeter. 1999. Gonze, Leloup, Goldbeter. 2000\n\nOriginal Python Code from the dysts package\n\nclass CircadianRhythm(DynSys):\n @staticjit\n def _rhs(\n m,\n fc,\n fs,\n fn,\n th,\n t,\n Ki,\n k,\n k1,\n k2,\n kd,\n kdn,\n km,\n ks,\n n,\n vd,\n vdn,\n vm,\n vmax,\n vmin,\n v,\n ):\n vs = 2.5 * ((0.5 + 0.5 * np.cos(th)) + vmin) * (vmax - vmin)\n mdot = vs * (Ki ** n) / (Ki ** n + fn ** n) - vm * m / (km + m)\n fcdot = ks * m - k1 * fc + k2 * fn - k * fc\n fsdot = k * fc - vd * fs / (kd + fs)\n fndot = k1 * fc - k2 * fn - vdn * fn / (kdn + fn)\n thdot = 2 * np.pi / 24\n return mdot, fcdot, fsdot, fndot, thdot\n\n @staticjit\n def _postprocessing(m, fc, fs, fn, th):\n return m, fc, fs, fn, np.cos(th)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.CoevolvingPredatorPrey","page":"Home","title":"Dysts.CoevolvingPredatorPrey","text":" CoevolvingPredatorPrey()\n\nA system of predator-prey equations with co-evolving prey.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0026539580881706\nLyapunov spectrum (estimated): Any[0.0034879428860592264, 0.0017391810332012293, -1.9676998121970224]\nMaximum Lyapunov exponent (estimated): 0.00732698039035783\nMultiscale Entropy: 1.006450625395487\nNon-autonomous: false\nPeriod: 122.77\nPesin entropy: 0.005379136261058156\n\nCitation\n\nGilpin & Feldman (2017). PLOS Comp Biol\n\nOriginal Python Code from the dysts package\n\nclass CoevolvingPredatorPrey(DynSys):\n @staticjit\n def _rhs(x, y, alpha, t, a1, a2, a3, b1, b2, d1, d2, delta, k1, k2, k4, vv):\n xdot = x * (\n -((a3 * y) / (1 + b2 * x))\n + (a1 * alpha * (1 - k1 * x * (-alpha + alpha * delta))) / (1 + b1 * alpha)\n - d1\n * (\n 1\n - k2 * (-(alpha ** 2) + (alpha * delta) ** 2)\n + k4 * (-(alpha ** 4) + (alpha * delta) ** 4)\n )\n )\n ydot = (-d2 + (a2 * x) / (1 + b2 * x)) * y\n alphadot = vv * (\n -((a1 * k1 * x * alpha * delta) / (1 + b1 * alpha))\n - d1 * (-2 * k2 * alpha * delta ** 2 + 4 * k4 * alpha ** 3 * delta ** 4)\n )\n return xdot, ydot, alphadot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Colpitts","page":"Home","title":"Dysts.Colpitts","text":" Colpitts()\n\nAn electrical circuit used as a signal generator.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.134959031654127\nLyapunov spectrum (estimated): Any[0.13750285290669104, -0.0021021241106901328, -1.0014048629576902]\nMaximum Lyapunov exponent (estimated): 0.3736008151161584\nMultiscale Entropy: 0.7266290517682168\nNon-autonomous: false\nPeriod: 4.2676\nPesin entropy: 0.1390751886208436\n\nCitation\n\nKennedy (2007). IEEE Trans Circuits & Systems. 1994: 771-774. Li, Zhou, Yang. Chaos, Solitons, & Fractals.\n\nOriginal Python Code from the dysts package\n\nclass Colpitts(DynSys):\n def rhs(self, X, t):\n x, y, z = X\n u = z - (self.e - 1)\n fz = -u * (1 - np.heaviside(u, 0))\n xdot = y - self.a * fz\n ydot = self.c - x - self.b * y - z\n zdot = y - self.d * z\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Coullet","page":"Home","title":"Dysts.Coullet","text":" Coullet()\n\nA variant of the Arneodo attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.252992250313846\nLyapunov spectrum (estimated): Any[0.14890693833273877, -6.504894715051528e-5, -0.5873658566804084]\nMaximum Lyapunov exponent (estimated): 0.13718762024831516\nMultiscale Entropy: 0.9055084400125268\nNon-autonomous: false\nPeriod: 5.7663\nPesin entropy: 0.14919849957322842\n\nCitation\n\nArneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional volterra equations. Physics Letters A 79, 259–263 (1980)\n\nOriginal Python Code from the dysts package\n\nclass Rucklidge(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x + b * y - y * z\n ydot = x\n zdot = -z + y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Dadras","page":"Home","title":"Dysts.Dadras","text":" Dadras()\n\nAn electronic circuit capable of producing multiple attractors under bifurcations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0430103150081735\nLyapunov spectrum (estimated): Any[0.47344233096408, 0.0064562720284210215, -11.157725546142514]\nMaximum Lyapunov exponent (estimated): 0.40591521049520063\nMultiscale Entropy: 0.8537980405426894\nNon-autonomous: false\nPeriod: 3.2886\nPesin entropy: 0.4798988304624064\n\nCitation\n\nS Dadras, HR Momeni (2009). A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Physics Letters A.\n\nOriginal Python Code from the dysts package\n\nclass Dadras(DynSys):\n @staticjit\n def _rhs(x, y, z, t, c, e, o, p, r):\n xdot = y - p * x + o * y * z\n ydot = r * y - x * z + z\n zdot = c * x * y - e * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.DequanLi","page":"Home","title":"Dysts.DequanLi","text":" DequanLi()\n\nRelated to the Three Scroll unified attractor TSUCS-2\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.6848604732863537\nLyapunov spectrum (estimated): Any[1.8601963396884453, 0.09866428540326228, -2.867185439620747]\nMaximum Lyapunov exponent (estimated): 0.739747976641268\nMultiscale Entropy: 0.8319369667525882\nNon-autonomous: false\nPeriod: 0.083806\nPesin entropy: 1.9588606250917073\n\nCitation\n\nLi, Phys Lett A. 2008: 387-393.\n\nOriginal Python Code from the dysts package\n\nclass DequanLi(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, c, d, eps, f, k):\n xdot = a * y - a * x + d * x * z\n ydot = k * x + f * y - x * z\n zdot = c * z + x * y - eps * x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.DoubleGyre","page":"Home","title":"Dysts.DoubleGyre","text":" DoubleGyre()\n\nA time-dependent fluid flow exhibiting Lagrangian coherent structures\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.881525629972418\nLyapunov spectrum (estimated): Any[0.06285936079346421, -0.0036702088073069326, -0.06633987533400756]\nMaximum Lyapunov exponent (estimated): 0.12185254082886439\nMultiscale Entropy: 0.66519273658672\nNon-autonomous: true\nPeriod: 13.953\nPesin entropy: 0.06285936079346421\n\nCitation\n\nShadden, Lekien, Marsden (2005). Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D.\n\nOriginal Python Code from the dysts package\n\nclass DoubleGyre(DynSys):\n @staticjit\n def _rhs(x, y, z, t, alpha, eps, omega):\n a = eps * np.sin(z)\n b = 1 - 2 * eps * np.sin(z)\n f = a * x ** 2 + b * x\n dx = -alpha * np.pi * np.sin(np.pi * f) * np.cos(np.pi * y)\n dy = alpha * np.pi * np.cos(np.pi * f) * np.sin(np.pi * y) * (2 * a * x + b)\n dz = omega\n return dx, dy, dz\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.DoublePendulum","page":"Home","title":"Dysts.DoublePendulum","text":" DoublePendulum()\n\nTwo coupled rigid pendula without damping.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: true\nKaplan-Yorke dimension: 3.810221473882809\nLyapunov spectrum (estimated): Any[8.464443924343541, 0.46411672696182493, -0.17345525567611142, -10.805817517613212]\nMaximum Lyapunov exponent (estimated): 3.985298067514186\nMultiscale Entropy: 0.5591104791210428\nNon-autonomous: false\nPeriod: 4.0234\nPesin entropy: 8.928560651305366\n\nCitation\n\nSee, for example: Marion (2013). Classical dynamics of particles and systems.\n\nOriginal Python Code from the dysts package\n\nclass DoublePendulum(DynSys):\n @staticjit\n def _rhs(th1, th2, p1, p2, t, d, m):\n g = 9.82\n pre = 6 / (m * d ** 2)\n denom = 16 - 9 * np.cos(th1 - th2) ** 2\n th1_dot = pre * (2 * p1 - 3 * np.cos(th1 - th2) * p2) / denom\n th2_dot = pre * (8 * p2 - 3 * np.cos(th1 - th2) * p1) / denom\n p1_dot = (\n -0.5\n * (m * d ** 2)\n * (th1_dot * th2_dot * np.sin(th1 - th2) + 3 * (g / d) * np.sin(th1))\n )\n p2_dot = (\n -0.5\n * (m * d ** 2)\n * (-th1_dot * th2_dot * np.sin(th1 - th2) + 3 * (g / d) * np.sin(th2))\n )\n return th1_dot, th2_dot, p1_dot, p2_dot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Duffing","page":"Home","title":"Dysts.Duffing","text":" Duffing()\n\nA monochromatically-forced rigid pendulum, with the nonlinearity approximated as cubic.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.561219319140712\nLyapunov spectrum (estimated): Any[0.09952054532941002, 0.0003077550037828977, -0.17595301607629793]\nMaximum Lyapunov exponent (estimated): 0.14026545752865122\nMultiscale Entropy: 0.8686568999981737\nNon-autonomous: true\nPeriod: 7.4496\nPesin entropy: 0.10008657388524086\n\nCitation\n\nDuffing, G. (1918), Forced oscillations with variable natural frequency and their technical relevance, Heft 41/42, Braunschweig: Vieweg.\n\nOriginal Python Code from the dysts package\n\nclass Duffing(DynSys):\n @staticjit\n def _rhs(x, y, z, t, alpha, beta, delta, gamma, omega):\n xdot = y\n ydot = -delta * y - beta * x - alpha * x ** 3 + gamma * np.cos(z)\n zdot = omega\n return xdot, ydot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.cos(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ExcitableCell","page":"Home","title":"Dysts.ExcitableCell","text":" ExcitableCell()\n\nA reduced-order variant of the Hodgkin-Huxley model. The parameter gkc controls the onset of chaos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1287842177704674\nLyapunov spectrum (estimated): Any[17.030315084501463, -0.3267053778962445, -129.70230355691595]\nMaximum Lyapunov exponent (estimated): 0.7164303367057223\nMultiscale Entropy: 0.7807524009298388\nNon-autonomous: false\nPeriod: 0.5528\nPesin entropy: 17.030315084501463\n\nCitation\n\nTeresa Chay. Chaos In A Three-variable Model Of An Excitable Cell. Physica D 1985\n\nOriginal Python Code from the dysts package\n\nclass ExcitableCell(DynSys):\n def rhs(self, X, t):\n v, n, c = X\n\n alpham = 0.1 * (25 + v) / (1 - np.exp(-0.1 * v - 2.5))\n betam = 4 * np.exp(-(v + 50) / 18)\n minf = alpham / (alpham + betam)\n\n alphah = 0.07 * np.exp(-0.05 * v - 2.5)\n betah = 1 / (1 + np.exp(-0.1 * v - 2))\n hinf = alphah / (alphah + betah)\n\n alphan = 0.01 * (20 + v) / (1 - np.exp(-0.1 * v - 2))\n betan = 0.125 * np.exp(-(v + 30) / 80)\n ninf = alphan / (alphan + betan)\n tau = 1 / (230 * (alphan + betan))\n\n ca = c / (1 + c)\n\n vdot = (\n self.gi * minf ** 3 * hinf * (self.vi - v)\n + self.gkv * n ** 4 * (self.vk - v)\n + self.gkc * ca * (self.vk - v)\n + self.gl * (self.vl - v)\n )\n ndot = (ninf - n) / tau\n cdot = self.rho * (minf ** 3 * hinf * (self.vc - v) - self.kc * c)\n return vdot, ndot, cdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Finance","page":"Home","title":"Dysts.Finance","text":" Finance()\n\nStock fluctuations under varying investment demand and interest rates\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.172040321306389\nLyapunov spectrum (estimated): Any[0.10708598383274774, 0.0009835797228343356, -0.6279525877233353]\nMaximum Lyapunov exponent (estimated): 0.09354106442670211\nMultiscale Entropy: 0.7817861077722251\nNon-autonomous: false\nPeriod: 9.8932\nPesin entropy: 0.1084488028544319\n\nCitation\n\nGuoliang Cai, Juanjuan Huang. International Journal of Nonlinear Science, Vol. 3 (2007)\n\nOriginal Python Code from the dysts package\n\nclass Finance(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = (1 / b - a) * x + z + x * y\n ydot = -b * y - x ** 2\n zdot = -x - c * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.FluidTrampoline","page":"Home","title":"Dysts.FluidTrampoline","text":" FluidTrampoline()\n\nA droplet bouncing on a horizontal soap film.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.8462636972682116\nLyapunov spectrum (estimated): Any[0.08081301489037329, -0.0011783910877478326, -0.09403052188090477]\nMaximum Lyapunov exponent (estimated): 0.22311237411461174\nMultiscale Entropy: 0.796312585554829\nNon-autonomous: true\nPeriod: 7.7351\nPesin entropy: 0.08094585172323163\n\nCitation\n\nGilet, Bush. The fluid trampoline: droplets bouncing on a soap film. JFM 2009.\n\nOriginal Python Code from the dysts package\n\nclass FluidTrampoline(DynSys):\n @staticmethod\n def _rhs(x, y, th, t, gamma, psi, w):\n xdot = y\n ydot = -1 - np.heaviside(-x, 0) * (x + psi * y * np.abs(y)) + gamma * np.cos(th)\n thdot = w\n return (xdot, ydot, thdot)\n\n @staticjit\n def _postprocessing(x, y, th):\n return x, y, np.cos(th)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ForcedBrusselator","page":"Home","title":"Dysts.ForcedBrusselator","text":" ForcedBrusselator()\n\nAn autocatalytic chemical system.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.190019824057157\nLyapunov spectrum (estimated): Any[0.0028307087627546254, -0.014898681909916936, -0.23304448314240073]\nMaximum Lyapunov exponent (estimated): 0.015326007974604663\nMultiscale Entropy: 0.47299463051887286\nNon-autonomous: true\nPeriod: 20.411\nPesin entropy: 0.0028307087627546254\n\nCitation\n\nI. Prigogine, From Being to Becoming: Time and Complexity in the Physical Sciences, New York: W.H. Freeman and Company, 1980.\n\nOriginal Python Code from the dysts package\n\nclass ForcedBrusselator(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, f, w):\n xdot = a + x ** 2 * y - (b + 1) * x + f * np.cos(z)\n ydot = b * x - x ** 2 * y\n zdot = w\n return xdot, ydot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ForcedFitzHughNagumo","page":"Home","title":"Dysts.ForcedFitzHughNagumo","text":" ForcedFitzHughNagumo()\n\nA driven neuron model sustaining both quiesent and oscillating states.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.5705852319395741\nLyapunov spectrum (estimated): Any[0.0006271045837994497, -0.0017541532237019502, -0.9544351506941636]\nMaximum Lyapunov exponent (estimated): 0.007561278655908403\nMultiscale Entropy: 0.946734422406113\nNon-autonomous: true\nPeriod: 42.12235294117647\nPesin entropy: 0.000828176101772673\n\nCitation\n\nFitzHugh, Richard (1961). Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal. 1 (6).\n\nOriginal Python Code from the dysts package\n\nclass ForcedFitzHughNagumo(DynSys):\n @staticjit\n def _rhs(v, w, z, t, a, b, curr, f, gamma, omega):\n vdot = v - v ** 3 / 3 - w + curr + f * np.sin(z)\n wdot = gamma * (v + a - b * w)\n zdot = omega\n return vdot, wdot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ForcedVanDerPol","page":"Home","title":"Dysts.ForcedVanDerPol","text":" ForcedVanDerPol()\n\nAn electronic circuit containing a triode.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.012828649055009\nLyapunov spectrum (estimated): Any[0.2651824388815116, -0.0008450617218488057, -20.60632249066986]\nMaximum Lyapunov exponent (estimated): 0.0035097622587646343\nMultiscale Entropy: 0.39443058905146283\nNon-autonomous: true\nPeriod: 14.476875\nPesin entropy: 0.26518243888151155\n\nCitation\n\nB. van der Pol (1920). A theory of the amplitude of free and forced triode vibrations. Radio Review 1.\n\nOriginal Python Code from the dysts package\n\nclass ForcedVanDerPol(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, mu, w):\n ydot = mu * (1 - x ** 2) * y - x + a * np.sin(z)\n xdot = y\n zdot = w\n return xdot, ydot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.GenesioTesi","page":"Home","title":"Dysts.GenesioTesi","text":" GenesioTesi()\n\nA nonlinear control system with feedback.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1892507433089325\nLyapunov spectrum (estimated): Any[0.09974381765927798, 0.00038146676582121423, -0.5282231255530121]\nMaximum Lyapunov exponent (estimated): 0.10509061137519304\nMultiscale Entropy: 0.7003576241153363\nNon-autonomous: false\nPeriod: 5.9431\nPesin entropy: 0.10027488648885732\n\nCitation\n\nGenesio, Tesi (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28.3 (1992): 531-548.\n\nOriginal Python Code from the dysts package\n\nclass GenesioTesi(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = y\n ydot = z\n zdot = -c * x - b * y - a * z + x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.GuckenheimerHolmes","page":"Home","title":"Dysts.GuckenheimerHolmes","text":" GuckenheimerHolmes()\n\nA nonlinear oscillator.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.3663164666051353\nLyapunov spectrum (estimated): Any[0.34082003567895874, -0.0032191807339425173, -0.9216135618098852]\nMaximum Lyapunov exponent (estimated): 0.7885729042404602\nMultiscale Entropy: 0.5385101182847213\nNon-autonomous: false\nPeriod: 0.3076923076923077\nPesin entropy: 0.3408200356941137\n\nCitation\n\nGuckenheimer, John, and Philip Holmes (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42. Springer Science & Business Media.\n\nOriginal Python Code from the dysts package\n\nclass GuckenheimerHolmes(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, e, f):\n xdot = a * x - b * y + c * z * x + d * z * x ** 2 + d * z * y ** 2\n ydot = a * y + b * x + c * z * y\n zdot = e - z ** 2 - f * x ** 2 - f * y ** 2 - a * z ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Hadley","page":"Home","title":"Dysts.Hadley","text":" Hadley()\n\nAn atmospheric convective cell.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.7273838319442127\nLyapunov spectrum (estimated): Any[0.31166133927315953, -0.0010838827053467116, -0.42739317979015523]\nMaximum Lyapunov exponent (estimated): 0.2386587703845738\nMultiscale Entropy: 0.9351389987191853\nNon-autonomous: false\nPeriod: 1.4541\nPesin entropy: 0.3144504334934165\n\nCitation\n\nG. Hadley (1735). On the cause of the general trade winds. Philosophical Transactions of the Royal Society, vol. 34, pp. 58–62.\n\nOriginal Python Code from the dysts package\n\nclass Hadley(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, f, g):\n xdot = -y ** 2 - z ** 2 - a * x + a * f\n ydot = x * y - b * x * z - y + g\n zdot = b * x * y + x * z - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Halvorsen","page":"Home","title":"Dysts.Halvorsen","text":" Halvorsen()\n\nAn algebraically-simple chaotic system with quadratic nonlinearity\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.118235268907861\nLyapunov spectrum (estimated): Any[0.586197150399321, -0.031953945256851884, -4.6876300977027565]\nMaximum Lyapunov exponent (estimated): 0.6963711539102683\nMultiscale Entropy: 0.7527831765757674\nNon-autonomous: false\nPeriod: 1.4889\nPesin entropy: 0.586197150399321\n\nCitation\n\nSprott, Julien C (2010). Elegant chaos: algebraically simple chaotic flows. World Scientific, 2010.\n\nOriginal Python Code from the dysts package\n\nclass Halvorsen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x - b * y - b * z - y ** 2\n ydot = -a * y - b * z - b * x - z ** 2\n zdot = -a * z - b * x - b * y - x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HastingsPowell","page":"Home","title":"Dysts.HastingsPowell","text":" HastingsPowell()\n\nA three species food web.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.027757892627057\nLyapunov spectrum (estimated): Any[0.008585012825774563, 0.005765040490702786, -0.5169232002224432]\nMaximum Lyapunov exponent (estimated): 0.012532796124563227\nMultiscale Entropy: 0.47829796264231095\nNon-autonomous: false\nPeriod: 135.19\nPesin entropy: 0.014350053316477346\n\nCitation\n\nHastings, Powell. Ecology 1991\n\nOriginal Python Code from the dysts package\n\nclass HastingsPowell(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a1, a2, b1, b2, d1, d2):\n xdot = x * (1 - x) - y * a1 * x / (1 + b1 * x)\n ydot = y * a1 * x / (1 + b1 * x) - z * a2 * y / (1 + b2 * y) - d1 * y\n zdot = z * a2 * y / (1 + b2 * y) - d2 * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HenonHeiles","page":"Home","title":"Dysts.HenonHeiles","text":" HenonHeiles()\n\nA star's motion around the galactic center.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 6.81141589838772\nLyapunov spectrum (estimated): Any[0.026747154244887203, 0.011500891097175434, -1.4057113933414076e-5, -0.011151041415888588]\nMaximum Lyapunov exponent (estimated): 0.03551120315239528\nMultiscale Entropy: 0.7137794258156454\nNon-autonomous: false\nPeriod: 6.37\nPesin entropy: 0.03870865064107525\n\nCitation\n\nHenon, M.; Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astronomical Journal. 69: 73.\n\nOriginal Python Code from the dysts package\n\nclass HenonHeiles(DynSys):\n @staticjit\n def _rhs(x, y, px, py, t, lam):\n xdot = px\n ydot = py\n pxdot = -x - 2 * lam * x * y\n pydot = -y - lam * x ** 2 + lam * y ** 2\n return xdot, ydot, pxdot, pydot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HindmarshRose","page":"Home","title":"Dysts.HindmarshRose","text":" HindmarshRose()\n\nA neuron model exhibiting spiking and bursting.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0405369895957373\nLyapunov spectrum (estimated): Any[0.2621464274609406, 0.0017705383465481184, -6.510663065519547]\nMaximum Lyapunov exponent (estimated): 0.6535393283242747\nMultiscale Entropy: 0.8692605688019491\nNon-autonomous: false\nPeriod: 1.752\nPesin entropy: 0.2639169709155772\n\nCitation\n\nMarhl, Perc. Chaos, Solitons, Fractals 2005.\n\nOriginal Python Code from the dysts package\n\nclass HindmarshRose(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, s, tx, tz):\n xdot = -x + 1 / tx * y - a / tx * x ** 3 + b / tx * x ** 2 + 1 / tx * z\n ydot = -a * x ** 3 - (d - b) * x ** 2 + z\n zdot = -s / tz * x - 1 / tz * z + c / tz\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Hopfield","page":"Home","title":"Dysts.Hopfield","text":" Hopfield()\n\nA neural network with frustrated connectivity\n\nStats\n\nEmbedding dimension: 6\nHamiltonian: false\nKaplan-Yorke dimension: 2.264479110541058\nLyapunov spectrum (estimated): Any[0.07726437124258394, -0.00018152933777384922, -0.29083324580448194, -0.36906925337474944, -0.40604718386075256, -1.4425404449511432]\nMaximum Lyapunov exponent (estimated): 0.5684266951234997\nMultiscale Entropy: 0.9160145554679957\nNon-autonomous: false\nPeriod: 10.865\nPesin entropy: 0.07748903749076723\n\nCitation\n\nLewis & Glass, Neur Comp (1992)\n\nOriginal Python Code from the dysts package\n\nclass Hopfield(DynSys):\n def f(self, x):\n return (1 + np.tanh(x)) / 2\n\n def rhs(self, X, t):\n Xdot = -X / self.tau + self.f(self.eps * np.matmul(self.k, X)) - self.beta\n return Xdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperBao","page":"Home","title":"Dysts.HyperBao","text":" HyperBao()\n\nHyperchaos in the Lu system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.0460429607933572\nLyapunov spectrum (estimated): Any[0.6959739496471207, 0.2076688444499756, 0.01646766869238271, -19.983737503321315]\nMaximum Lyapunov exponent (estimated): 0.7941925747845208\nMultiscale Entropy: 0.8910215561062699\nNon-autonomous: false\nPeriod: 0.5831505997603473\nPesin entropy: 0.9201105075982862\n\nCitation\n\nBao, Liu (2008). A hyperchaotic attractor coined from the chaotic Lu system. Chin. Physics Letters, Vol. 25, pp 2396-2399.\n\nOriginal Python Code from the dysts package\n\nclass HyperBao(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d, e):\n xdot = a * y - a * x + w\n ydot = c * y - x * z\n zdot = x * y - b * z\n wdot = e * x + d * y * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperCai","page":"Home","title":"Dysts.HyperCai","text":" HyperCai()\n\nA hyperchaotic variant of the Finance system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.138030987910704\nLyapunov spectrum (estimated): Any[1.634443623908957, 0.12845528254618424, -0.003328688360603957, -12.747646341795276]\nMaximum Lyapunov exponent (estimated): 1.6777584899044127\nMultiscale Entropy: 1.1155413734604414\nNon-autonomous: false\nPeriod: 0.6113936187028436\nPesin entropy: 1.7628989064551457\n\nCitation\n\nGuoliang, Huang (2007). A New Finance Chaotic Attractor. International Journal of Nonlinear Science.\n\nOriginal Python Code from the dysts package\n\nclass HyperCai(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d, e):\n xdot = a * y - a * x\n ydot = b * x + c * y - x * z + w\n zdot = -d * z + y ** 2\n wdot = -e * x\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperJha","page":"Home","title":"Dysts.HyperJha","text":" HyperJha()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.053181782412727\nLyapunov spectrum (estimated): Any[0.3949499860671474, 0.2920214751184925, 0.005568431034995312, -13.022125042186726]\nMaximum Lyapunov exponent (estimated): 0.38695361636760767\nMultiscale Entropy: 0.8934166449122942\nNon-autonomous: false\nPeriod: 0.8564467295817622\nPesin entropy: 0.6925398922206352\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperJha(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + w\n ydot = -x * z + b * x - y\n zdot = x * y - c * z\n wdot = -x * z + d * w\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperLorenz","page":"Home","title":"Dysts.HyperLorenz","text":" HyperLorenz()\n\nA hyperchaotic variant of the Lorenz attractor.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.044709906668086\nLyapunov spectrum (estimated): Any[0.2925643599949083, 0.24691553258701357, 0.04704434899468292, -13.118440302968889]\nMaximum Lyapunov exponent (estimated): 0.3288809596909799\nMultiscale Entropy: 0.8026458791332332\nNon-autonomous: false\nPeriod: 1.277546730110058\nPesin entropy: 0.5865242415766048\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperLorenz(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + w\n ydot = -x * z + c * x - y\n zdot = -b * z + x * y\n wdot = d * w - x * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperLu","page":"Home","title":"Dysts.HyperLu","text":" HyperLu()\n\nA hyperchaotic variant of the Lu attractor.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.0822889656900583\nLyapunov spectrum (estimated): Any[1.0855770127041702, 0.4753850931706396, 0.02849951886859947, -19.313823203887658]\nMaximum Lyapunov exponent (estimated): 1.042961263138283\nMultiscale Entropy: 1.0455788608813867\nNon-autonomous: false\nPeriod: 0.7162856981996777\nPesin entropy: 1.5894616248138722\n\nCitation\n\nChen, A., Lu, J., Lü, J., & Yu, S. (2006). Generating hyperchaotic Lü attractor via state feedback control. Physica A: Statistical Mechanics and its Applications, 364, 103-110.\n\nOriginal Python Code from the dysts package\n\nclass HyperLu(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=36, b=3, c=20, d=1.3):\n xdot = a * y - a * x + w\n ydot = -x * z + c * y\n zdot = x * y - b * z\n wdot = d * w + x * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperPang","page":"Home","title":"Dysts.HyperPang","text":" HyperPang()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.08902125132107\nLyapunov spectrum (estimated): Any[1.586932998381499, 0.28444192185263556, -0.0011541322141891326, -21.007696151519628]\nMaximum Lyapunov exponent (estimated): 1.4465315511406867\nMultiscale Entropy: 0.9172158388570456\nNon-autonomous: false\nPeriod: 0.8439374360017028\nPesin entropy: 1.871913394885623\n\nCitation\n\nPang, S., & Liu, Y. (2011). A new hyperchaotic system from the Lü system and its control. Journal of Computational and Applied Mathematics, 235(8), 2775-2789.\n\nOriginal Python Code from the dysts package\n\nclass HyperPang(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=36, b=3, c=20, d=2):\n xdot = a * y - a * x\n ydot = -x * z + c * y + w\n zdot = x * y - b * z\n wdot = -d * x - d * y\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperQi","page":"Home","title":"Dysts.HyperQi","text":" HyperQi()\n\nA hyperchaotic variant of the Qi system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.2816770286126777\nLyapunov spectrum (estimated): Any[13.908799876284396, 2.8519979827853357, -0.04957904444451209, -59.32758839771818]\nMaximum Lyapunov exponent (estimated): 12.787249321134263\nMultiscale Entropy: 0.9988440965886223\nNon-autonomous: false\nPeriod: 0.06186038560173437\nPesin entropy: 16.760797859069733\n\nCitation\n\nG. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.\n\nOriginal Python Code from the dysts package\n\nclass HyperQi(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d, e, f):\n xdot = a * y - a * x + y * z\n ydot = b * x + b * y - x * z\n zdot = -c * z - e * w + x * y\n wdot = -d * w + f * z + x * y\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperRossler","page":"Home","title":"Dysts.HyperRossler","text":" HyperRossler()\n\nA hyperchaotic variant of the Rossler system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.916384112225743\nLyapunov spectrum (estimated): Any[0.27859233648384435, -7.749198930652314e-5, -0.3014660985555502, -70.26609894565237]\nMaximum Lyapunov exponent (estimated): 0.1718849632568779\nMultiscale Entropy: 0.9144754695112332\nNon-autonomous: false\nPeriod: 1.1157476694793766\nPesin entropy: 0.2789695684743801\n\nCitation\n\nRossler, O. E. (1979). An equation for hyperchaos.\n\nOriginal Python Code from the dysts package\n\nclass HyperRossler(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = -y - z\n ydot = x + a * y + w\n zdot = b + x * z\n wdot = -c * z + d * w\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperWang","page":"Home","title":"Dysts.HyperWang","text":" HyperWang()\n\nA hyperchaotic variant of the Wang system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.105321722354005\nLyapunov spectrum (estimated): Any[1.2224015619377895, 0.25827623672371736, -0.0075512136517132625, -13.986919236227278]\nMaximum Lyapunov exponent (estimated): 1.2930140887300063\nMultiscale Entropy: 0.9969261837212393\nNon-autonomous: false\nPeriod: 0.6481520567576332\nPesin entropy: 1.480677798661507\n\nCitation\n\nWang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.\n\nOriginal Python Code from the dysts package\n\nclass HyperWang(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=10, b=40, c=2.5, d=10.6, e=4):\n xdot = a * y - a * x\n ydot = -x * z + b * x + w\n zdot = -c * z + e * x ** 2\n wdot = -d * x\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperXu","page":"Home","title":"Dysts.HyperXu","text":" HyperXu()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.0909471864561056\nLyapunov spectrum (estimated): Any[1.0420068843753962, 0.20717763316365526, -0.0033988115975040426, -13.697896294059921]\nMaximum Lyapunov exponent (estimated): 0.9819830018599975\nMultiscale Entropy: 0.9312323303231519\nNon-autonomous: false\nPeriod: 1.1328151255687646\nPesin entropy: 1.2491913312241498\n\nCitation\n\nLetellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.\n\nOriginal Python Code from the dysts package\n\nclass HyperXu(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=10, b=40, c=2.5, d=2, e=16):\n xdot = a * y - a * x + w\n ydot = b * x + e * x * z\n zdot = -c * z - x * y\n wdot = x * z - d * y\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperYan","page":"Home","title":"Dysts.HyperYan","text":" HyperYan()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.0711118082329567\nLyapunov spectrum (estimated): Any[0.9120352761173002, 0.15303345953001665, -14.977382134147613, -39.36365990052555]\nMaximum Lyapunov exponent (estimated): 1.3945968455812914\nMultiscale Entropy: 0.8166157518949907\nNon-autonomous: false\nPeriod: 0.9127028626966961\nPesin entropy: 1.0650687356473167\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperYan(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=37, b=3, c=26, d=38):\n xdot = a * y - a * x\n ydot = (c - a) * x - x * z + c * y\n zdot = -b * z + x * y - y * z + x * z - w\n wdot = -d * w + y * z - x * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.HyperYangChen","page":"Home","title":"Dysts.HyperYangChen","text":" HyperYangChen()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.026605264361547\nLyapunov spectrum (estimated): Any[0.5742688696462735, 0.327365255173604, 0.023510192276255986, -34.77284163099938]\nMaximum Lyapunov exponent (estimated): 0.42042732521941667\nMultiscale Entropy: 0.9730322217427024\nNon-autonomous: false\nPeriod: 0.9127028626966961\nPesin entropy: 0.9255171125652686\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperYangChen(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=30, b=3, c=35, d=8):\n xdot = a * y - a * x\n ydot = c * x - x * z + w\n zdot = -b * z + x * y\n wdot = -d * x\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.IkedaDelay","page":"Home","title":"Dysts.IkedaDelay","text":" IkedaDelay()\n\nA passive optical resonator system. A standard delay dimension of three is used.\n\nStats\n\nEmbedding dimension: 10\nHamiltonian: false\nKaplan-Yorke dimension: 8.284295212220359\nLyapunov spectrum (estimated): Any[0.7139744431963619, 0.03258370277261943, -0.06766062243103049, -0.07744565521894307, -0.09054735418410836, -0.10900317234818346, -0.13696057562417352, -0.18439894250590672, -0.2833034824174493, -0.6392438255039592]\nMaximum Lyapunov exponent (estimated): 0.448298365567425\nNon-autonomous: false\nPeriod: 1.486528\nPesin entropy: 0.7465581459689814\n\nCitation\n\nK. Ikeda and K. Matsumoto (1987). High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29 (1-2), 223-235\n\nOriginal Python Code from the dysts package\n\nclass IkedaDelay(DynSysDelay):\n @staticjit\n def _rhs(x, xt, t, c, mu, tau, x0):\n xdot = mu * np.sin(xt - x0) - c * x\n return xdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.IsothermalChemical","page":"Home","title":"Dysts.IsothermalChemical","text":" IsothermalChemical()\n\nAn isothermal chemical system with mixed-mode oscillations\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0385823810364654\nLyapunov spectrum (estimated): Any[0.3365871343421436, -0.05136527222093221, -7.392541736904205]\nMaximum Lyapunov exponent (estimated): 0.450090505359164\nMultiscale Entropy: 0.6589422003608746\nNon-autonomous: false\nPeriod: 0.7302799999999999\nPesin entropy: 0.3365871343421436\n\nCitation\n\nPetrov, Scott, Showalter. Mixed-mode oscillations in chemical systems. J Chem Phys 1992\n\nOriginal Python Code from the dysts package\n\nclass IsothermalChemical(DynSys):\n @staticmethod\n def _rhs(alpha, beta, gamma, t, delta, kappa, mu, sigma):\n alphadot = mu * (kappa + gamma) - alpha * beta ** 2 - alpha\n betadot = (alpha * beta ** 2 + alpha - beta) / sigma\n gammadot = (beta - gamma) / delta\n return alphadot, betadot, gammadot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ItikBanksTumor","page":"Home","title":"Dysts.ItikBanksTumor","text":" ItikBanksTumor()\n\nA model of cancer cell populations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0152772613355485\nLyapunov spectrum (estimated): Any[0.022156938955395417, -0.0135394920593605, -0.5610833889492404]\nMaximum Lyapunov exponent (estimated): 0.03621509684991421\nMultiscale Entropy: 0.5883298730506914\nNon-autonomous: false\nPeriod: 49.74\nPesin entropy: 0.022156938955395417\n\nCitation\n\nItik, Banks. Int J Bifurcat Chaos 2010\n\nOriginal Python Code from the dysts package\n\nclass ItikBanksTumor(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a12, a13, a21, a31, d3, k3, r2, r3):\n xdot = x * (1 - x) - a12 * x * y - a13 * x * z\n ydot = r2 * y * (1 - y) - a21 * x * y\n zdot = r3 * x * z / (x + k3) - a31 * x * z - d3 * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.JerkCircuit","page":"Home","title":"Dysts.JerkCircuit","text":" JerkCircuit()\n\nAn electronic circuit with nonlinearity provided by a diode.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.5082428902531144\nLyapunov spectrum (estimated): Any[0.394872929553187, 0.1849026306096344, -1.1407351759248092]\nMaximum Lyapunov exponent (estimated): 0.2922722771543928\nMultiscale Entropy: 0.38139916676685187\nNon-autonomous: false\nPeriod: 19.4546\nPesin entropy: 0.5797755601628214\n\nCitation\n\nSprott (2011). A new chaotic jerk circuit. IEEE Circ. Sys. 2011.\n\nOriginal Python Code from the dysts package\n\nclass JerkCircuit(DynSys):\n @staticjit\n def _rhs(x, y, z, t, eps, y0):\n xdot = y\n ydot = z\n zdot = -z - x - eps * (np.exp(y / y0) - 1)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.KawczynskiStrizhak","page":"Home","title":"Dysts.KawczynskiStrizhak","text":" KawczynskiStrizhak()\n\nA chemical oscillator model describing mixed-modes in the BZ equations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.4235221965773541\nLyapunov spectrum (estimated): Any[0.025607519929323416, -0.06145426210744069, -0.31683393262481707]\nMaximum Lyapunov exponent (estimated): 0.0757060839542753\nMultiscale Entropy: 0.5935098791328474\nNon-autonomous: false\nPeriod: 7.991666666666667\nPesin entropy: 0.02560751992932341\n\nCitation\n\nP. E. Strizhak and A. L. Kawczynski, J. Phys. Chem. 99, 10830 (1995).\n\nOriginal Python Code from the dysts package\n\nclass KawczynskiStrizhak(DynSys):\n @staticjit\n def _rhs(x, y, z, t, beta, gamma, kappa, mu):\n xdot = gamma * y - gamma * x ** 3 + 3 * mu * gamma * x\n ydot = -2 * mu * x - y - z + beta\n zdot = kappa * x - kappa * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Laser","page":"Home","title":"Dysts.Laser","text":" Laser()\n\nA semiconductor laser model\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.049321615812801\nLyapunov spectrum (estimated): Any[0.7637435195473311, 0.020943250302999646, -15.909545615344538]\nMaximum Lyapunov exponent (estimated): 0.6080602180196161\nMultiscale Entropy: 1.004437967414151\nNon-autonomous: false\nPeriod: 0.94667\nPesin entropy: 0.7848314215142518\n\nCitation\n\nAbooee, Yaghini-Bonabi, Jahed-Motlagh (2013). Analysis and circuitry realization of a novel three-dimensional chaotic system. Comm Nonline Sci 2013\n\nOriginal Python Code from the dysts package\n\nclass Laser(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, h, k):\n xdot = a * y - a * x + b * y * z ** 2\n ydot = c * x + d * x * z ** 2\n zdot = h * z + k * x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.LidDrivenCavityFlow","page":"Home","title":"Dysts.LidDrivenCavityFlow","text":" LidDrivenCavityFlow()\n\nA two-dimensional flow alternately driven by translation of the left and right walls.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: true\nKaplan-Yorke dimension: 2.9856669109882814\nLyapunov spectrum (estimated): Any[1.00757312, -0.00276829, -1.01941623]\nMaximum Lyapunov exponent (estimated): 2.172601412109087\nMultiscale Entropy: 0.7815824521381879\nNon-autonomous: true\nPeriod: 2.7333134945588955\nPesin entropy: 1.007573134065856\n\nCitation\n\nGrover et al. Chaos (2012). Topological chaos, braiding and bifurcation of almost-cyclic sets.\n\nOriginal Python Code from the dysts package\n\nclass LidDrivenCavityFlow(DynSys):\n @staticjit\n def _lid(x, y, a, b, tau, u1, u2):\n \"\"\"The velocity field when the left domain drives\"\"\"\n prefactor1 = 2 * u1 * np.sin(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n prefactor2 = 2 * u2 * np.sin(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n vx1 = -b * np.pi * np.sinh(np.pi * b / a) * np.sinh(np.pi * y / a) + np.cosh(np.pi * b / a) * (np.pi * y * np.cosh(np.pi * y /a) + a * np.sinh(np.pi * y / a))\n vx2 = -2 * b * np.pi * np.sinh(2 * np.pi * b / a) * np.sinh(2 * np.pi * y / a) + np.cosh(2 * np.pi * b / a) * (2 * np.pi * y * np.cosh(2 * np.pi * y / a) + a * np.sinh(2 * np.pi * y / a))\n vx = prefactor1 * vx1 + prefactor2 * vx2\n\n prefactor1 = 2 * np.pi * u1 * np.cos(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n prefactor2 = 4 * np.pi * u2 * np.cos(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n vy1 = b * np.sinh(np.pi * b / a) * np.cosh(np.pi * y / a) - np.cosh(np.pi * b / a) * y * np.sinh(np.pi * y / a)\n vy2 = b * np.sinh(2 * np.pi * b / a) * np.cosh(2 * np.pi * y / a) - np.cosh(2 * np.pi * b / a) * y * np.sinh(2 * np.pi * y / a)\n vy = prefactor1 * vy1 + prefactor2 * vy2\n\n # vy1 = b * np.sinh(np.pi * b / a) * np.cosh(np.pi * y / a) - np.cosh(np.pi * b / a) * y * np.sinh(np.pi * y / a)\n # vy2 = b * np.sinh(2 * np.pi * b / a) * np.cosh(2 * np.pi * y / a) - np.cosh(2 * np.pi * b / a) * y * np.sinh(2 * np.pi * y / a)\n # vy = np.pi * prefactor1 * vy1 + 2 * np.pi * prefactor2 * vy2\n\n return vx, vy\n\n # @staticjit\n # def _right(x, y, a, b, tau, u1, u2):\n # \"\"\"The velocity field when the right domain drives\"\"\"\n # prefactor1 = 2 * u1 * np.sin(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n # prefactor2 = 2 * u2 * np.sin(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n # vx1 = -b * np.pi * np.sinh(np.pi * b / a) * np.sinh(np.pi * y / a) - np.cosh(np.pi * b / a) * (np.pi * y * np.cosh(np.pi * y /a) + a * np.sinh(np.pi * y /a))\n # vx2 = -4 * b * np.pi * np.sinh(2 * np.pi * b / a) * np.sinh(2 * np.pi * y / a) - np.cosh(2 * np.pi * b / a) * (2 * np.pi * y * np.cosh(2 * np.pi * y /a) + a * np.sinh(2 * np.pi * y /a))\n # vx = prefactor1 * vx1 - prefactor2 * vx2\n\n # prefactor1 = 2 * np.pi * u1 * np.cos(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n # prefactor2 = 4 * np.pi * u2 * np.cos(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n # vy1 = -b * np.sinh(np.pi * b / a) * np.cosh(np.pi * y / a) + np.cosh(np.pi * b / a) * y * np.sinh(np.pi * y / a)\n # vy2 = -2 * b * np.sinh(2 * np.pi * b / a) * np.cosh(2 * np.pi * y / a) + np.cosh(2 * np.pi * b / a) * 2 * y * np.sinh(2 * np.pi * y / a)\n # vy = prefactor1 * vy1 + prefactor2 * vy2\n\n # return vx, vy\n\n @staticjit\n def _protocol(t, tau, stiffness=20):\n return 0.5 + 0.5 * np.tanh(tau * stiffness * np.sin(2 * np.pi * t / tau))\n\n def rhs(self, X, t):\n x, y, tt = X\n weight = self._protocol(tt, self.tau)\n dx1, dy1 = self._lid(x, y, self.a, self.b, self.tau, self.u1, self.u2)\n dx2, dy2 = self._lid(x, y, self.a, self.b, self.tau, -self.u1, self.u2)\n dx = weight * dx1 + (1 - weight) * dx2\n dy = weight * dy1 + (1 - weight) * dy2\n dtt = 1\n return dx, dy, dtt\n\n def _postprocessing(self, x, y, tt):\n return x, y, np.sin(2 * np.pi * tt / self.tau)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.LiuChen","page":"Home","title":"Dysts.LiuChen","text":" LiuChen()\n\nDerived from Sakarya.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.2188137323261288\nLyapunov spectrum (estimated): Any[0.21332142584135133, -0.9748996261500176, -18.807059276226646]\nMaximum Lyapunov exponent (estimated): 0.2457084451352168\nMultiscale Entropy: 0.7641473604711979\nNon-autonomous: false\nPeriod: 8.0164\nPesin entropy: 0.2133214258413513\n\nCitation\n\nLiu, Chen. Int J Bifurcat Chaos. 2004: 1395-1403.\n\nOriginal Python Code from the dysts package\n\nclass RayleighBenard(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, r):\n xdot = a * y - a * x\n ydot = r * y - x * z\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Lorenz-Tuple{}","page":"Home","title":"Dysts.Lorenz","text":" Lorenz()\n\nA minimal weather model based on atmospheric convection.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.075158758095728\nLyapunov spectrum (estimated): Any[1.0910931847726466, 0.02994120961308413, -14.915552395875103]\nMaximum Lyapunov exponent (estimated): 0.8917098035724058\nMultiscale Entropy: 1.1541457906835575\nNon-autonomous: false\nPeriod: 1.5008\nPesin entropy: 1.121034394385731\n\nCitation\n\nLorenz, Edward N (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.\n\nOriginal Python Code from the dysts package\n\nclass Lorenz(DynSys):\n @staticjit\n def _rhs(x, y, z, t, beta, rho, sigma):\n xdot = sigma * y - sigma * x\n ydot = rho * x - x * z - y\n zdot = x * y - beta * z\n return xdot, ydot, zdot\n @staticjit\n def _jac(x, y, z, t, beta, rho, sigma):\n row1 = [-sigma, sigma, 0]\n row2 = [rho - z, -1, -x]\n row3 = [y, x, -beta]\n return [row1, row2, row3]\n\n\n\n\n\n","category":"method"},{"location":"#Dysts.Lorenz84","page":"Home","title":"Dysts.Lorenz84","text":" Lorenz84()\n\nAtmospheric circulation analogous to Hadley convection.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.4449761695022096\nLyapunov spectrum (estimated): Any[1.0474807193950644, -0.045398035352994866, -2.251992293230755]\nMaximum Lyapunov exponent (estimated): 0.46146683857936616\nMultiscale Entropy: 0.6685802317710752\nNon-autonomous: false\nPeriod: 6.2586\nPesin entropy: 1.0474807193950644\n\nCitation\n\nE. Lorenz (1984). Irregularity: a fundamental property of the atmosphere. Tellus A, vol. 36, no. 2, pp. 98–110, 1984.\n\nOriginal Python Code from the dysts package\n\nclass Lorenz84(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, f, g):\n xdot = -a * x - y ** 2 - z ** 2 + a * f\n ydot = -y + x * y - b * x * z + g\n zdot = -z + b * x * y + x * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Lorenz96","page":"Home","title":"Dysts.Lorenz96","text":" Lorenz96()\n\nA climate model containing fluid-like advective nonlinearities, and multiscale coupling of fast and slow variables.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.1698163415908196\nLyapunov spectrum (estimated): Any[1.6247012663091516, 0.17185232259745578, -1.1936282999960617, -3.550455057872212]\nMaximum Lyapunov exponent (estimated): 1.3361667787286362\nMultiscale Entropy: 0.8046726640855828\nNon-autonomous: false\nPeriod: 2.1969\nPesin entropy: 1.7965535889066073\n\nCitation\n\nLorenz, Edward (1996). Predictability: A problem partly solved. Seminar on Predictability, Vol. I, ECMWF.\n\nOriginal Python Code from the dysts package\n\nclass Lorenz96(DynSys):\n def rhs(self, X, t):\n Xdot = np.zeros_like(X)\n Xdot[0] = (X[1] - X[-2]) * X[-1] - X[0] + self.f\n Xdot[1] = (X[2] - X[-1]) * X[0] - X[1] + self.f\n Xdot[-1] = (X[0] - X[-3]) * X[-2] - X[-1] + self.f\n Xdot[2:-1] = (X[3:] - X[:-3]) * X[1:-2] - X[2:-1] + self.f\n return Xdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.LorenzBounded-Tuple{}","page":"Home","title":"Dysts.LorenzBounded","text":" LorenzBounded()\n\nThe Lorenz attractor in the presence of a confining potential.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0704391073439155\nLyapunov spectrum (estimated): Any[0.8497533717540197, 0.001413707285345669, -12.120634680260636]\nMaximum Lyapunov exponent (estimated): 0.7124135270827626\nMultiscale Entropy: 1.1983193915488475\nNon-autonomous: false\nPeriod: 1.3716\nPesin entropy: 0.8513198508911037\n\nCitation\n\nSprott & Xiong (2015). Chaos.\n\nOriginal Python Code from the dysts package\n\nclass LorenzBounded(DynSys):\n @staticjit\n def _rhs(x, y, z, t, beta, r, rho, sigma):\n xdot = sigma * y - sigma * x - sigma/r**2 * y * x ** 2 - sigma/r**2 * y ** 3 - sigma/r**2 * y * z ** 2 + sigma/r**2 * x ** 3 + sigma/r**2 * x * y ** 2 + sigma/r**2 * x * z ** 2\n ydot = rho * x - x * z - y - rho/r**2 * x ** 3 - rho/r**2 * x * y ** 2 - rho/r**2 * x * z ** 2 + 1/r**2 * z * x ** 3 + 1/r**2 * x * z * y ** 2 + 1/r**2 * x * z ** 3 + 1/r**2 * y * x ** 2 + 1/r**2 * y ** 3 + 1/r**2 * y * z ** 2\n zdot = x * y - beta * z - 1/r**2 * y * x ** 3 - 1/r**2 * x * y ** 3 - 1/r**2 * x * y * z ** 2 + beta/r**2 * z * x ** 2 + beta/r**2 * z * y ** 2 + beta/r**2 * z ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"method"},{"location":"#Dysts.LorenzCoupled-Tuple{}","page":"Home","title":"Dysts.LorenzCoupled","text":" LorenzCoupled()\n\nTwo coupled Lorenz attractors.\n\nStats\n\nEmbedding dimension: 6\nHamiltonian: false\nKaplan-Yorke dimension: 3.0408069075490425\nLyapunov spectrum (estimated): Any[1.2218762961515068, 0.1222254715421154, -1.265209618630438, -1.9333037412103384, -13.618111101381306, -15.073328947851422]\nMaximum Lyapunov exponent (estimated): 1.0284398414904556\nMultiscale Entropy: 1.1717493753138832\nNon-autonomous: false\nPeriod: 1.6207\nPesin entropy: 1.3441017676936224\n\nCitation\n\nLorenz, Edward N. Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.\n\nOriginal Python Code from the dysts package\n\nclass LorenzCoupled(DynSys):\n @staticjit\n def _rhs(x1, y1, z1, x2, y2, z2, t, beta, eps, rho, rho1, rho2, sigma):\n x1dot = sigma * y1 - sigma * x1\n y1dot = rho1 * x1 - x1 * z1 - y1\n z1dot = x1 * y1 - beta * z1\n x2dot = sigma * y2 - sigma * x2 + eps * x1 - eps * x2\n y2dot = rho2 * x2 - x2 * z2 - y2\n z2dot = x2 * y2 - beta * z2\n return x1dot, y1dot, z1dot, x2dot, y2dot, z2dot\n\n\n\n\n\n","category":"method"},{"location":"#Dysts.LorenzStenflo","page":"Home","title":"Dysts.LorenzStenflo","text":" LorenzStenflo()\n\nAtmospheric acoustic-gravity waves.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.1738121741011174\nLyapunov spectrum (estimated): Any[0.44944459925783387, 0.026077542739630446, -2.7358399678801906, -3.4213273148340293]\nMaximum Lyapunov exponent (estimated): 0.3807742646421621\nMultiscale Entropy: 1.0299236916146275\nNon-autonomous: false\nPeriod: 3.8877142857142855\nPesin entropy: 0.47552214199746434\n\nCitation\n\nLetellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.\n\nOriginal Python Code from the dysts package\n\nclass LorenzStenflo(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + d * w\n ydot = c * x - x * z - y\n zdot = x * y - b * z\n wdot = -x - a * w\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.LuChen","page":"Home","title":"Dysts.LuChen","text":" LuChen()\n\nA system that switches shapes between the Lorenz and Chen attractors as the parameter c is varied from 12.7 to 28.5. Also called the Chen Celikovsky attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0358980309317456\nLyapunov spectrum (estimated): Any[0.6651845739025707, 0.1450338199952451, -22.56999542505058]\nMaximum Lyapunov exponent (estimated): 0.3266868574498627\nMultiscale Entropy: 0.4483903807918106\nNon-autonomous: false\nPeriod: 2.3385\nPesin entropy: 0.8102183938978158\n\nCitation\n\nLu, Chen. Int J Bifurcat Chaos. 2002: 659-661.\n\nOriginal Python Code from the dysts package\n\nclass LuChen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * y - a * x\n ydot = -x * z + c * y\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.LuChenCheng","page":"Home","title":"Dysts.LuChenCheng","text":" LuChenCheng()\n\nA four scroll attractor that reduces to Lorenz-like dynamics under bifurcations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0293036172873244\nLyapunov spectrum (estimated): Any[0.35106210160478324, -0.012414362698175424, -11.556516575635818]\nMaximum Lyapunov exponent (estimated): 0.2612775880231121\nMultiscale Entropy: 0.6029763775293518\nNon-autonomous: false\nPeriod: 0.92333\nPesin entropy: 0.3510621518701211\n\nCitation\n\nLu, Chen, Cheng. Int J Bifurcat Chaos. 2004: 1507–1537.\n\nOriginal Python Code from the dysts package\n\nclass LuChenCheng(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = -(a * b) / (a + b) * x - y * z + c\n ydot = a * y + x * z\n zdot = b * z + x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.MacArthur","page":"Home","title":"Dysts.MacArthur","text":" MacArthur()\n\nPopulation abundances in a plankton community, based on MacArthur's consumer resource model. Growth rates are determined using Liebig's law of the maximum\n\nStats\n\nEmbedding dimension: 10\nHamiltonian: false\nKaplan-Yorke dimension: 3.8961444775216614\nLyapunov spectrum (estimated): Any[0.1934617869736995, 0.004866584152339167, -0.0431132227766039, -0.17665092099562724, -0.30357752906614904, -0.30392441898183636, -0.30474608910196055, -0.3051840587615213, -0.3058908399877519, -17.37453227551654]\nMaximum Lyapunov exponent (estimated): 0.2494638510848618\nMultiscale Entropy: 0.7844902548776685\nNon-autonomous: false\nPeriod: 43.406\nPesin entropy: 0.19832837112603868\n\nCitation\n\nMacArthur, R. 1969. Species packing, and what competition minimizes. PNAS 64:1369. Parameter values taken from a model of phytoplankton in Huisman & Weissing. Nature 1999.\n\nOriginal Python Code from the dysts package\n\nclass MacArthur(DynSys):\n def growth_rate(self, rr):\n u0 = rr / (self.k.T + rr)\n u = self.r * u0.T\n return np.min(u.T, axis=1)\n\n def rhs(self, X, t):\n nn, rr = X[:5], X[5:]\n mu = self.growth_rate(rr)\n nndot = nn * (mu - self.m)\n rrdot = self.d * (self.s - rr) - np.matmul(self.c, (mu * nn))\n return np.hstack([nndot, rrdot])\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.MackeyGlass","page":"Home","title":"Dysts.MackeyGlass","text":" MackeyGlass()\n\nA physiological circuit with time-delayed feedback. A standard delay dimension of three is used.\n\nStats\n\nEmbedding dimension: 10\nHamiltonian: false\nKaplan-Yorke dimension: 8.012917999650943\nLyapunov spectrum (estimated): Any[0.21081023801544096, 0.16332731438425996, -0.03580777756738295, -0.041336844485590406, -0.04889078450243534, -0.059836905162829496, -0.07714200039885702, -0.10872244658769878, -0.18584871959889662, -2.2412562118189827]\nMaximum Lyapunov exponent (estimated): 0.07287527454565443\nNon-autonomous: false\nPeriod: 5.9612230962292045\nPesin entropy: 0.3741375523997009\n\nCitation\n\nGlass, L. and Mackey, M. C. (1979). Pathological physiological conditions resulting from instabilities in physiological control systems. Ann. NY. Acad. Sci.\n\nOriginal Python Code from the dysts package\n\nclass MackeyGlass(DynSysDelay):\n @staticjit\n def _rhs(x, xt, t, beta, gamma, n, tau):\n xdot = beta * (xt / (1 + xt ** n)) - gamma * x\n return xdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.MooreSpiegel","page":"Home","title":"Dysts.MooreSpiegel","text":" MooreSpiegel()\n\nA thermo-mechanical oscillator.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.3217713120052443\nLyapunov spectrum (estimated): Any[0.4074353399833356, -0.022079955878999143, -1.1975751283606624]\nMaximum Lyapunov exponent (estimated): 0.6976623003008144\nMultiscale Entropy: 0.5401952179209213\nNon-autonomous: false\nPeriod: 1.2887\nPesin entropy: 0.4076334003995097\n\nCitation\n\nMoore, Spiegel. A Thermally Excited Nonlinear Oscillator, Ap. J., 143, 871 (1966)\n\nOriginal Python Code from the dysts package\n\nclass MooreSpiegel(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, eps):\n xdot = y\n ydot = a * z\n zdot = -z + eps * y - y * x ** 2 - b * x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.MultiChua","page":"Home","title":"Dysts.MultiChua","text":" MultiChua()\n\nMultiple interacting Chua electronic circuits.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0710214570482988\nLyapunov spectrum (estimated): Any[0.2210571234274344, -0.00140279701621115, -3.10264254580399]\nMaximum Lyapunov exponent (estimated): 1.462134784614238\nMultiscale Entropy: 0.8118881305702679\nNon-autonomous: false\nPeriod: 2.2586\nPesin entropy: 0.22151080996231726\n\nCitation\n\nMufcstak E. Yalcin, Johan A. K. Suykens, Joos Vandewalle. Cellular Neural Networks, Multi-Scroll Chaos And Synchronization. \n\nOriginal Python Code from the dysts package\n\nclass MultiChua(DynSys):\n def diode(self, x):\n m, c = self.m, self.c\n total = m[-1] * x\n for i in range(1, 6):\n total += 0.5 * (m[i - 1] - m[i]) * (np.abs(x + c[i]) - np.abs(x - c[i]))\n return total\n\n def rhs(self, X, t):\n x, y, z = X\n xdot = self.a * (y - self.diode(x))\n ydot = x - y + z\n zdot = -self.b * y\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.NewtonLiepnik","page":"Home","title":"Dysts.NewtonLiepnik","text":" NewtonLiepnik()\n\nEuler's equations for a rigid body, augmented with a linear feedback term.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2206092778719664\nLyapunov spectrum (estimated): Any[0.1723564207468104, -0.00036976661148194504, -0.7795789311467557]\nMaximum Lyapunov exponent (estimated): 0.14945341784639976\nMultiscale Entropy: 0.6322918700854281\nNon-autonomous: false\nPeriod: 8.5095\nPesin entropy: 0.17242608410350868\n\nCitation\n\nLeipnik, R. B., and T. A. Newton (1981). Double strange attractors in rigid body motion with linear feedback control. Physics Letters A 86.2 (1981): 63-67.\n\nOriginal Python Code from the dysts package\n\nclass NewtonLiepnik(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x + y + 10 * y * z\n ydot = -x - 0.4 * y + 5 * x * z\n zdot = b * z - 5 * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.NoseHoover","page":"Home","title":"Dysts.NoseHoover","text":" NoseHoover()\n\nFixed temperature molecular dynamics for a strained flow under a reduced order model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 6.213204677134776\nLyapunov spectrum (estimated): Any[0.04757010871740569, 0.0013844486958287268, -0.011653762286922777]\nMaximum Lyapunov exponent (estimated): 0.010464383779400952\nMultiscale Entropy: 0.5656074689683536\nNon-autonomous: false\nPeriod: 5.8673\nPesin entropy: 0.04895455741323442\n\nCitation\n\nNose, S (1985). A unified formulation of the constant temperature molecular-dynamics methods. Journal of Chemical Physics (1984). Hoover, William G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A. Hoover, William G., Anthony JC Ladd, and Bill Moran (1982). High-strain-rate plastic flow studied via nonequilibrium molecular dynamics. Physical Review Letters.\n\nOriginal Python Code from the dysts package\n\nclass NoseHoover(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y\n ydot = -x + y * z\n zdot = a - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.NuclearQuadrupole","page":"Home","title":"Dysts.NuclearQuadrupole","text":" NuclearQuadrupole()\n\nA quadrupole boson Hamiltonian that produces chaos via a resonance overlap mechanism\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: true\nKaplan-Yorke dimension: 4.280996083113027\nLyapunov spectrum (estimated): Any[0.27858365099437626, 0.06629535725881477, -0.00449888101008936, -0.2658004759134054]\nMaximum Lyapunov exponent (estimated): 0.29554197767663787\nMultiscale Entropy: 0.8209529097052416\nNon-autonomous: false\nPeriod: 6.023076923076923\nPesin entropy: 0.3448827969293456\n\nCitation\n\nBaran V. and Raduta A. A. (1998), International Journal of Modern Physics E\n\nOriginal Python Code from the dysts package\n\nclass NuclearQuadrupole(DynSys):\n @staticjit\n def _rhs(q1, q2, p1, p2, t, a, b, d):\n q1dot = a * p1\n q2dot = a * p2\n p1dot = - a * q1 + 3 / np.sqrt(2) * b * q1 ** 2 - 3 / np.sqrt(2) * b * q2 ** 2 - d * q1 ** 3 - d * q1 * q2 ** 2\n p2dot = -a * q2 - 3 * np.sqrt(2) * b * q1 * q2 - d * q2 * q1 ** 2 - d * q2 ** 3\n return q1dot, q2dot, p1dot, p2dot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.OscillatingFlow","page":"Home","title":"Dysts.OscillatingFlow","text":" OscillatingFlow()\n\nA model fluid flow that produces KAM tori. Original parameters have been rescaled by k\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.962629189215532\nLyapunov spectrum (estimated): Any[0.4274159768365289, -0.005029615144529367, -0.43888500460322705]\nMaximum Lyapunov exponent (estimated): 0.13470179997834794\nMultiscale Entropy: 0.6177682255070645\nNon-autonomous: true\nPeriod: 23.857\nPesin entropy: 0.4274159768365288\n\nCitation\n\nT. H. Solomon and J. P. Gollub, Phys. Rev. A 38, 6280 (1988).\n\nOriginal Python Code from the dysts package\n\nclass OscillatingFlow(DynSys):\n @staticjit\n def _rhs(x, y, z, t, b, k, omega, u):\n f = x + b * np.sin(z)\n dx = u * np.cos(k * y) * np.sin(k * f)\n dy = -u * np.sin(k * y) * np.cos(k * f)\n dz = omega\n return dx, dy, dz\n\n def _postprocessing(self, x, y, z):\n return np.cos(self.k * x), y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.PehlivanWei","page":"Home","title":"Dysts.PehlivanWei","text":" PehlivanWei()\n\nA system with quadratic nonlinearity, which undergoes Hopf bifurcations of codimension 1 and 2.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1563635563177295\nLyapunov spectrum (estimated): Any[0.09512327528959069, 0.0015323304198873254, -0.6168618108533457]\nMaximum Lyapunov exponent (estimated): 0.10415076085878888\nMultiscale Entropy: 0.6547493000137818\nNon-autonomous: false\nPeriod: 3.7487330316742087\nPesin entropy: 0.09665562724246173\n\nCitation\n\nPehlivan, Ihsan, and Wei Zhouchao (2012). Analysis, nonlinear control, and chaos generator circuit of another strange chaotic system.\n\nOriginal Python Code from the dysts package\n\nclass PehlivanWei(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y - y * z\n ydot = y + y * z - 2 * x\n zdot = 2 - x * y - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Qi","page":"Home","title":"Dysts.Qi","text":" Qi()\n\nA hyperchaotic system with a wide power spectrum.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.498645226832379\nLyapunov spectrum (estimated): Any[2.0360429519545677, 0.1320125824983059, -4.347891883424509, -44.356041824978305]\nMaximum Lyapunov exponent (estimated): 2.1916060472989556\nMultiscale Entropy: 0.8855652233700573\nNon-autonomous: false\nPeriod: 0.40820999999999996\nPesin entropy: 2.168055534452874\n\nCitation\n\nG. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.\n\nOriginal Python Code from the dysts package\n\nclass Qi(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + y * z * w\n ydot = b * x + b * y - x * z * w\n zdot = -c * z + x * y * w\n wdot = -d * w + x * y * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.QiChen","page":"Home","title":"Dysts.QiChen","text":" QiChen()\n\nA double-wing chaotic attractor that arises from two bistable attractors. The system has quadratic nonlinearity.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0917466081236316\nLyapunov spectrum (estimated): Any[3.8880075026301206, 0.13379585631170307, -43.8359895934497]\nMaximum Lyapunov exponent (estimated): 4.0275703123027435\nMultiscale Entropy: 0.846453868214798\nNon-autonomous: false\nPeriod: 0.39179\nPesin entropy: 4.021803358941824\n\nCitation\n\nQi et al. Chaos, Solitons & Fractals 2008.\n\nOriginal Python Code from the dysts package\n\nclass QiChen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * y - a * x + y * z\n ydot = c * x + y - x * z\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.RabinovichFabrikant","page":"Home","title":"Dysts.RabinovichFabrikant","text":" RabinovichFabrikant()\n\nA reduced-order model of propagating waves in monequilibrium media.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.36076141745315\nLyapunov spectrum (estimated): Any[0.24233580922074083, 1.7068504600119996e-5, -0.6717710576786404]\nMaximum Lyapunov exponent (estimated): 0.2541443669572464\nMultiscale Entropy: 0.9189339042839536\nNon-autonomous: false\nPeriod: 2.3356\nPesin entropy: 0.24239107527154555\n\nCitation\n\nRabinovich, Mikhail I.; Fabrikant, A. L. (1979). Stochastic Self-Modulation of Waves in Nonequilibrium Media. Sov. Phys. JETP.\n\nOriginal Python Code from the dysts package\n\nclass RabinovichFabrikant(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, g):\n xdot = y * z - y + y * x ** 2 + g * x\n ydot = 3 * x * z + x - x ** 3 + g * y\n zdot = -2 * a * z - 2 * x * y * z\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.RikitakeDynamo","page":"Home","title":"Dysts.RikitakeDynamo","text":" RikitakeDynamo()\n\nElectric current and magnetic field of two coupled disk dynamos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0672834652602137\nLyapunov spectrum (estimated): Any[0.14571192786308473, 0.0007222098218782572, -2.1743990404663296]\nMaximum Lyapunov exponent (estimated): 0.1318127385277225\nMultiscale Entropy: 0.9664471212975273\nNon-autonomous: false\nPeriod: 5.8975\nPesin entropy: 0.14655639292019562\n\nCitation\n\nRikitake, T., Oscillations of a system of disk dynamos, Mathematical Proceedings of the Cambridge Philosophical Society, 1958.\n\nOriginal Python Code from the dysts package\n\nclass RikitakeDynamo(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, mu):\n xdot = -mu * x + y * z\n ydot = -mu * y - a * x + x * z\n zdot = 1 - x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Rossler","page":"Home","title":"Dysts.Rossler","text":" Rossler()\n\nSpiral-type chaos in a simple oscillator model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0146095059018845\nLyapunov spectrum (estimated): Any[0.0763301393780693, 0.007522476799147231, -5.741024338716747]\nMaximum Lyapunov exponent (estimated): 0.15059688939547888\nMultiscale Entropy: 0.605878349292794\nNon-autonomous: false\nPeriod: 5.9086\nPesin entropy: 0.08385261617721652\n\nCitation\n\nRossler, O. E. (1976), An Equation for Continuous Chaos, Physics Letters, 57A (5): 3972013398,\n\nOriginal Python Code from the dysts package\n\nclass Rossler(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = -y - z\n ydot = x + a * y\n zdot = b + z * x - c * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Sakarya","page":"Home","title":"Dysts.Sakarya","text":" Sakarya()\n\nAn attractor that arises due to merging of two disjoint bistable attractors.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.245095314885476\nLyapunov spectrum (estimated): Any[0.3080921246264039, 0.00035292037656675234, -1.2584697568737606]\nMaximum Lyapunov exponent (estimated): 0.26501036485491514\nMultiscale Entropy: 0.6082767822064958\nNon-autonomous: false\nPeriod: 4.9844\nPesin entropy: 0.3084450450029706\n\nCitation\n\nLi, Chunbiao, et al (2015). A novel four-wing strange attractor born in bistability. IEICE Electronics Express 12.4.\n\nOriginal Python Code from the dysts package\n\nclass Sakarya(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, h, p, q, r, s):\n xdot = a * x + h * y + s * y * z\n ydot = -b * y - p * x + q * x * z\n zdot = c * z - r * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SaltonSea","page":"Home","title":"Dysts.SaltonSea","text":" SaltonSea()\n\nAn eco-epidemiological model of bird and fish abundances in the Salton sea of southern California.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0843843542474496\nLyapunov spectrum (estimated): Any[0.3914445015077359, -0.0037578848522063194, -4.592867667006148]\nMaximum Lyapunov exponent (estimated): 1.0702552017260383\nMultiscale Entropy: 0.9335269725171643\nNon-autonomous: false\nPeriod: 0.96738\nPesin entropy: 0.39160860195935765\n\nCitation\n\nUpadhyay, Bairagi, Kundu, Chattopadhyay (2007).Chaos in eco-epidemiological problem of the Salton Sea and its possible control. Applied mathematics and computation.\n\nOriginal Python Code from the dysts package\n\nclass SaltonSea(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, d, k, lam, m, mu, r, th):\n xdot = r * x * (1 - (x + y) / k) - lam * x * y\n ydot = lam * x * y - m * y * z / (y + a) - mu * y\n zdot = th * y * z / (y + a) - d * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SanUmSrisuchinwong","page":"Home","title":"Dysts.SanUmSrisuchinwong","text":" SanUmSrisuchinwong()\n\nA two-scroll attractor arising from dynamical equations with piecewise nonlinearities. Equivalent to the Wimol-Banlue system\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.194219372064799\nLyapunov spectrum (estimated): Any[0.23913890446639047, -0.0002916165416126154, -1.2296759041605134]\nMaximum Lyapunov exponent (estimated): 0.2606135207962152\nMultiscale Entropy: 0.9947542017597873\nNon-autonomous: false\nPeriod: 7.4654\nPesin entropy: 0.23946267672516214\n\nCitation\n\nSan-Um, Srisuchinwong. J. Comp 2012\n\nOriginal Python Code from the dysts package\n\nclass SanUmSrisuchinwong(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y - x\n ydot = -z * np.tanh(x)\n zdot = -a + x * y + np.abs(y)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ShimizuMorioka","page":"Home","title":"Dysts.ShimizuMorioka","text":" ShimizuMorioka()\n\nA system that bifurcates from a symmetric limit cycle to an asymmetric one, similar to the Lorenz attractor.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0312187919639926\nLyapunov spectrum (estimated): Any[0.044032234819090736, -1.3942528300804223e-5, -1.409991553164768]\nMaximum Lyapunov exponent (estimated): 0.03577425334463406\nMultiscale Entropy: 0.7156563118943259\nNon-autonomous: false\nPeriod: 12.038\nPesin entropy: 0.044239420179671494\n\nCitation\n\nShimizu, Morioka. Phys Lett A. 1980: 201-204\n\nOriginal Python Code from the dysts package\n\nclass ShimizuMorioka(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = y\n ydot = x - a * y - x * z\n zdot = -b * z + x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottA","page":"Home","title":"Dysts.SprottA","text":" SprottA()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 60.66528307527488\nLyapunov spectrum (estimated): Any[0.029259920155059096, 0.004990159457687409, -0.0015561317158397701]\nMaximum Lyapunov exponent (estimated): 0.012993052652429771\nMultiscale Entropy: 0.5748267225365974\nNon-autonomous: false\nPeriod: 6.3728461538461545\nPesin entropy: 0.0343008088722448\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottA(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y\n ydot = -x + y * z\n zdot = 1 - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottB","page":"Home","title":"Dysts.SprottB","text":" SprottB()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1852994894688953\nLyapunov spectrum (estimated): Any[0.21987503866971636, 0.005187437926969091, -1.2145826242115807]\nMaximum Lyapunov exponent (estimated): 0.20674275256006883\nMultiscale Entropy: 0.8826029903839796\nNon-autonomous: false\nPeriod: 11.050142857142857\nPesin entropy: 0.2250624765966855\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottB(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y * z\n ydot = x - y\n zdot = 1 - x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottC","page":"Home","title":"Dysts.SprottC","text":" SprottC()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1529236029587073\nLyapunov spectrum (estimated): Any[0.17898394736414464, -0.0007304674803429964, -1.1644550651924115]\nMaximum Lyapunov exponent (estimated): 0.15806924766944033\nMultiscale Entropy: 1.14839462464542\nNon-autonomous: false\nPeriod: 8.8852\nPesin entropy: 0.17933456833596748\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottC(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y * z\n ydot = x - y\n zdot = 1 - x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottD","page":"Home","title":"Dysts.SprottD","text":" SprottD()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0807565309303526\nLyapunov spectrum (estimated): Any[0.11415315682442645, 0.001787998513315278, -1.4601870646976078]\nMaximum Lyapunov exponent (estimated): 0.10170941886932346\nMultiscale Entropy: 1.0030259093693934\nNon-autonomous: false\nPeriod: 4.8333\nPesin entropy: 0.11594526886491094\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottD(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = -y\n ydot = x + z\n zdot = x * z + 3 * y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottE","page":"Home","title":"Dysts.SprottE","text":" SprottE()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0894640654250147\nLyapunov spectrum (estimated): Any[0.09834470642927143, -0.00031069798260046597, -1.0953799295090543]\nMaximum Lyapunov exponent (estimated): 0.09016067590084897\nMultiscale Entropy: 0.8387888716672988\nNon-autonomous: false\nPeriod: 7.6275\nPesin entropy: 0.09866310814686535\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottE(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y * z\n ydot = x ** 2 - y\n zdot = 1 - 4 * x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottF","page":"Home","title":"Dysts.SprottF","text":" SprottF()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2516153966506316\nLyapunov spectrum (estimated): Any[0.13246006583273318, 0.0192716855937824, -0.603011686855317]\nMaximum Lyapunov exponent (estimated): 0.1183090101484132\nMultiscale Entropy: 0.7674306147320477\nNon-autonomous: false\nPeriod: 19.244\nPesin entropy: 0.1517317514265156\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottF(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y + z\n ydot = -x + a * y\n zdot = x ** 2 - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottG","page":"Home","title":"Dysts.SprottG","text":" SprottG()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0519020031307744\nLyapunov spectrum (estimated): Any[0.03349131570257798, -0.0012749369106980868, -0.6182204265060449]\nMaximum Lyapunov exponent (estimated): 0.09067832253388744\nMultiscale Entropy: 0.6185493944257671\nNon-autonomous: false\nPeriod: 8.7097\nPesin entropy: 0.03403484153064641\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottG(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = a * x + z\n ydot = x * z - y\n zdot = -x + y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottH","page":"Home","title":"Dysts.SprottH","text":" SprottH()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2414973487697853\nLyapunov spectrum (estimated): Any[0.13871794302385684, 0.003501431527456669, -0.5889065665901029]\nMaximum Lyapunov exponent (estimated): 0.13100182470690253\nMultiscale Entropy: 0.6325364503922819\nNon-autonomous: false\nPeriod: 20.229\nPesin entropy: 0.14221937455131348\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottH(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = -y + z ** 2\n ydot = x + a * y\n zdot = x - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottI","page":"Home","title":"Dysts.SprottI","text":" SprottI()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.019844528960972\nLyapunov spectrum (estimated): Any[0.020459651934364873, 9.478591964272237e-6, -1.031514712617076]\nMaximum Lyapunov exponent (estimated): 0.023218704376506836\nMultiscale Entropy: 0.5893805046352655\nNon-autonomous: false\nPeriod: 10.93\nPesin entropy: 0.02077411702709478\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottI(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = -a * y\n ydot = x + z\n zdot = x + y ** 2 - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottJ","page":"Home","title":"Dysts.SprottJ","text":" SprottJ()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0221468017281006\nLyapunov spectrum (estimated): Any[0.04451808076432957, 0.0010621228852282804, -2.055842238064512]\nMaximum Lyapunov exponent (estimated): 0.12290789099642041\nMultiscale Entropy: 0.6352974298685767\nNon-autonomous: false\nPeriod: 4.836\nPesin entropy: 0.04588958821017871\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottJ(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = 2 * z\n ydot = -2 * y + z\n zdot = -x + y + y ** 2\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottJerk","page":"Home","title":"Dysts.SprottJerk","text":" SprottJerk()\n\nAn algebraidally simple flow depending on a third time derivative. Numerical integration of this attractor is extremely sensitive to the initial conditions\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.031190248069004\nLyapunov spectrum (estimated): Any[0.06738258231535933, -0.0007831911114099333, -2.1347047232043663]\nMaximum Lyapunov exponent (estimated): 0.08785190282162628\nMultiscale Entropy: 0.5314887786603959\nNon-autonomous: false\nPeriod: 11.921\nPesin entropy: 0.06776939078007471\n\nCitation\n\nSprott, J. C. Simplest dissipative chaotic flow. Physics Letters A (1997).\n\nOriginal Python Code from the dysts package\n\nclass SprottJerk(DynSys):\n @staticjit\n def _rhs(x, y, z, t, mu):\n xdot = y\n ydot = z\n zdot = -x + y ** 2 - mu * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottK","page":"Home","title":"Dysts.SprottK","text":" SprottK()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0417036451446204\nLyapunov spectrum (estimated): Any[0.03778259119323125, -0.0005109461395399151, -0.8934722146671759]\nMaximum Lyapunov exponent (estimated): 0.05962593786799159\nMultiscale Entropy: 0.6562812206790845\nNon-autonomous: false\nPeriod: 7.006\nPesin entropy: 0.03788160511007809\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottK(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = x * y - z\n ydot = x - y\n zdot = x + a * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottL","page":"Home","title":"Dysts.SprottL","text":" SprottL()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0329815956000377\nLyapunov spectrum (estimated): Any[0.03407149258791976, -0.000391471796353918, -1.012962157697122]\nMaximum Lyapunov exponent (estimated): 0.09216197583296963\nMultiscale Entropy: 0.5605870039438788\nNon-autonomous: false\nPeriod: 6.7262\nPesin entropy: 0.0343958205328805\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottL(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = y + b * z\n ydot = a * x ** 2 - y\n zdot = 1 - x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottM","page":"Home","title":"Dysts.SprottM","text":" SprottM()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.04996202872584\nLyapunov spectrum (estimated): Any[0.05230117379497565, -0.00033063791832836577, -1.0391447706177515]\nMaximum Lyapunov exponent (estimated): 0.08422043130700084\nMultiscale Entropy: 0.7481178461456197\nNon-autonomous: false\nPeriod: 5.3644\nPesin entropy: 0.05234518101225785\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottM(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = -z\n ydot = -x ** 2 - y\n zdot = a + a * x + y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottMore","page":"Home","title":"Dysts.SprottMore","text":" SprottMore()\n\nA multifractal system with a nearly 3D attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.999952495039482\nLyapunov spectrum (estimated): Any[0.32499834596046606, -0.0001641583999746131, -0.31918544080126926]\nMaximum Lyapunov exponent (estimated): 1.0946909441433958\nMultiscale Entropy: 0.6370154977546665\nNon-autonomous: false\nPeriod: 6.62095890410959\nPesin entropy: 0.32515558132375566\n\nCitation\n\nSprott, J. C. (2020). Do We Need More Chaos Examples? Chaos Theory and Applications 2(2),1-3, 2020.\n\nOriginal Python Code from the dysts package\n\nclass SprottMore(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y\n ydot = -x - np.sign(z) * y\n zdot = y ** 2 - np.exp(-(x ** 2))\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottN","page":"Home","title":"Dysts.SprottN","text":" SprottN()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.028951405806632\nLyapunov spectrum (estimated): Any[0.06095861449194353, 6.560209716837452e-5, -2.101648044064128]\nMaximum Lyapunov exponent (estimated): 0.11869612209728901\nMultiscale Entropy: 0.6385218945868028\nNon-autonomous: false\nPeriod: 4.8135\nPesin entropy: 0.061478272616693266\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottN(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = -2 * y\n ydot = x + z ** 2\n zdot = 1 + y - 2 * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottO","page":"Home","title":"Dysts.SprottO","text":" SprottO()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2474645346215256\nLyapunov spectrum (estimated): Any[0.06685684927644238, 0.011533284505120292, -0.3165440349833235]\nMaximum Lyapunov exponent (estimated): 0.09141077854590315\nMultiscale Entropy: 0.4161154943097029\nNon-autonomous: false\nPeriod: 9.755\nPesin entropy: 0.07839013378156268\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottO(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y\n ydot = x - z\n zdot = x + x * z + a * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottP","page":"Home","title":"Dysts.SprottP","text":" SprottP()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2122245136720187\nLyapunov spectrum (estimated): Any[0.1026165705645786, 0.00025136328640969387, -0.4843551814513015]\nMaximum Lyapunov exponent (estimated): 0.1064676723437159\nMultiscale Entropy: 0.8486396469370624\nNon-autonomous: false\nPeriod: 5.3414\nPesin entropy: 0.10329419803150507\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottP(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = a * y + z\n ydot = -x + y ** 2\n zdot = x + y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottQ","page":"Home","title":"Dysts.SprottQ","text":" SprottQ()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2246852931286063\nLyapunov spectrum (estimated): Any[0.13698250147166127, 0.0010386716745898255, -0.6140516336902575]\nMaximum Lyapunov exponent (estimated): 0.17169784338063596\nMultiscale Entropy: 0.8105158145249107\nNon-autonomous: false\nPeriod: 4.8443\nPesin entropy: 0.13852213114060488\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottQ(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -z\n ydot = x - y\n zdot = a * x + y ** 2 + b * z\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottR","page":"Home","title":"Dysts.SprottR","text":" SprottR()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.071744469374985\nLyapunov spectrum (estimated): Any[0.07608481335025485, 0.0013697096816499383, -1.0784000269941107]\nMaximum Lyapunov exponent (estimated): 0.09661415277799693\nMultiscale Entropy: 0.5877166245477188\nNon-autonomous: false\nPeriod: 7.0133\nPesin entropy: 0.07840404228136309\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottR(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = a - y\n ydot = b + z\n zdot = x * y - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottS","page":"Home","title":"Dysts.SprottS","text":" SprottS()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.182390107009358\nLyapunov spectrum (estimated): Any[0.2149312025337423, 0.0030194370306177213, -1.1945344436340624]\nMaximum Lyapunov exponent (estimated): 0.2737629235389973\nMultiscale Entropy: 0.6690400524585065\nNon-autonomous: false\nPeriod: 3.3505\nPesin entropy: 0.21876911249150574\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottS(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = -x - 4 * y\n ydot = x + z ** 2\n zdot = 1 + x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SprottTorus","page":"Home","title":"Dysts.SprottTorus","text":" SprottTorus()\n\nA multiattractor system that goes to a torus or a complex attractor depending on the initial conditions.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.759693065535405\nLyapunov spectrum (estimated): Any[0.29656066211909027, 0.0005468818242604863, -0.3910853761095065]\nMaximum Lyapunov exponent (estimated): 0.1056964344650892\nMultiscale Entropy: 0.9562901894463037\nNon-autonomous: false\nPeriod: 13.607\nPesin entropy: 0.29710937347161515\n\nCitation\n\nSprott Physics Letters A 2014\n\nOriginal Python Code from the dysts package\n\nclass SprottTorus(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y + 2 * x * y + x * z\n ydot = 1 - 2 * x ** 2 + y * z\n zdot = x - x ** 2 - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.StickSlipOscillator","page":"Home","title":"Dysts.StickSlipOscillator","text":" StickSlipOscillator()\n\nA weakly forced (quasiautonomous) oscillator with dynamics similar to stick-slip friction. The parameter gamma controls the transition to chaos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 5.682155291949877\nLyapunov spectrum (estimated): Any[0.07073329226101453, 3.356854711815694e-5, -0.02131392884593247]\nMaximum Lyapunov exponent (estimated): 0.06406131604734494\nMultiscale Entropy: 0.9213029740441252\nNon-autonomous: true\nPeriod: 1.6129\nPesin entropy: 0.07108101456222432\n\nCitation\n\nAwrejcewicz, Jan, and M. M. Holicke (1999). Int J of Bifurcation and Chaos.\n\nOriginal Python Code from the dysts package\n\nclass StickSlipOscillator(DynSys):\n def _t(self, v):\n return self.t0 * np.sign(v) - self.alpha * v + self.beta * v ** 3\n\n @staticjit\n def _rhs(x, v, th, t, a, alpha, b, beta, eps, gamma, t0, vs, w):\n tq = t0 * np.sign(v - vs) - alpha * v + beta * (v - vs) ** 3\n xdot = v\n vdot = eps * (gamma * np.cos(th) - tq) + a * x - b * x ** 3\n thdot = w\n return xdot, vdot, thdot\n\n @staticjit\n def _postprocessing(x, v, th):\n return x, v, np.cos(th)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.SwingingAtwood","page":"Home","title":"Dysts.SwingingAtwood","text":" SwingingAtwood()\n\nA mechanical system consisting of two swinging weights connected by ropes and pulleys. This is only chaotic when m2 is sufficiently larger than m1, and there are nonzero initial momenta\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: true\nKaplan-Yorke dimension: 4.0198847313975445\nLyapunov spectrum (estimated): Any[7.735006558021491, 3.0380875684439133, -3.002075505122432, -7.664246463289054]\nMaximum Lyapunov exponent (estimated): 1.0111284961954359\nNon-autonomous: false\nPeriod: 0.3417647058823529\nPesin entropy: 10.773094126465404\n\nCitation\n\nTufillaro, Nicholas B.; Abbott, Tyler A.; Griffiths, David J. (1984). Swinging Atwood's Machine. American Journal of Physics. 52 (10): 895–903.\n\nOriginal Python Code from the dysts package\n\nclass SwingingAtwood(DynSys):\n @staticjit\n def _rhs(r, th, pr, pth, t, m1, m2):\n g = 9.82\n rdot = pr / (m1 + m2)\n thdot = pth / (m1 * r ** 2)\n prdot = pth ** 2 / (m1 * r ** 3) - m2 * g + m1 * g * np.cos(th)\n pthdot = -m1 * g * r * np.sin(th)\n return rdot, thdot, prdot, pthdot\n\n @staticjit\n def _postprocessing(r, th, pr, pth):\n return r, np.sin(th), pr, pth\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Thomas","page":"Home","title":"Dysts.Thomas","text":" Thomas()\n\nA cyclically-symmetric attractor correspondng to a frictionally-damped particle traversing a 3D lattice of forces\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1271136075295507\nLyapunov spectrum (estimated): Any[0.7589685797630259, 0.057355308262307496, -6.421950974462397]\nMaximum Lyapunov exponent (estimated): 0.6324054000339867\nMultiscale Entropy: 0.9921951854111759\nNon-autonomous: false\nPeriod: 4.969\nPesin entropy: 0.8163238880253334\n\nCitation\n\nThomas, Rene (1999). Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, labyrinth chaos. Int. J. Bifurc. Chaos.\n\nOriginal Python Code from the dysts package\n\nclass Thomas(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x + b * np.sin(y)\n ydot = -a * y + b * np.sin(z)\n zdot = -a * z + b * np.sin(x)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.Torus","page":"Home","title":"Dysts.Torus","text":" Torus()\n\nA minimal quasiperiodic flow on a torus. All lyapunov exponents and related quantities are zero\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0\nLyapunov spectrum (estimated): Any[0.0, 0.0, 0.0]\nMaximum Lyapunov exponent (estimated): 0.00817284128815307\nMultiscale Entropy: 0.47330913201518593\nNon-autonomous: false\nPeriod: 6.36\nPesin entropy: 0.0\n\nCitation\n\nSee, for example, Strogatz (1994). Nonlinear Dynamics and Chaos.\n\nOriginal Python Code from the dysts package\n\nclass Torus(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, n, r):\n xdot = (-a * n * np.sin(n * t)) * np.cos(t) - (r + a * np.cos(n * t)) * np.sin(\n t\n )\n ydot = (-a * n * np.sin(n * t)) * np.sin(t) + (r + a * np.cos(n * t)) * np.cos(\n t\n )\n zdot = a * n * np.cos(n * t)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.TurchinHanski","page":"Home","title":"Dysts.TurchinHanski","text":" TurchinHanski()\n\nA chaotic three species food web. The species growth rate has been increased from the default parameters in the original paper, in order to ensure that the system exhibits sustained chaotic oscillations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.2053069221786739\nLyapunov spectrum (estimated): Any[0.03719830596656144, -0.181183885924572, -2.522759887828014]\nMaximum Lyapunov exponent (estimated): 0.12906862565697674\nMultiscale Entropy: 0.4989636261373046\nNon-autonomous: true\nPeriod: 4.40529411764706\nPesin entropy: 0.03719830596656144\n\nCitation\n\nTurchin, Hanski. The American Naturalist 1997. Turchin, Hanski. Ecology 2000\n\nOriginal Python Code from the dysts package\n\nclass TurchinHanski(DynSys):\n @staticjit\n def _rhs(n, p, z, t, a, d, e, g, h, r, s):\n ndot = (\n r * (1 - e * np.sin(z)) * n\n - r * (n ** 2)\n - g * (n ** 2) / (n ** 2 + h ** 2)\n - a * n * p / (n + d)\n )\n pdot = s * (1 - e * np.sin(z)) * p - s * (p ** 2) / n\n zdot = 2 * np.pi\n return ndot, pdot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.VallisElNino","page":"Home","title":"Dysts.VallisElNino","text":" VallisElNino()\n\nAtmospheric temperature fluctuations with annual forcing.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1205065379188333\nLyapunov spectrum (estimated): Any[0.6561566154431996, 0.022689184568283626, -5.6332694618503885]\nMaximum Lyapunov exponent (estimated): 0.5477586276153029\nMultiscale Entropy: 0.9058707360850027\nNon-autonomous: false\nPeriod: 2.2087\nPesin entropy: 0.6788458000114831\n\nCitation\n\nVallis GK. Conceptual models of El Nio and the southern oscillation. J Geophys Res 1988\n\nOriginal Python Code from the dysts package\n\nclass VallisElNino(DynSys):\n @staticmethod\n def _rhs(x, y, z, t, b, c, p):\n xdot = b * y - c * x - c * p\n ydot = -y + x * z\n zdot = -z - x * y + 1\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.WangSun","page":"Home","title":"Dysts.WangSun","text":" WangSun()\n\nA four-scroll attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0595560098996604\nLyapunov spectrum (estimated): Any[0.07990874520682271, -0.001043918775422804, -1.3240808432405]\nMaximum Lyapunov exponent (estimated): 0.06735463002480017\nMultiscale Entropy: 0.9045422097501858\nNon-autonomous: false\nPeriod: 26.958\nPesin entropy: 0.07994326076367073\n\nCitation\n\nWang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.\n\nOriginal Python Code from the dysts package\n\nclass WangSun(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, d, e, f, q):\n xdot = a * x + q * y * z\n ydot = b * x + d * y - x * z\n zdot = e * z + f * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.WindmiReduced","page":"Home","title":"Dysts.WindmiReduced","text":" WindmiReduced()\n\nEnergy transfer into the ionosphere and magnetosphere by the solar wind. The parameter vsw controls the onset of chaos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0947817139683895\nLyapunov spectrum (estimated): Any[0.15595238514031914, 9.695760899839663e-5, -1.6450480911820564]\nMaximum Lyapunov exponent (estimated): 0.403521000622694\nMultiscale Entropy: 0.5896070173085494\nNon-autonomous: false\nPeriod: 4.6733\nPesin entropy: 0.15635149449218705\n\nCitation\n\nSmith, Thiffeault, Horton. J Geophys Res. 2000. Horton, Weigel, Sprott. Chaos 2001.\n\nOriginal Python Code from the dysts package\n\nclass WindmiReduced(DynSys):\n @staticjit\n def _rhs(i, v, p, t, a1, b1, b2, b3, d1, vsw):\n idot = a1 * (vsw - v)\n vdot = b1 * i - b2 * p ** 1 / 2 - b3 * v\n pdot = (\n vsw ** 2 - p ** (5 / 4) * vsw ** (1 / 2) * (1 + np.tanh(d1 * (i - 1))) / 2\n )\n return idot, vdot, pdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.YuWang","page":"Home","title":"Dysts.YuWang","text":" YuWang()\n\nA temperature-compensation circuit with an operational amplifier\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1703721122420614\nLyapunov spectrum (estimated): Any[2.459880656234634, -0.07297917814211805, -14.00993065520751]\nMaximum Lyapunov exponent (estimated): 1.8525516537708415\nMultiscale Entropy: 0.6340382196367295\nNon-autonomous: false\nPeriod: 1.3366\nPesin entropy: 2.459880656234634\n\nCitation\n\nYu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.\n\nOriginal Python Code from the dysts package\n\nclass YuWang(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d):\n xdot = a * (y - x)\n ydot = b * x - c * x * z\n zdot = np.exp(x * y) - d * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.YuWang2","page":"Home","title":"Dysts.YuWang2","text":" YuWang2()\n\nAn alternative temperature-compensation circuit with an operational amplifier\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0619881661890873\nLyapunov spectrum (estimated): Any[0.802715215563475, 0.007966964664720393, -13.078014115524866]\nMaximum Lyapunov exponent (estimated): 1.1576066800869989\nMultiscale Entropy: 0.9335570471768684\nNon-autonomous: false\nPeriod: 0.39265\nPesin entropy: 0.8106827804626555\n\nCitation\n\nYu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.\n\nOriginal Python Code from the dysts package\n\nclass YuWang2(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d):\n xdot = a * (y - x)\n ydot = b * x - c * x * z\n zdot = np.cosh(x * y) - d * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#Dysts.ZhouChen","page":"Home","title":"Dysts.ZhouChen","text":" ZhouChen()\n\nA feedback circuit model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.045920730005392\nLyapunov spectrum (estimated): Any[0.5917259125455694, 0.005203946388490041, -12.999237688229242]\nMaximum Lyapunov exponent (estimated): 0.6828000288089435\nMultiscale Entropy: 0.7337896982114851\nNon-autonomous: false\nPeriod: 0.87518\nPesin entropy: 0.5969734164748293\n\nCitation\n\nZhou, Chen (2004). A simple smooth chaotic system with a 3-layer attractor. Int J Bifurcat Chaos, 2004\n\nOriginal Python Code from the dysts package\n\nclass ZhouChen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, e):\n xdot = a * x + b * y + y * z\n ydot = c * y - x * z + d * y * z\n zdot = e * z - x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"}] +[{"location":"","page":"Home","title":"Home","text":"CurrentModule = ChaoticDynamicalSystemsLibrary","category":"page"},{"location":"#ChaoticDynamicalSystemsLibrary.jl","page":"Home","title":"ChaoticDynamicalSystemsLibrary.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"ChaoticDynamicalSystemLibrary.jl implements a collection of chaotic dynamical systems in the Julia SciML ecosystem.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The package is a partial port of the dysts Python package developed by William Gilpin. It implements many (but not all) of the same systems, but does not provide any functionality to simulate them. In contrast to dysts, the main focus of this package is not to be a benchmark for general time-series ML models, but only to provide a collection of ODEs that can be used with the DifferentialEquations.jl ecosystem.","category":"page"},{"location":"","page":"Home","title":"Home","text":"","category":"page"},{"location":"","page":"Home","title":"Home","text":"William Gilpin deserves most of the credit for this package. He is the original author of the dysts, and without dysts this package would not exist. –-","category":"page"},{"location":"#Full-list-of-all-dynamical-systems:","page":"Home","title":"Full list of all dynamical systems:","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Modules = [ChaoticDynamicalSystemsLibrary]\nPublic = false\nPages = [\"chaotic_attractors.jl\"]","category":"page"},{"location":"#ChaoticDynamicalSystemsLibrary.Aizawa","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Aizawa","text":" Aizawa()\n\nA torus-like attractor related to the forced Lorenz system.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.3501297618370773\nLyapunov spectrum (estimated): Any[0.08947878317195473, 0.020305496211675024, -0.3090729926541944]\nMaximum Lyapunov exponent (estimated): 0.13489555530106362\nMultiscale Entropy: 0.514512246123007\nNon-autonomous: false\nPeriod: 2.5837\nPesin entropy: 0.10978427938362978\n\nCitation\n\nAizawa, Yoji, and Tatsuya Uezu (1982). Topological Aspects in Chaos and in 2 k-Period Doubling Cascade. Progress of Theoretical Physics 67.3 (1982): 982-985. See also Langford (1984). Numerical studies of torus bifurcations. Numerical Methods for Bifurcation Problems. Birkhauser, Basel, 1984. 285-295.\n\nOriginal Python Code from the dysts package\n\nclass Aizawa(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, e, f):\n xdot = x * z - b * x - d * y\n ydot = d * x + y * z - b * y\n zdot = c + a * z - 0.333333333333333333 * z ** 3 - x ** 2 - y ** 2 - e * z * x ** 2 - e * z * y ** 2 + f * z * x ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.AnishchenkoAstakhov","page":"Home","title":"ChaoticDynamicalSystemsLibrary.AnishchenkoAstakhov","text":" AnishchenkoAstakhov()\n\nStochastic resonance in forced oscillators.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0916403758729327\nLyapunov spectrum (estimated): Any[0.039777554890210017, 0.0008785544495802295, -0.44025308728606666]\nMaximum Lyapunov exponent (estimated): 0.10107726791527546\nMultiscale Entropy: 0.7612343892027325\nNon-autonomous: false\nPeriod: 6.814\nPesin entropy: 0.040715925817307534\n\nCitation\n\nAnishchenko, et al. Nonlinear dynamics of chaotic and stochastic systems: tutorial and modern developments. 2007.\n\nOriginal Python Code from the dysts package\n\nclass AnishchenkoAstakhov(DynSys):\n def rhs(self, X, t):\n x, y, z = X\n mu, eta = self.mu, self.eta\n xdot = mu * x + y - x * z\n ydot = -x\n zdot = -eta * z + eta * np.heaviside(x, 0) * x ** 2\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Arneodo","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Arneodo","text":" Arneodo()\n\nA modified Lotka-Volterra ecosystem, also known as the ACT attractor.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.198607392233255\nLyapunov spectrum (estimated): Any[0.24285661379003065, -0.0006168721448237418, -1.2169777901647385]\nMaximum Lyapunov exponent (estimated): 0.23737468667051287\nMultiscale Entropy: 0.7388404221540498\nNon-autonomous: false\nPeriod: 3.1641\nPesin entropy: 0.2428749592514372\n\nCitation\n\nArneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional Volterra equations. Physics Letters A 79, 259–263 (1980)\n\nOriginal Python Code from the dysts package\n\nclass Arneodo(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d):\n xdot = y\n ydot = z\n zdot = -a * x - b * y - c * z + d * x ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ArnoldBeltramiChildress","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ArnoldBeltramiChildress","text":" ArnoldBeltramiChildress()\n\nAn exact solution of Euler's equation for inviscid flow in 3D. Often referred to as the ABC flow.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.612281260580053\nLyapunov spectrum (estimated): Any[1.47222087191395, -0.01824039630029517, -2.378562986862767]\nMaximum Lyapunov exponent (estimated): 0.3667300080060703\nMultiscale Entropy: 1.2759526834024162\nNon-autonomous: false\nPeriod: 2.8508\nPesin entropy: 1.47222087191395\n\nCitation\n\nV. I. Arnold, Journal of Applied Mathematics and Mechanics 30, 223 (1966).\n\nOriginal Python Code from the dysts package\n\nclass ArnoldBeltramiChildress(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n dx = a * np.sin(z) + c * np.cos(y)\n dy = b * np.sin(x) + a * np.cos(z)\n dz = c * np.sin(y) + b * np.cos(x)\n return dx, dy, dz\n\n @staticjit\n def _postprocessing(x, y, z):\n return np.sin(x), np.cos(y), np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ArnoldWeb","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ArnoldWeb","text":" ArnoldWeb()\n\nA quasi-integrable system that transitions to Chirikov diffusive dynamics when perturbed.\n\nStats\n\nEmbedding dimension: 5\nHamiltonian: true\nKaplan-Yorke dimension: 22.554733912627043\nLyapunov spectrum (estimated): Any[0.06266308160064184, 0.0032177165561558833, 0.0011281526899086728, -0.00233619507966634, -0.0039196281630406]\nMaximum Lyapunov exponent (estimated): 0.06514413995171335\nMultiscale Entropy: 0.620621428966386\nNon-autonomous: true\nPeriod: 67.25\nPesin entropy: 0.06706808893310089\n\nCitation\n\nFroeschle, C., Guzzo, M. & Legga, E (2000). Graphical evolution of the Arnold web: From order to chaos. Science 289.\n\nOriginal Python Code from the dysts package\n\nclass ArnoldWeb(DynSys):\n @staticjit\n def _rhs(p1, p2, x1, x2, z, t, mu, w):\n denom = 4 + np.cos(z) + np.cos(x1) + np.cos(x2)\n p1dot = -mu * np.sin(x1) / denom ** 2\n p2dot = -mu * np.sin(x2) / denom ** 2\n x1dot = p1\n x2dot = p2\n zdot = w\n return p1dot, p2dot, x1dot, x2dot, zdot\n\n @staticjit\n def _postprocessing(p1, p2, x1, x2, z):\n return p1, p2, np.sin(x1), np.sin(x2), np.cos(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.AtmosphericRegime","page":"Home","title":"ChaoticDynamicalSystemsLibrary.AtmosphericRegime","text":" AtmosphericRegime()\n\nA model of regime transition in atmospheric flows. Contains widely-separated timescales due a quadratic term, which itself arises from a conservation law.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1859596374283137\nLyapunov spectrum (estimated): nothing\nMaximum Lyapunov exponent (estimated): 0.010543274932274862\nMultiscale Entropy: 0.4278776303819699\nNon-autonomous: false\nPeriod: 89.59496756697331\nPesin entropy: 0.3055574134566322\n\nCitation\n\nTuwankotta (2006). Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity\n\nOriginal Python Code from the dysts package\n\nclass AtmosphericRegime(DynSys):\n @staticjit\n def _rhs(\n x, y, z, t, alpha, beta, mu1, mu2, omega, sigma\n ):\n xdot = mu1 * x + sigma * x * y\n ydot = mu2 * y + omega * z + alpha * y * z + beta * z ** 2 - sigma * x ** 2\n zdot = mu2 * z - omega * y - alpha * y ** 2 - beta * y * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.BeerRNN","page":"Home","title":"ChaoticDynamicalSystemsLibrary.BeerRNN","text":" BeerRNN()\n\nA two-neuron minimal model nervous system.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0303797735862443\nLyapunov spectrum (estimated): Any[0.016274571192515633, -0.00011803510272572572, -0.5318869010933206]\nMaximum Lyapunov exponent (estimated): 0.038767800131815223\nMultiscale Entropy: 0.5244807630539362\nNon-autonomous: false\nPeriod: 181.58\nPesin entropy: 0.01647451962951782\n\nCitation\n\nBeer, R. D. (1995). On the dynamics of small continuous-time recurrent neural networks. Adapt. Behav., 3(4), 4692013509.\n\nOriginal Python Code from the dysts package\n\nclass BeerRNN(DynSys):\n @staticjit\n def _sig(x):\n return 1.0 / (1.0 + np.exp(-x))\n\n def rhs(self, X, t):\n Xdot = (-X + np.matmul(self.w, self._sig(X + self.theta))) / self.tau\n return Xdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.BelousovZhabotinsky","page":"Home","title":"ChaoticDynamicalSystemsLibrary.BelousovZhabotinsky","text":" BelousovZhabotinsky()\n\nA reduced-order model of the BZ reaction that exhibits period doubling. The bifurcation parameter for controlling the onset of chaos is kf. The system undergoes regular cycling when kf=3e-4, and chaotic oscillations when kf=3.5e-4\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0206505961796535\nLyapunov spectrum (estimated): Any[114.64513768787073, -0.4669861532822692, -5529.048679334775]\nMaximum Lyapunov exponent (estimated): 132.08766434104578\nMultiscale Entropy: 0.8764191281600929\nNon-autonomous: false\nPeriod: 0.018257\nPesin entropy: 114.64513768787073\n\nCitation\n\nGyorgyi and Field (1992). A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction. Nature.\n\nOriginal Python Code from the dysts package\n\nclass BelousovZhabotinsky(DynSys):\n @staticjit\n def _rhs(\n x,\n z,\n v,\n t,\n c1,\n c10,\n c11,\n c12,\n c13,\n c2,\n c3,\n c4,\n c5,\n c6,\n c7,\n c8,\n c9,\n ci,\n kf,\n t0,\n y0,\n yb1,\n yb2,\n yb3,\n z0,\n ):\n ybar = (1 / y0) * yb1 * z * v / (yb2 * x + yb3 + kf)\n if x < 0.0:\n x = 0\n rf = (ci - z0 * z) * np.sqrt(x)\n xdot = c1 * x * ybar + c2 * ybar + c3 * x ** 2 + c4 * rf + c5 * x * z - kf * x\n zdot = (c6 / z0) * rf + c7 * x * z + c8 * z * v + c9 * z - kf * z\n vdot = c10 * x * ybar + c11 * ybar + c12 * x ** 2 + c13 * z * v - kf * v\n return xdot * t0, zdot * t0, vdot * t0\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.BickleyJet","page":"Home","title":"ChaoticDynamicalSystemsLibrary.BickleyJet","text":" BickleyJet()\n\nA zonal jet passing between two counter rotating vortices. A steady Hamiltonian with a time-dependent perturbation.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: true\nKaplan-Yorke dimension: 4.066710025133773\nLyapunov spectrum (estimated): Any[1.9651794377408363e-7, 2.5409880958463764e-9, -9.631983584415862e-8]\nMaximum Lyapunov exponent (estimated): 3.0318672544949706e-6\nMultiscale Entropy: 0.733566708397195\nNon-autonomous: false\nPeriod: 87826.57\nPesin entropy: 1.99059498764899e-7\n\nCitation\n\nHadjighasem, Karrasch, Teramoto, Haller (2016). A Spectral Clustering Approach to Lagrangian Vortex Detection Phys Rev E.\n\nOriginal Python Code from the dysts package\n\nclass BickleyJet(DynSys):\n @staticjit\n def _rhs(y, x, z, t, ell, eps, k, omega, sigma, u):\n sechy = 1 / np.cosh(y / ell)\n inds = np.arange(3)\n un = k[inds] * (x - z * sigma[inds])\n dx = u * sechy ** 2 * (-1 - 2 * np.dot(np.cos(un), eps) * np.tanh(y / ell))\n dy = ell * u * sechy ** 2 * np.dot(eps * k, np.sin(un))\n dz = omega\n return dy, dx, dz\n\n def _postprocessing(self, x, y, z):\n km = np.min(self.k)\n sm = np.min(self.sigma)\n return x, np.sin(km * y), np.sin(self.omega * z * km * sm)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Blasius","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Blasius","text":" Blasius()\n\nA chaotic food web composed of interacting predators,, consumers, and vegetation.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.016657959617314\nLyapunov spectrum (estimated): Any[0.04712755400915782, 0.001803137034352918, -2.9362739640325555]\nMaximum Lyapunov exponent (estimated): 0.12597339364128174\nMultiscale Entropy: 0.5917316835585642\nNon-autonomous: false\nPeriod: 6.6038\nPesin entropy: 0.04893069104351076\n\nCitation\n\nBlasius, Huppert, Stone. Nature 1999\n\nOriginal Python Code from the dysts package\n\nclass Blasius(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, alpha1, alpha2, b, c, k1, k2, zs):\n xdot = a * x - alpha1 * x * y / (1 + k1 * x)\n ydot = -b * y + alpha1 * x * y / (1 + k1 * x) - alpha2 * y * z / (1 + k2 * y)\n zdot = -c * (z - zs) + alpha2 * y * z / (1 + k2 * y)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.BlinkingRotlet","page":"Home","title":"ChaoticDynamicalSystemsLibrary.BlinkingRotlet","text":" BlinkingRotlet()\n\nThe location of the mixer is chosen so that there is a stagnation point at its symmetric complement. Solution is given in polar coordinates (r, theta).\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: true\nKaplan-Yorke dimension: 2.910902586026695\nLyapunov spectrum (estimated): Any[5.609358269449125, 0.007963961496808246, -6.16801583110239]\nMaximum Lyapunov exponent (estimated): 2.120338159738003\nMultiscale Entropy: 1.2895030395332947\nNon-autonomous: true\nPeriod: 2.9520625\nPesin entropy: 5.617567417119514\n\nCitation\n\nMeleshko & Aref. A blinking rotlet model for chaotic advection. Physics of Fluids 1996. See also Tallapragada and Sudarsanam PRE 2019\n\nOriginal Python Code from the dysts package\n\nclass BlinkingRotlet(DynSys):\n @staticjit\n def _rotlet(r, theta, a, b, bc):\n \"\"\"A rotlet velocity field\"\"\"\n kappa = a ** 2 + (b ** 2 * r ** 2) / a ** 2 - 2 * b * r * np.cos(theta)\n gamma = (1 - r ** 2 / a ** 2) * (a ** 2 - (b ** 2 * r ** 2) / a ** 2)\n iota = (b ** 2 * r) / a ** 2 - b * np.cos(theta)\n zeta = b ** 2 + r ** 2 - 2 * b * r * np.cos(theta)\n nu = a ** 2 + b ** 2 - (2 * b ** 2 * r ** 2) / a ** 2\n vr = b * np.sin(theta) * (-bc * (gamma / kappa ** 2) - 1 / kappa + 1 / zeta)\n vth = (\n bc * (gamma * iota) / kappa ** 2\n + bc * r * nu / (a ** 2 * kappa)\n + iota / kappa\n - (r - b * np.cos(theta)) / zeta\n )\n return vr, vth\n\n @staticjit\n def _protocol(t, tau, stiffness=20):\n return 0.5 + 0.5 * np.tanh(tau * stiffness * np.sin(2 * np.pi * t / tau))\n\n def rhs(self, X, t):\n r, theta, tt = X\n weight = self._protocol(tt, self.tau)\n dr1, dth1 = self._rotlet(r, theta, self.a, self.b, self.bc)\n dr2, dth2 = self._rotlet(r, theta, self.a, -self.b, self.bc)\n dr = weight * dr1 + (1 - weight) * dr2\n dth = (weight * dth1 + (1 - weight) * dth2) / r\n dtt = 1\n return self.sigma * dr, self.sigma * dth, dtt\n\n def _postprocessing(self, r, th, tt):\n return r * np.cos(th), r * np.sin(th), np.sin(2 * np.pi * tt / self.tau)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Bouali2","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Bouali2","text":" Bouali2()\n\nA modified economic cycle model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 16.471858210544983\nLyapunov spectrum (estimated): Any[0.020389225004628773, 0.0033455417942589724, -0.009602768742817189]\nMaximum Lyapunov exponent (estimated): 0.07681730213975516\nMultiscale Entropy: 0.4520009826326611\nNon-autonomous: false\nPeriod: 2.6779\nPesin entropy: 0.02439941982137089\n\nCitation\n\nBouali (1999). Feedback loop in extended Van der Pols equation applied to an economic model of cycles. International Journal of Bifurkation and Chaos, Vol. 9, No. 4\n\nOriginal Python Code from the dysts package\n\nclass Bouali2(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, bb, c, g, m, y0):\n xdot = a * y0 * x - a * x * y - b * z\n ydot = -g * y + g * y * x ** 2\n zdot = -1.5 * m * x + m * bb * x * z - c * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.BurkeShaw","page":"Home","title":"ChaoticDynamicalSystemsLibrary.BurkeShaw","text":" BurkeShaw()\n\nA scroll-like attractor with unique symmetry along the z axis.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2113447891991065\nLyapunov spectrum (estimated): Any[2.3453555864112117, 0.027056468824027846, -11.225316054517004]\nMaximum Lyapunov exponent (estimated): 2.324596351534631\nMultiscale Entropy: 0.906432166797223\nNon-autonomous: false\nPeriod: 0.78333\nPesin entropy: 2.37241205523524\n\nCitation\n\nShaw (1981). Zeitschrift fur Naturforschung.\n\nOriginal Python Code from the dysts package\n\nclass BurkeShaw(DynSys):\n @staticjit\n def _rhs(x, y, z, t, e, n):\n xdot = -n * x - n * y\n ydot = y - n * x * z\n zdot = n * x * y + e\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.CaTwoPlus","page":"Home","title":"ChaoticDynamicalSystemsLibrary.CaTwoPlus","text":" CaTwoPlus()\n\nIntracellular calcium ion oscillations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0647813731079956\nLyapunov spectrum (estimated): Any[0.5788123099350391, 0.022652249040266133, -9.284523260554844]\nMaximum Lyapunov exponent (estimated): 0.5802865327706731\nMultiscale Entropy: 0.5770296569860646\nNon-autonomous: false\nPeriod: 1.343\nPesin entropy: 0.6014645592138562\n\nCitation\n\nHouart, Dupont, Goldbeter. Bull Math Biol 1999.\n\nOriginal Python Code from the dysts package\n\nclass CaTwoPlus(DynSys):\n def rhs(self, X, t):\n z, y, a = X\n Vin = self.V0 + self.V1 * self.beta\n V2 = self.Vm2 * (z ** 2) / (self.K2 ** 2 + z ** 2)\n V3 = (\n (self.Vm3 * (z ** self.m) / (self.Kz ** self.m + z ** self.m))\n * (y ** 2 / (self.Ky ** 2 + y ** 2))\n * (a ** 4 / (self.Ka ** 4 + a ** 4))\n )\n V5 = (\n self.Vm5\n * (a ** self.p / (self.K5 ** self.p + a ** self.p))\n * (z ** self.n / (self.Kd ** self.n + z ** self.n))\n )\n zdot = Vin - V2 + V3 + self.kf * y - self.k * z\n ydot = V2 - V3 - self.kf * y\n adot = self.beta * self.V4 - V5 - self.eps * a\n return (zdot, ydot, adot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.CellCycle","page":"Home","title":"ChaoticDynamicalSystemsLibrary.CellCycle","text":" CellCycle()\n\nA simplified model of the cell cycle. The parameter Kim controls the bifurcations.\n\nStats\n\nEmbedding dimension: 6\nHamiltonian: false\nKaplan-Yorke dimension: 2.020810681242375\nLyapunov spectrum (estimated): Any[0.00469985418122158, -0.0003567612310661562, -0.20915644240415007, -0.43394905365820846, -2.850536314608979, -6.767614340964144]\nMaximum Lyapunov exponent (estimated): 0.010172128409635746\nMultiscale Entropy: 1.021736994304444\nNon-autonomous: false\nPeriod: 204.71\nPesin entropy: 0.004765100518900255\n\nCitation\n\nRomond, Rustici, Gonze, Goldbeter. 1999.\n\nOriginal Python Code from the dysts package\n\nclass CellCycle(DynSys):\n def rhs(self, X, t):\n c1, m1, x1, c2, m2, x2 = X\n Vm1, Um1 = 2 * [self.Vm1]\n vi1, vi2 = 2 * [self.vi]\n H1, H2, H3, H4 = 4 * [self.K]\n K1, K2, K3, K4 = 4 * [self.K]\n V2, U2 = 2 * [self.V2]\n Vm3, Um3 = 2 * [self.Vm3]\n V4, U4 = 2 * [self.V4]\n Kc1, Kc2 = 2 * [self.Kc]\n vd1, vd2 = 2 * [self.vd]\n Kd1, Kd2 = 2 * [self.Kd1]\n kd1, kd2 = 2 * [self.kd1]\n Kim1, Kim2 = 2 * [self.Kim]\n V1 = Vm1 * c1 / (Kc1 + c1)\n U1 = Um1 * c2 / (Kc2 + c2)\n V3 = m1 * Vm3\n U3 = m2 * Um3\n c1dot = vi1 * Kim1 / (Kim1 + m2) - vd1 * x1 * c1 / (Kd1 + c1) - kd1 * c1\n c2dot = vi2 * Kim2 / (Kim2 + m1) - vd2 * x2 * c2 / (Kd2 + c2) - kd2 * c2\n m1dot = V1 * (1 - m1) / (K1 + (1 - m1)) - V2 * m1 / (K2 + m1)\n m2dot = U1 * (1 - m2) / (H1 + (1 - m2)) - U2 * m2 / (H2 + m2)\n x1dot = V3 * (1 - x1) / (K3 + (1 - x1)) - V4 * x1 / (K4 + x1)\n x2dot = U3 * (1 - x2) / (H3 + (1 - x2)) - U4 * x2 / (H4 + x2)\n return c1dot, m1dot, x1dot, c2dot, m2dot, x2dot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.CellularNeuralNetwork","page":"Home","title":"ChaoticDynamicalSystemsLibrary.CellularNeuralNetwork","text":" CellularNeuralNetwork()\n\nCellular neural network dynamics.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.167668286708633\nLyapunov spectrum (estimated): Any[0.09799656528996928, 0.0036354952626342415, -0.6061495728157424]\nMaximum Lyapunov exponent (estimated): 2.3258468977096562\nMultiscale Entropy: 1.2414841874061011\nNon-autonomous: false\nPeriod: 3.2\nPesin entropy: 0.1016320623247746\n\nCitation\n\nArena, Caponetto, Fortuna, and Porto., Int J Bifurcat Chaos. 1998: 1527-1539.\n\nOriginal Python Code from the dysts package\n\nclass CellularNeuralNetwork(DynSys):\n @staticjit\n def f(x):\n return 0.5 * (np.abs(x + 1) - np.abs(x - 1))\n\n def rhs(self, X, t):\n x, y, z = X\n xdot = -x + self.d * self.f(x) - self.b * self.f(y) - self.b * self.f(z)\n ydot = -y - self.b * self.f(x) + self.c * self.f(y) - self.a * self.f(z)\n zdot = -z - self.b * self.f(x) + self.a * self.f(y) + self.f(z)\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Chen","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Chen","text":" Chen()\n\nA system based on feedback anti-control in engineering.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.168621926035506\nLyapunov spectrum (estimated): Any[2.0471208808332557, -0.05031354288886987, -11.841816202859391]\nMaximum Lyapunov exponent (estimated): 2.005961349488129\nMultiscale Entropy: 0.7604529380690989\nNon-autonomous: false\nPeriod: 0.58689\nPesin entropy: 2.047313967129477\n\nCitation\n\nChen (1997). Proc. First Int. Conf. Control of Oscillations and Chaos.\n\nOriginal Python Code from the dysts package\n\nclass Chen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * y - a * x\n ydot = (c - a) * x - x * z + c * y\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ChenLee","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ChenLee","text":" ChenLee()\n\nA rigid body with feedback anti-control.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0300416878282137\nLyapunov spectrum (estimated): Any[0.16460631102343384, 0.00018225769802326938, -5.485329903201408]\nMaximum Lyapunov exponent (estimated): 0.16363115430160305\nMultiscale Entropy: 0.6224325755254609\nNon-autonomous: false\nPeriod: 2.8043\nPesin entropy: 0.16478856872146466\n\nCitation\n\nChen HK, Lee CI (2004). Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals\n\nOriginal Python Code from the dysts package\n\nclass ChenLee(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * x - y * z\n ydot = b * y + x * z\n zdot = c * z + 0.3333333333333333333333333 * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Chua","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Chua","text":" Chua()\n\nAn electronic circuit with a diode providing negative resistance.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.10727056025829\nLyapunov spectrum (estimated): Any[0.4605474829443073, 0.0022048281168492866, -4.319076735248446]\nMaximum Lyapunov exponent (estimated): 1.2301163591725595\nMultiscale Entropy: 1.2693527097629784\nNon-autonomous: false\nPeriod: 1.4235\nPesin entropy: 0.4645707969482971\n\nCitation\n\nChua, L. O. (1969) Introduction to Nonlinear Network Theory, McGraw-Hill, New York.\n\nOriginal Python Code from the dysts package\n\nclass Chua(DynSys):\n @staticjit\n def _rhs(x, y, z, t, alpha, beta, m0, m1):\n ramp_x = m1 * x + 0.5 * (m0 - m1) * (np.abs(x + 1) - np.abs(x - 1))\n xdot = alpha * (y - x - ramp_x)\n ydot = x - y + z\n zdot = -beta * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.CircadianRhythm","page":"Home","title":"ChaoticDynamicalSystemsLibrary.CircadianRhythm","text":" CircadianRhythm()\n\nThe Drosophila circadian rhythm under periodic light/dark forcing.\n\nStats\n\nEmbedding dimension: 5\nHamiltonian: false\nKaplan-Yorke dimension: 2.640554412377753\nLyapunov spectrum (estimated): Any[0.09298369588063138, 0.00046501992305171574, -0.14586025062784352, -0.3598453092044599, -1.091393481159416]\nMaximum Lyapunov exponent (estimated): 0.006263277393890506\nMultiscale Entropy: 0.7054585345031309\nNon-autonomous: true\nPeriod: 27.0\nPesin entropy: 0.09372724379002938\n\nCitation\n\nLeloup, Gonze, Goldbeter. 1999. Gonze, Leloup, Goldbeter. 2000\n\nOriginal Python Code from the dysts package\n\nclass CircadianRhythm(DynSys):\n @staticjit\n def _rhs(\n m,\n fc,\n fs,\n fn,\n th,\n t,\n Ki,\n k,\n k1,\n k2,\n kd,\n kdn,\n km,\n ks,\n n,\n vd,\n vdn,\n vm,\n vmax,\n vmin,\n v,\n ):\n vs = 2.5 * ((0.5 + 0.5 * np.cos(th)) + vmin) * (vmax - vmin)\n mdot = vs * (Ki ** n) / (Ki ** n + fn ** n) - vm * m / (km + m)\n fcdot = ks * m - k1 * fc + k2 * fn - k * fc\n fsdot = k * fc - vd * fs / (kd + fs)\n fndot = k1 * fc - k2 * fn - vdn * fn / (kdn + fn)\n thdot = 2 * np.pi / 24\n return mdot, fcdot, fsdot, fndot, thdot\n\n @staticjit\n def _postprocessing(m, fc, fs, fn, th):\n return m, fc, fs, fn, np.cos(th)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.CoevolvingPredatorPrey","page":"Home","title":"ChaoticDynamicalSystemsLibrary.CoevolvingPredatorPrey","text":" CoevolvingPredatorPrey()\n\nA system of predator-prey equations with co-evolving prey.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0026539580881706\nLyapunov spectrum (estimated): Any[0.0034879428860592264, 0.0017391810332012293, -1.9676998121970224]\nMaximum Lyapunov exponent (estimated): 0.00732698039035783\nMultiscale Entropy: 1.006450625395487\nNon-autonomous: false\nPeriod: 122.77\nPesin entropy: 0.005379136261058156\n\nCitation\n\nGilpin & Feldman (2017). PLOS Comp Biol\n\nOriginal Python Code from the dysts package\n\nclass CoevolvingPredatorPrey(DynSys):\n @staticjit\n def _rhs(x, y, alpha, t, a1, a2, a3, b1, b2, d1, d2, delta, k1, k2, k4, vv):\n xdot = x * (\n -((a3 * y) / (1 + b2 * x))\n + (a1 * alpha * (1 - k1 * x * (-alpha + alpha * delta))) / (1 + b1 * alpha)\n - d1\n * (\n 1\n - k2 * (-(alpha ** 2) + (alpha * delta) ** 2)\n + k4 * (-(alpha ** 4) + (alpha * delta) ** 4)\n )\n )\n ydot = (-d2 + (a2 * x) / (1 + b2 * x)) * y\n alphadot = vv * (\n -((a1 * k1 * x * alpha * delta) / (1 + b1 * alpha))\n - d1 * (-2 * k2 * alpha * delta ** 2 + 4 * k4 * alpha ** 3 * delta ** 4)\n )\n return xdot, ydot, alphadot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Colpitts","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Colpitts","text":" Colpitts()\n\nAn electrical circuit used as a signal generator.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.134959031654127\nLyapunov spectrum (estimated): Any[0.13750285290669104, -0.0021021241106901328, -1.0014048629576902]\nMaximum Lyapunov exponent (estimated): 0.3736008151161584\nMultiscale Entropy: 0.7266290517682168\nNon-autonomous: false\nPeriod: 4.2676\nPesin entropy: 0.1390751886208436\n\nCitation\n\nKennedy (2007). IEEE Trans Circuits & Systems. 1994: 771-774. Li, Zhou, Yang. Chaos, Solitons, & Fractals.\n\nOriginal Python Code from the dysts package\n\nclass Colpitts(DynSys):\n def rhs(self, X, t):\n x, y, z = X\n u = z - (self.e - 1)\n fz = -u * (1 - np.heaviside(u, 0))\n xdot = y - self.a * fz\n ydot = self.c - x - self.b * y - z\n zdot = y - self.d * z\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Coullet","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Coullet","text":" Coullet()\n\nA variant of the Arneodo attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.252992250313846\nLyapunov spectrum (estimated): Any[0.14890693833273877, -6.504894715051528e-5, -0.5873658566804084]\nMaximum Lyapunov exponent (estimated): 0.13718762024831516\nMultiscale Entropy: 0.9055084400125268\nNon-autonomous: false\nPeriod: 5.7663\nPesin entropy: 0.14919849957322842\n\nCitation\n\nArneodo, A., Coullet, P. & Tresser, C. Occurence of strange attractors in three-dimensional volterra equations. Physics Letters A 79, 259–263 (1980)\n\nOriginal Python Code from the dysts package\n\nclass Rucklidge(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x + b * y - y * z\n ydot = x\n zdot = -z + y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Dadras","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Dadras","text":" Dadras()\n\nAn electronic circuit capable of producing multiple attractors under bifurcations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0430103150081735\nLyapunov spectrum (estimated): Any[0.47344233096408, 0.0064562720284210215, -11.157725546142514]\nMaximum Lyapunov exponent (estimated): 0.40591521049520063\nMultiscale Entropy: 0.8537980405426894\nNon-autonomous: false\nPeriod: 3.2886\nPesin entropy: 0.4798988304624064\n\nCitation\n\nS Dadras, HR Momeni (2009). A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Physics Letters A.\n\nOriginal Python Code from the dysts package\n\nclass Dadras(DynSys):\n @staticjit\n def _rhs(x, y, z, t, c, e, o, p, r):\n xdot = y - p * x + o * y * z\n ydot = r * y - x * z + z\n zdot = c * x * y - e * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.DequanLi","page":"Home","title":"ChaoticDynamicalSystemsLibrary.DequanLi","text":" DequanLi()\n\nRelated to the Three Scroll unified attractor TSUCS-2\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.6848604732863537\nLyapunov spectrum (estimated): Any[1.8601963396884453, 0.09866428540326228, -2.867185439620747]\nMaximum Lyapunov exponent (estimated): 0.739747976641268\nMultiscale Entropy: 0.8319369667525882\nNon-autonomous: false\nPeriod: 0.083806\nPesin entropy: 1.9588606250917073\n\nCitation\n\nLi, Phys Lett A. 2008: 387-393.\n\nOriginal Python Code from the dysts package\n\nclass DequanLi(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, c, d, eps, f, k):\n xdot = a * y - a * x + d * x * z\n ydot = k * x + f * y - x * z\n zdot = c * z + x * y - eps * x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.DoubleGyre","page":"Home","title":"ChaoticDynamicalSystemsLibrary.DoubleGyre","text":" DoubleGyre()\n\nA time-dependent fluid flow exhibiting Lagrangian coherent structures\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.881525629972418\nLyapunov spectrum (estimated): Any[0.06285936079346421, -0.0036702088073069326, -0.06633987533400756]\nMaximum Lyapunov exponent (estimated): 0.12185254082886439\nMultiscale Entropy: 0.66519273658672\nNon-autonomous: true\nPeriod: 13.953\nPesin entropy: 0.06285936079346421\n\nCitation\n\nShadden, Lekien, Marsden (2005). Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D.\n\nOriginal Python Code from the dysts package\n\nclass DoubleGyre(DynSys):\n @staticjit\n def _rhs(x, y, z, t, alpha, eps, omega):\n a = eps * np.sin(z)\n b = 1 - 2 * eps * np.sin(z)\n f = a * x ** 2 + b * x\n dx = -alpha * np.pi * np.sin(np.pi * f) * np.cos(np.pi * y)\n dy = alpha * np.pi * np.cos(np.pi * f) * np.sin(np.pi * y) * (2 * a * x + b)\n dz = omega\n return dx, dy, dz\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.DoublePendulum","page":"Home","title":"ChaoticDynamicalSystemsLibrary.DoublePendulum","text":" DoublePendulum()\n\nTwo coupled rigid pendula without damping.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: true\nKaplan-Yorke dimension: 3.810221473882809\nLyapunov spectrum (estimated): Any[8.464443924343541, 0.46411672696182493, -0.17345525567611142, -10.805817517613212]\nMaximum Lyapunov exponent (estimated): 3.985298067514186\nMultiscale Entropy: 0.5591104791210428\nNon-autonomous: false\nPeriod: 4.0234\nPesin entropy: 8.928560651305366\n\nCitation\n\nSee, for example: Marion (2013). Classical dynamics of particles and systems.\n\nOriginal Python Code from the dysts package\n\nclass DoublePendulum(DynSys):\n @staticjit\n def _rhs(th1, th2, p1, p2, t, d, m):\n g = 9.82\n pre = 6 / (m * d ** 2)\n denom = 16 - 9 * np.cos(th1 - th2) ** 2\n th1_dot = pre * (2 * p1 - 3 * np.cos(th1 - th2) * p2) / denom\n th2_dot = pre * (8 * p2 - 3 * np.cos(th1 - th2) * p1) / denom\n p1_dot = (\n -0.5\n * (m * d ** 2)\n * (th1_dot * th2_dot * np.sin(th1 - th2) + 3 * (g / d) * np.sin(th1))\n )\n p2_dot = (\n -0.5\n * (m * d ** 2)\n * (-th1_dot * th2_dot * np.sin(th1 - th2) + 3 * (g / d) * np.sin(th2))\n )\n return th1_dot, th2_dot, p1_dot, p2_dot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Duffing","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Duffing","text":" Duffing()\n\nA monochromatically-forced rigid pendulum, with the nonlinearity approximated as cubic.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.561219319140712\nLyapunov spectrum (estimated): Any[0.09952054532941002, 0.0003077550037828977, -0.17595301607629793]\nMaximum Lyapunov exponent (estimated): 0.14026545752865122\nMultiscale Entropy: 0.8686568999981737\nNon-autonomous: true\nPeriod: 7.4496\nPesin entropy: 0.10008657388524086\n\nCitation\n\nDuffing, G. (1918), Forced oscillations with variable natural frequency and their technical relevance, Heft 41/42, Braunschweig: Vieweg.\n\nOriginal Python Code from the dysts package\n\nclass Duffing(DynSys):\n @staticjit\n def _rhs(x, y, z, t, alpha, beta, delta, gamma, omega):\n xdot = y\n ydot = -delta * y - beta * x - alpha * x ** 3 + gamma * np.cos(z)\n zdot = omega\n return xdot, ydot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.cos(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ExcitableCell","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ExcitableCell","text":" ExcitableCell()\n\nA reduced-order variant of the Hodgkin-Huxley model. The parameter gkc controls the onset of chaos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1287842177704674\nLyapunov spectrum (estimated): Any[17.030315084501463, -0.3267053778962445, -129.70230355691595]\nMaximum Lyapunov exponent (estimated): 0.7164303367057223\nMultiscale Entropy: 0.7807524009298388\nNon-autonomous: false\nPeriod: 0.5528\nPesin entropy: 17.030315084501463\n\nCitation\n\nTeresa Chay. Chaos In A Three-variable Model Of An Excitable Cell. Physica D 1985\n\nOriginal Python Code from the dysts package\n\nclass ExcitableCell(DynSys):\n def rhs(self, X, t):\n v, n, c = X\n\n alpham = 0.1 * (25 + v) / (1 - np.exp(-0.1 * v - 2.5))\n betam = 4 * np.exp(-(v + 50) / 18)\n minf = alpham / (alpham + betam)\n\n alphah = 0.07 * np.exp(-0.05 * v - 2.5)\n betah = 1 / (1 + np.exp(-0.1 * v - 2))\n hinf = alphah / (alphah + betah)\n\n alphan = 0.01 * (20 + v) / (1 - np.exp(-0.1 * v - 2))\n betan = 0.125 * np.exp(-(v + 30) / 80)\n ninf = alphan / (alphan + betan)\n tau = 1 / (230 * (alphan + betan))\n\n ca = c / (1 + c)\n\n vdot = (\n self.gi * minf ** 3 * hinf * (self.vi - v)\n + self.gkv * n ** 4 * (self.vk - v)\n + self.gkc * ca * (self.vk - v)\n + self.gl * (self.vl - v)\n )\n ndot = (ninf - n) / tau\n cdot = self.rho * (minf ** 3 * hinf * (self.vc - v) - self.kc * c)\n return vdot, ndot, cdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Finance","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Finance","text":" Finance()\n\nStock fluctuations under varying investment demand and interest rates\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.172040321306389\nLyapunov spectrum (estimated): Any[0.10708598383274774, 0.0009835797228343356, -0.6279525877233353]\nMaximum Lyapunov exponent (estimated): 0.09354106442670211\nMultiscale Entropy: 0.7817861077722251\nNon-autonomous: false\nPeriod: 9.8932\nPesin entropy: 0.1084488028544319\n\nCitation\n\nGuoliang Cai, Juanjuan Huang. International Journal of Nonlinear Science, Vol. 3 (2007)\n\nOriginal Python Code from the dysts package\n\nclass Finance(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = (1 / b - a) * x + z + x * y\n ydot = -b * y - x ** 2\n zdot = -x - c * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.FluidTrampoline","page":"Home","title":"ChaoticDynamicalSystemsLibrary.FluidTrampoline","text":" FluidTrampoline()\n\nA droplet bouncing on a horizontal soap film.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.8462636972682116\nLyapunov spectrum (estimated): Any[0.08081301489037329, -0.0011783910877478326, -0.09403052188090477]\nMaximum Lyapunov exponent (estimated): 0.22311237411461174\nMultiscale Entropy: 0.796312585554829\nNon-autonomous: true\nPeriod: 7.7351\nPesin entropy: 0.08094585172323163\n\nCitation\n\nGilet, Bush. The fluid trampoline: droplets bouncing on a soap film. JFM 2009.\n\nOriginal Python Code from the dysts package\n\nclass FluidTrampoline(DynSys):\n @staticmethod\n def _rhs(x, y, th, t, gamma, psi, w):\n xdot = y\n ydot = -1 - np.heaviside(-x, 0) * (x + psi * y * np.abs(y)) + gamma * np.cos(th)\n thdot = w\n return (xdot, ydot, thdot)\n\n @staticjit\n def _postprocessing(x, y, th):\n return x, y, np.cos(th)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ForcedBrusselator","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ForcedBrusselator","text":" ForcedBrusselator()\n\nAn autocatalytic chemical system.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.190019824057157\nLyapunov spectrum (estimated): Any[0.0028307087627546254, -0.014898681909916936, -0.23304448314240073]\nMaximum Lyapunov exponent (estimated): 0.015326007974604663\nMultiscale Entropy: 0.47299463051887286\nNon-autonomous: true\nPeriod: 20.411\nPesin entropy: 0.0028307087627546254\n\nCitation\n\nI. Prigogine, From Being to Becoming: Time and Complexity in the Physical Sciences, New York: W.H. Freeman and Company, 1980.\n\nOriginal Python Code from the dysts package\n\nclass ForcedBrusselator(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, f, w):\n xdot = a + x ** 2 * y - (b + 1) * x + f * np.cos(z)\n ydot = b * x - x ** 2 * y\n zdot = w\n return xdot, ydot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ForcedFitzHughNagumo","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ForcedFitzHughNagumo","text":" ForcedFitzHughNagumo()\n\nA driven neuron model sustaining both quiesent and oscillating states.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.5705852319395741\nLyapunov spectrum (estimated): Any[0.0006271045837994497, -0.0017541532237019502, -0.9544351506941636]\nMaximum Lyapunov exponent (estimated): 0.007561278655908403\nMultiscale Entropy: 0.946734422406113\nNon-autonomous: true\nPeriod: 42.12235294117647\nPesin entropy: 0.000828176101772673\n\nCitation\n\nFitzHugh, Richard (1961). Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal. 1 (6).\n\nOriginal Python Code from the dysts package\n\nclass ForcedFitzHughNagumo(DynSys):\n @staticjit\n def _rhs(v, w, z, t, a, b, curr, f, gamma, omega):\n vdot = v - v ** 3 / 3 - w + curr + f * np.sin(z)\n wdot = gamma * (v + a - b * w)\n zdot = omega\n return vdot, wdot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ForcedVanDerPol","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ForcedVanDerPol","text":" ForcedVanDerPol()\n\nAn electronic circuit containing a triode.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.012828649055009\nLyapunov spectrum (estimated): Any[0.2651824388815116, -0.0008450617218488057, -20.60632249066986]\nMaximum Lyapunov exponent (estimated): 0.0035097622587646343\nMultiscale Entropy: 0.39443058905146283\nNon-autonomous: true\nPeriod: 14.476875\nPesin entropy: 0.26518243888151155\n\nCitation\n\nB. van der Pol (1920). A theory of the amplitude of free and forced triode vibrations. Radio Review 1.\n\nOriginal Python Code from the dysts package\n\nclass ForcedVanDerPol(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, mu, w):\n ydot = mu * (1 - x ** 2) * y - x + a * np.sin(z)\n xdot = y\n zdot = w\n return xdot, ydot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.GenesioTesi","page":"Home","title":"ChaoticDynamicalSystemsLibrary.GenesioTesi","text":" GenesioTesi()\n\nA nonlinear control system with feedback.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1892507433089325\nLyapunov spectrum (estimated): Any[0.09974381765927798, 0.00038146676582121423, -0.5282231255530121]\nMaximum Lyapunov exponent (estimated): 0.10509061137519304\nMultiscale Entropy: 0.7003576241153363\nNon-autonomous: false\nPeriod: 5.9431\nPesin entropy: 0.10027488648885732\n\nCitation\n\nGenesio, Tesi (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28.3 (1992): 531-548.\n\nOriginal Python Code from the dysts package\n\nclass GenesioTesi(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = y\n ydot = z\n zdot = -c * x - b * y - a * z + x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.GuckenheimerHolmes","page":"Home","title":"ChaoticDynamicalSystemsLibrary.GuckenheimerHolmes","text":" GuckenheimerHolmes()\n\nA nonlinear oscillator.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.3663164666051353\nLyapunov spectrum (estimated): Any[0.34082003567895874, -0.0032191807339425173, -0.9216135618098852]\nMaximum Lyapunov exponent (estimated): 0.7885729042404602\nMultiscale Entropy: 0.5385101182847213\nNon-autonomous: false\nPeriod: 0.3076923076923077\nPesin entropy: 0.3408200356941137\n\nCitation\n\nGuckenheimer, John, and Philip Holmes (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42. Springer Science & Business Media.\n\nOriginal Python Code from the dysts package\n\nclass GuckenheimerHolmes(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, e, f):\n xdot = a * x - b * y + c * z * x + d * z * x ** 2 + d * z * y ** 2\n ydot = a * y + b * x + c * z * y\n zdot = e - z ** 2 - f * x ** 2 - f * y ** 2 - a * z ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Hadley","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Hadley","text":" Hadley()\n\nAn atmospheric convective cell.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.7273838319442127\nLyapunov spectrum (estimated): Any[0.31166133927315953, -0.0010838827053467116, -0.42739317979015523]\nMaximum Lyapunov exponent (estimated): 0.2386587703845738\nMultiscale Entropy: 0.9351389987191853\nNon-autonomous: false\nPeriod: 1.4541\nPesin entropy: 0.3144504334934165\n\nCitation\n\nG. Hadley (1735). On the cause of the general trade winds. Philosophical Transactions of the Royal Society, vol. 34, pp. 58–62.\n\nOriginal Python Code from the dysts package\n\nclass Hadley(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, f, g):\n xdot = -y ** 2 - z ** 2 - a * x + a * f\n ydot = x * y - b * x * z - y + g\n zdot = b * x * y + x * z - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Halvorsen","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Halvorsen","text":" Halvorsen()\n\nAn algebraically-simple chaotic system with quadratic nonlinearity\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.118235268907861\nLyapunov spectrum (estimated): Any[0.586197150399321, -0.031953945256851884, -4.6876300977027565]\nMaximum Lyapunov exponent (estimated): 0.6963711539102683\nMultiscale Entropy: 0.7527831765757674\nNon-autonomous: false\nPeriod: 1.4889\nPesin entropy: 0.586197150399321\n\nCitation\n\nSprott, Julien C (2010). Elegant chaos: algebraically simple chaotic flows. World Scientific, 2010.\n\nOriginal Python Code from the dysts package\n\nclass Halvorsen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x - b * y - b * z - y ** 2\n ydot = -a * y - b * z - b * x - z ** 2\n zdot = -a * z - b * x - b * y - x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HastingsPowell","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HastingsPowell","text":" HastingsPowell()\n\nA three species food web.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.027757892627057\nLyapunov spectrum (estimated): Any[0.008585012825774563, 0.005765040490702786, -0.5169232002224432]\nMaximum Lyapunov exponent (estimated): 0.012532796124563227\nMultiscale Entropy: 0.47829796264231095\nNon-autonomous: false\nPeriod: 135.19\nPesin entropy: 0.014350053316477346\n\nCitation\n\nHastings, Powell. Ecology 1991\n\nOriginal Python Code from the dysts package\n\nclass HastingsPowell(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a1, a2, b1, b2, d1, d2):\n xdot = x * (1 - x) - y * a1 * x / (1 + b1 * x)\n ydot = y * a1 * x / (1 + b1 * x) - z * a2 * y / (1 + b2 * y) - d1 * y\n zdot = z * a2 * y / (1 + b2 * y) - d2 * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HenonHeiles","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HenonHeiles","text":" HenonHeiles()\n\nA star's motion around the galactic center.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 6.81141589838772\nLyapunov spectrum (estimated): Any[0.026747154244887203, 0.011500891097175434, -1.4057113933414076e-5, -0.011151041415888588]\nMaximum Lyapunov exponent (estimated): 0.03551120315239528\nMultiscale Entropy: 0.7137794258156454\nNon-autonomous: false\nPeriod: 6.37\nPesin entropy: 0.03870865064107525\n\nCitation\n\nHenon, M.; Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astronomical Journal. 69: 73.\n\nOriginal Python Code from the dysts package\n\nclass HenonHeiles(DynSys):\n @staticjit\n def _rhs(x, y, px, py, t, lam):\n xdot = px\n ydot = py\n pxdot = -x - 2 * lam * x * y\n pydot = -y - lam * x ** 2 + lam * y ** 2\n return xdot, ydot, pxdot, pydot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HindmarshRose","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HindmarshRose","text":" HindmarshRose()\n\nA neuron model exhibiting spiking and bursting.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0405369895957373\nLyapunov spectrum (estimated): Any[0.2621464274609406, 0.0017705383465481184, -6.510663065519547]\nMaximum Lyapunov exponent (estimated): 0.6535393283242747\nMultiscale Entropy: 0.8692605688019491\nNon-autonomous: false\nPeriod: 1.752\nPesin entropy: 0.2639169709155772\n\nCitation\n\nMarhl, Perc. Chaos, Solitons, Fractals 2005.\n\nOriginal Python Code from the dysts package\n\nclass HindmarshRose(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, s, tx, tz):\n xdot = -x + 1 / tx * y - a / tx * x ** 3 + b / tx * x ** 2 + 1 / tx * z\n ydot = -a * x ** 3 - (d - b) * x ** 2 + z\n zdot = -s / tz * x - 1 / tz * z + c / tz\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Hopfield","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Hopfield","text":" Hopfield()\n\nA neural network with frustrated connectivity\n\nStats\n\nEmbedding dimension: 6\nHamiltonian: false\nKaplan-Yorke dimension: 2.264479110541058\nLyapunov spectrum (estimated): Any[0.07726437124258394, -0.00018152933777384922, -0.29083324580448194, -0.36906925337474944, -0.40604718386075256, -1.4425404449511432]\nMaximum Lyapunov exponent (estimated): 0.5684266951234997\nMultiscale Entropy: 0.9160145554679957\nNon-autonomous: false\nPeriod: 10.865\nPesin entropy: 0.07748903749076723\n\nCitation\n\nLewis & Glass, Neur Comp (1992)\n\nOriginal Python Code from the dysts package\n\nclass Hopfield(DynSys):\n def f(self, x):\n return (1 + np.tanh(x)) / 2\n\n def rhs(self, X, t):\n Xdot = -X / self.tau + self.f(self.eps * np.matmul(self.k, X)) - self.beta\n return Xdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperBao","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperBao","text":" HyperBao()\n\nHyperchaos in the Lu system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.0460429607933572\nLyapunov spectrum (estimated): Any[0.6959739496471207, 0.2076688444499756, 0.01646766869238271, -19.983737503321315]\nMaximum Lyapunov exponent (estimated): 0.7941925747845208\nMultiscale Entropy: 0.8910215561062699\nNon-autonomous: false\nPeriod: 0.5831505997603473\nPesin entropy: 0.9201105075982862\n\nCitation\n\nBao, Liu (2008). A hyperchaotic attractor coined from the chaotic Lu system. Chin. Physics Letters, Vol. 25, pp 2396-2399.\n\nOriginal Python Code from the dysts package\n\nclass HyperBao(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d, e):\n xdot = a * y - a * x + w\n ydot = c * y - x * z\n zdot = x * y - b * z\n wdot = e * x + d * y * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperCai","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperCai","text":" HyperCai()\n\nA hyperchaotic variant of the Finance system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.138030987910704\nLyapunov spectrum (estimated): Any[1.634443623908957, 0.12845528254618424, -0.003328688360603957, -12.747646341795276]\nMaximum Lyapunov exponent (estimated): 1.6777584899044127\nMultiscale Entropy: 1.1155413734604414\nNon-autonomous: false\nPeriod: 0.6113936187028436\nPesin entropy: 1.7628989064551457\n\nCitation\n\nGuoliang, Huang (2007). A New Finance Chaotic Attractor. International Journal of Nonlinear Science.\n\nOriginal Python Code from the dysts package\n\nclass HyperCai(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d, e):\n xdot = a * y - a * x\n ydot = b * x + c * y - x * z + w\n zdot = -d * z + y ** 2\n wdot = -e * x\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperJha","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperJha","text":" HyperJha()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.053181782412727\nLyapunov spectrum (estimated): Any[0.3949499860671474, 0.2920214751184925, 0.005568431034995312, -13.022125042186726]\nMaximum Lyapunov exponent (estimated): 0.38695361636760767\nMultiscale Entropy: 0.8934166449122942\nNon-autonomous: false\nPeriod: 0.8564467295817622\nPesin entropy: 0.6925398922206352\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperJha(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + w\n ydot = -x * z + b * x - y\n zdot = x * y - c * z\n wdot = -x * z + d * w\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperLorenz","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperLorenz","text":" HyperLorenz()\n\nA hyperchaotic variant of the Lorenz attractor.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.044709906668086\nLyapunov spectrum (estimated): Any[0.2925643599949083, 0.24691553258701357, 0.04704434899468292, -13.118440302968889]\nMaximum Lyapunov exponent (estimated): 0.3288809596909799\nMultiscale Entropy: 0.8026458791332332\nNon-autonomous: false\nPeriod: 1.277546730110058\nPesin entropy: 0.5865242415766048\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperLorenz(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + w\n ydot = -x * z + c * x - y\n zdot = -b * z + x * y\n wdot = d * w - x * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperLu","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperLu","text":" HyperLu()\n\nA hyperchaotic variant of the Lu attractor.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.0822889656900583\nLyapunov spectrum (estimated): Any[1.0855770127041702, 0.4753850931706396, 0.02849951886859947, -19.313823203887658]\nMaximum Lyapunov exponent (estimated): 1.042961263138283\nMultiscale Entropy: 1.0455788608813867\nNon-autonomous: false\nPeriod: 0.7162856981996777\nPesin entropy: 1.5894616248138722\n\nCitation\n\nChen, A., Lu, J., Lü, J., & Yu, S. (2006). Generating hyperchaotic Lü attractor via state feedback control. Physica A: Statistical Mechanics and its Applications, 364, 103-110.\n\nOriginal Python Code from the dysts package\n\nclass HyperLu(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=36, b=3, c=20, d=1.3):\n xdot = a * y - a * x + w\n ydot = -x * z + c * y\n zdot = x * y - b * z\n wdot = d * w + x * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperPang","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperPang","text":" HyperPang()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.08902125132107\nLyapunov spectrum (estimated): Any[1.586932998381499, 0.28444192185263556, -0.0011541322141891326, -21.007696151519628]\nMaximum Lyapunov exponent (estimated): 1.4465315511406867\nMultiscale Entropy: 0.9172158388570456\nNon-autonomous: false\nPeriod: 0.8439374360017028\nPesin entropy: 1.871913394885623\n\nCitation\n\nPang, S., & Liu, Y. (2011). A new hyperchaotic system from the Lü system and its control. Journal of Computational and Applied Mathematics, 235(8), 2775-2789.\n\nOriginal Python Code from the dysts package\n\nclass HyperPang(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=36, b=3, c=20, d=2):\n xdot = a * y - a * x\n ydot = -x * z + c * y + w\n zdot = x * y - b * z\n wdot = -d * x - d * y\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperQi","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperQi","text":" HyperQi()\n\nA hyperchaotic variant of the Qi system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.2816770286126777\nLyapunov spectrum (estimated): Any[13.908799876284396, 2.8519979827853357, -0.04957904444451209, -59.32758839771818]\nMaximum Lyapunov exponent (estimated): 12.787249321134263\nMultiscale Entropy: 0.9988440965886223\nNon-autonomous: false\nPeriod: 0.06186038560173437\nPesin entropy: 16.760797859069733\n\nCitation\n\nG. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.\n\nOriginal Python Code from the dysts package\n\nclass HyperQi(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d, e, f):\n xdot = a * y - a * x + y * z\n ydot = b * x + b * y - x * z\n zdot = -c * z - e * w + x * y\n wdot = -d * w + f * z + x * y\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperRossler","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperRossler","text":" HyperRossler()\n\nA hyperchaotic variant of the Rossler system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.916384112225743\nLyapunov spectrum (estimated): Any[0.27859233648384435, -7.749198930652314e-5, -0.3014660985555502, -70.26609894565237]\nMaximum Lyapunov exponent (estimated): 0.1718849632568779\nMultiscale Entropy: 0.9144754695112332\nNon-autonomous: false\nPeriod: 1.1157476694793766\nPesin entropy: 0.2789695684743801\n\nCitation\n\nRossler, O. E. (1979). An equation for hyperchaos.\n\nOriginal Python Code from the dysts package\n\nclass HyperRossler(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = -y - z\n ydot = x + a * y + w\n zdot = b + x * z\n wdot = -c * z + d * w\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperWang","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperWang","text":" HyperWang()\n\nA hyperchaotic variant of the Wang system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.105321722354005\nLyapunov spectrum (estimated): Any[1.2224015619377895, 0.25827623672371736, -0.0075512136517132625, -13.986919236227278]\nMaximum Lyapunov exponent (estimated): 1.2930140887300063\nMultiscale Entropy: 0.9969261837212393\nNon-autonomous: false\nPeriod: 0.6481520567576332\nPesin entropy: 1.480677798661507\n\nCitation\n\nWang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.\n\nOriginal Python Code from the dysts package\n\nclass HyperWang(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=10, b=40, c=2.5, d=10.6, e=4):\n xdot = a * y - a * x\n ydot = -x * z + b * x + w\n zdot = -c * z + e * x ** 2\n wdot = -d * x\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperXu","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperXu","text":" HyperXu()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.0909471864561056\nLyapunov spectrum (estimated): Any[1.0420068843753962, 0.20717763316365526, -0.0033988115975040426, -13.697896294059921]\nMaximum Lyapunov exponent (estimated): 0.9819830018599975\nMultiscale Entropy: 0.9312323303231519\nNon-autonomous: false\nPeriod: 1.1328151255687646\nPesin entropy: 1.2491913312241498\n\nCitation\n\nLetellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.\n\nOriginal Python Code from the dysts package\n\nclass HyperXu(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=10, b=40, c=2.5, d=2, e=16):\n xdot = a * y - a * x + w\n ydot = b * x + e * x * z\n zdot = -c * z - x * y\n wdot = x * z - d * y\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperYan","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperYan","text":" HyperYan()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.0711118082329567\nLyapunov spectrum (estimated): Any[0.9120352761173002, 0.15303345953001665, -14.977382134147613, -39.36365990052555]\nMaximum Lyapunov exponent (estimated): 1.3945968455812914\nMultiscale Entropy: 0.8166157518949907\nNon-autonomous: false\nPeriod: 0.9127028626966961\nPesin entropy: 1.0650687356473167\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperYan(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=37, b=3, c=26, d=38):\n xdot = a * y - a * x\n ydot = (c - a) * x - x * z + c * y\n zdot = -b * z + x * y - y * z + x * z - w\n wdot = -d * w + y * z - x * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.HyperYangChen","page":"Home","title":"ChaoticDynamicalSystemsLibrary.HyperYangChen","text":" HyperYangChen()\n\nA hyperchaotic system.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.026605264361547\nLyapunov spectrum (estimated): Any[0.5742688696462735, 0.327365255173604, 0.023510192276255986, -34.77284163099938]\nMaximum Lyapunov exponent (estimated): 0.42042732521941667\nMultiscale Entropy: 0.9730322217427024\nNon-autonomous: false\nPeriod: 0.9127028626966961\nPesin entropy: 0.9255171125652686\n\nCitation\n\nJürgen Meier (2003). Presentation of Attractors with Cinema.\n\nOriginal Python Code from the dysts package\n\nclass HyperYangChen(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a=30, b=3, c=35, d=8):\n xdot = a * y - a * x\n ydot = c * x - x * z + w\n zdot = -b * z + x * y\n wdot = -d * x\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.IkedaDelay","page":"Home","title":"ChaoticDynamicalSystemsLibrary.IkedaDelay","text":" IkedaDelay()\n\nA passive optical resonator system. A standard delay dimension of three is used.\n\nStats\n\nEmbedding dimension: 10\nHamiltonian: false\nKaplan-Yorke dimension: 8.284295212220359\nLyapunov spectrum (estimated): Any[0.7139744431963619, 0.03258370277261943, -0.06766062243103049, -0.07744565521894307, -0.09054735418410836, -0.10900317234818346, -0.13696057562417352, -0.18439894250590672, -0.2833034824174493, -0.6392438255039592]\nMaximum Lyapunov exponent (estimated): 0.448298365567425\nNon-autonomous: false\nPeriod: 1.486528\nPesin entropy: 0.7465581459689814\n\nCitation\n\nK. Ikeda and K. Matsumoto (1987). High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D, 29 (1-2), 223-235\n\nOriginal Python Code from the dysts package\n\nclass IkedaDelay(DynSysDelay):\n @staticjit\n def _rhs(x, xt, t, c, mu, tau, x0):\n xdot = mu * np.sin(xt - x0) - c * x\n return xdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.IsothermalChemical","page":"Home","title":"ChaoticDynamicalSystemsLibrary.IsothermalChemical","text":" IsothermalChemical()\n\nAn isothermal chemical system with mixed-mode oscillations\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0385823810364654\nLyapunov spectrum (estimated): Any[0.3365871343421436, -0.05136527222093221, -7.392541736904205]\nMaximum Lyapunov exponent (estimated): 0.450090505359164\nMultiscale Entropy: 0.6589422003608746\nNon-autonomous: false\nPeriod: 0.7302799999999999\nPesin entropy: 0.3365871343421436\n\nCitation\n\nPetrov, Scott, Showalter. Mixed-mode oscillations in chemical systems. J Chem Phys 1992\n\nOriginal Python Code from the dysts package\n\nclass IsothermalChemical(DynSys):\n @staticmethod\n def _rhs(alpha, beta, gamma, t, delta, kappa, mu, sigma):\n alphadot = mu * (kappa + gamma) - alpha * beta ** 2 - alpha\n betadot = (alpha * beta ** 2 + alpha - beta) / sigma\n gammadot = (beta - gamma) / delta\n return alphadot, betadot, gammadot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ItikBanksTumor","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ItikBanksTumor","text":" ItikBanksTumor()\n\nA model of cancer cell populations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0152772613355485\nLyapunov spectrum (estimated): Any[0.022156938955395417, -0.0135394920593605, -0.5610833889492404]\nMaximum Lyapunov exponent (estimated): 0.03621509684991421\nMultiscale Entropy: 0.5883298730506914\nNon-autonomous: false\nPeriod: 49.74\nPesin entropy: 0.022156938955395417\n\nCitation\n\nItik, Banks. Int J Bifurcat Chaos 2010\n\nOriginal Python Code from the dysts package\n\nclass ItikBanksTumor(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a12, a13, a21, a31, d3, k3, r2, r3):\n xdot = x * (1 - x) - a12 * x * y - a13 * x * z\n ydot = r2 * y * (1 - y) - a21 * x * y\n zdot = r3 * x * z / (x + k3) - a31 * x * z - d3 * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.JerkCircuit","page":"Home","title":"ChaoticDynamicalSystemsLibrary.JerkCircuit","text":" JerkCircuit()\n\nAn electronic circuit with nonlinearity provided by a diode.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.5082428902531144\nLyapunov spectrum (estimated): Any[0.394872929553187, 0.1849026306096344, -1.1407351759248092]\nMaximum Lyapunov exponent (estimated): 0.2922722771543928\nMultiscale Entropy: 0.38139916676685187\nNon-autonomous: false\nPeriod: 19.4546\nPesin entropy: 0.5797755601628214\n\nCitation\n\nSprott (2011). A new chaotic jerk circuit. IEEE Circ. Sys. 2011.\n\nOriginal Python Code from the dysts package\n\nclass JerkCircuit(DynSys):\n @staticjit\n def _rhs(x, y, z, t, eps, y0):\n xdot = y\n ydot = z\n zdot = -z - x - eps * (np.exp(y / y0) - 1)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.KawczynskiStrizhak","page":"Home","title":"ChaoticDynamicalSystemsLibrary.KawczynskiStrizhak","text":" KawczynskiStrizhak()\n\nA chemical oscillator model describing mixed-modes in the BZ equations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.4235221965773541\nLyapunov spectrum (estimated): Any[0.025607519929323416, -0.06145426210744069, -0.31683393262481707]\nMaximum Lyapunov exponent (estimated): 0.0757060839542753\nMultiscale Entropy: 0.5935098791328474\nNon-autonomous: false\nPeriod: 7.991666666666667\nPesin entropy: 0.02560751992932341\n\nCitation\n\nP. E. Strizhak and A. L. Kawczynski, J. Phys. Chem. 99, 10830 (1995).\n\nOriginal Python Code from the dysts package\n\nclass KawczynskiStrizhak(DynSys):\n @staticjit\n def _rhs(x, y, z, t, beta, gamma, kappa, mu):\n xdot = gamma * y - gamma * x ** 3 + 3 * mu * gamma * x\n ydot = -2 * mu * x - y - z + beta\n zdot = kappa * x - kappa * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Laser","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Laser","text":" Laser()\n\nA semiconductor laser model\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.049321615812801\nLyapunov spectrum (estimated): Any[0.7637435195473311, 0.020943250302999646, -15.909545615344538]\nMaximum Lyapunov exponent (estimated): 0.6080602180196161\nMultiscale Entropy: 1.004437967414151\nNon-autonomous: false\nPeriod: 0.94667\nPesin entropy: 0.7848314215142518\n\nCitation\n\nAbooee, Yaghini-Bonabi, Jahed-Motlagh (2013). Analysis and circuitry realization of a novel three-dimensional chaotic system. Comm Nonline Sci 2013\n\nOriginal Python Code from the dysts package\n\nclass Laser(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, h, k):\n xdot = a * y - a * x + b * y * z ** 2\n ydot = c * x + d * x * z ** 2\n zdot = h * z + k * x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.LidDrivenCavityFlow","page":"Home","title":"ChaoticDynamicalSystemsLibrary.LidDrivenCavityFlow","text":" LidDrivenCavityFlow()\n\nA two-dimensional flow alternately driven by translation of the left and right walls.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: true\nKaplan-Yorke dimension: 2.9856669109882814\nLyapunov spectrum (estimated): Any[1.00757312, -0.00276829, -1.01941623]\nMaximum Lyapunov exponent (estimated): 2.172601412109087\nMultiscale Entropy: 0.7815824521381879\nNon-autonomous: true\nPeriod: 2.7333134945588955\nPesin entropy: 1.007573134065856\n\nCitation\n\nGrover et al. Chaos (2012). Topological chaos, braiding and bifurcation of almost-cyclic sets.\n\nOriginal Python Code from the dysts package\n\nclass LidDrivenCavityFlow(DynSys):\n @staticjit\n def _lid(x, y, a, b, tau, u1, u2):\n \"\"\"The velocity field when the left domain drives\"\"\"\n prefactor1 = 2 * u1 * np.sin(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n prefactor2 = 2 * u2 * np.sin(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n vx1 = -b * np.pi * np.sinh(np.pi * b / a) * np.sinh(np.pi * y / a) + np.cosh(np.pi * b / a) * (np.pi * y * np.cosh(np.pi * y /a) + a * np.sinh(np.pi * y / a))\n vx2 = -2 * b * np.pi * np.sinh(2 * np.pi * b / a) * np.sinh(2 * np.pi * y / a) + np.cosh(2 * np.pi * b / a) * (2 * np.pi * y * np.cosh(2 * np.pi * y / a) + a * np.sinh(2 * np.pi * y / a))\n vx = prefactor1 * vx1 + prefactor2 * vx2\n\n prefactor1 = 2 * np.pi * u1 * np.cos(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n prefactor2 = 4 * np.pi * u2 * np.cos(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n vy1 = b * np.sinh(np.pi * b / a) * np.cosh(np.pi * y / a) - np.cosh(np.pi * b / a) * y * np.sinh(np.pi * y / a)\n vy2 = b * np.sinh(2 * np.pi * b / a) * np.cosh(2 * np.pi * y / a) - np.cosh(2 * np.pi * b / a) * y * np.sinh(2 * np.pi * y / a)\n vy = prefactor1 * vy1 + prefactor2 * vy2\n\n # vy1 = b * np.sinh(np.pi * b / a) * np.cosh(np.pi * y / a) - np.cosh(np.pi * b / a) * y * np.sinh(np.pi * y / a)\n # vy2 = b * np.sinh(2 * np.pi * b / a) * np.cosh(2 * np.pi * y / a) - np.cosh(2 * np.pi * b / a) * y * np.sinh(2 * np.pi * y / a)\n # vy = np.pi * prefactor1 * vy1 + 2 * np.pi * prefactor2 * vy2\n\n return vx, vy\n\n # @staticjit\n # def _right(x, y, a, b, tau, u1, u2):\n # \"\"\"The velocity field when the right domain drives\"\"\"\n # prefactor1 = 2 * u1 * np.sin(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n # prefactor2 = 2 * u2 * np.sin(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n # vx1 = -b * np.pi * np.sinh(np.pi * b / a) * np.sinh(np.pi * y / a) - np.cosh(np.pi * b / a) * (np.pi * y * np.cosh(np.pi * y /a) + a * np.sinh(np.pi * y /a))\n # vx2 = -4 * b * np.pi * np.sinh(2 * np.pi * b / a) * np.sinh(2 * np.pi * y / a) - np.cosh(2 * np.pi * b / a) * (2 * np.pi * y * np.cosh(2 * np.pi * y /a) + a * np.sinh(2 * np.pi * y /a))\n # vx = prefactor1 * vx1 - prefactor2 * vx2\n\n # prefactor1 = 2 * np.pi * u1 * np.cos(np.pi * x / a) / (2 * b * np.pi + a * np.sinh(2 * np.pi * b / a))\n # prefactor2 = 4 * np.pi * u2 * np.cos(2 * np.pi * x / a) / (4 * b * np.pi + a * np.sinh(4 * np.pi * b / a))\n # vy1 = -b * np.sinh(np.pi * b / a) * np.cosh(np.pi * y / a) + np.cosh(np.pi * b / a) * y * np.sinh(np.pi * y / a)\n # vy2 = -2 * b * np.sinh(2 * np.pi * b / a) * np.cosh(2 * np.pi * y / a) + np.cosh(2 * np.pi * b / a) * 2 * y * np.sinh(2 * np.pi * y / a)\n # vy = prefactor1 * vy1 + prefactor2 * vy2\n\n # return vx, vy\n\n @staticjit\n def _protocol(t, tau, stiffness=20):\n return 0.5 + 0.5 * np.tanh(tau * stiffness * np.sin(2 * np.pi * t / tau))\n\n def rhs(self, X, t):\n x, y, tt = X\n weight = self._protocol(tt, self.tau)\n dx1, dy1 = self._lid(x, y, self.a, self.b, self.tau, self.u1, self.u2)\n dx2, dy2 = self._lid(x, y, self.a, self.b, self.tau, -self.u1, self.u2)\n dx = weight * dx1 + (1 - weight) * dx2\n dy = weight * dy1 + (1 - weight) * dy2\n dtt = 1\n return dx, dy, dtt\n\n def _postprocessing(self, x, y, tt):\n return x, y, np.sin(2 * np.pi * tt / self.tau)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.LiuChen","page":"Home","title":"ChaoticDynamicalSystemsLibrary.LiuChen","text":" LiuChen()\n\nDerived from Sakarya.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.2188137323261288\nLyapunov spectrum (estimated): Any[0.21332142584135133, -0.9748996261500176, -18.807059276226646]\nMaximum Lyapunov exponent (estimated): 0.2457084451352168\nMultiscale Entropy: 0.7641473604711979\nNon-autonomous: false\nPeriod: 8.0164\nPesin entropy: 0.2133214258413513\n\nCitation\n\nLiu, Chen. Int J Bifurcat Chaos. 2004: 1395-1403.\n\nOriginal Python Code from the dysts package\n\nclass RayleighBenard(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, r):\n xdot = a * y - a * x\n ydot = r * y - x * z\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Lorenz-Tuple{}","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Lorenz","text":" Lorenz()\n\nA minimal weather model based on atmospheric convection.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.075158758095728\nLyapunov spectrum (estimated): Any[1.0910931847726466, 0.02994120961308413, -14.915552395875103]\nMaximum Lyapunov exponent (estimated): 0.8917098035724058\nMultiscale Entropy: 1.1541457906835575\nNon-autonomous: false\nPeriod: 1.5008\nPesin entropy: 1.121034394385731\n\nCitation\n\nLorenz, Edward N (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.\n\nOriginal Python Code from the dysts package\n\nclass Lorenz(DynSys):\n @staticjit\n def _rhs(x, y, z, t, beta, rho, sigma):\n xdot = sigma * y - sigma * x\n ydot = rho * x - x * z - y\n zdot = x * y - beta * z\n return xdot, ydot, zdot\n @staticjit\n def _jac(x, y, z, t, beta, rho, sigma):\n row1 = [-sigma, sigma, 0]\n row2 = [rho - z, -1, -x]\n row3 = [y, x, -beta]\n return [row1, row2, row3]\n\n\n\n\n\n","category":"method"},{"location":"#ChaoticDynamicalSystemsLibrary.Lorenz84","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Lorenz84","text":" Lorenz84()\n\nAtmospheric circulation analogous to Hadley convection.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.4449761695022096\nLyapunov spectrum (estimated): Any[1.0474807193950644, -0.045398035352994866, -2.251992293230755]\nMaximum Lyapunov exponent (estimated): 0.46146683857936616\nMultiscale Entropy: 0.6685802317710752\nNon-autonomous: false\nPeriod: 6.2586\nPesin entropy: 1.0474807193950644\n\nCitation\n\nE. Lorenz (1984). Irregularity: a fundamental property of the atmosphere. Tellus A, vol. 36, no. 2, pp. 98–110, 1984.\n\nOriginal Python Code from the dysts package\n\nclass Lorenz84(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, f, g):\n xdot = -a * x - y ** 2 - z ** 2 + a * f\n ydot = -y + x * y - b * x * z + g\n zdot = -z + b * x * y + x * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Lorenz96","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Lorenz96","text":" Lorenz96()\n\nA climate model containing fluid-like advective nonlinearities, and multiscale coupling of fast and slow variables.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 3.1698163415908196\nLyapunov spectrum (estimated): Any[1.6247012663091516, 0.17185232259745578, -1.1936282999960617, -3.550455057872212]\nMaximum Lyapunov exponent (estimated): 1.3361667787286362\nMultiscale Entropy: 0.8046726640855828\nNon-autonomous: false\nPeriod: 2.1969\nPesin entropy: 1.7965535889066073\n\nCitation\n\nLorenz, Edward (1996). Predictability: A problem partly solved. Seminar on Predictability, Vol. I, ECMWF.\n\nOriginal Python Code from the dysts package\n\nclass Lorenz96(DynSys):\n def rhs(self, X, t):\n Xdot = np.zeros_like(X)\n Xdot[0] = (X[1] - X[-2]) * X[-1] - X[0] + self.f\n Xdot[1] = (X[2] - X[-1]) * X[0] - X[1] + self.f\n Xdot[-1] = (X[0] - X[-3]) * X[-2] - X[-1] + self.f\n Xdot[2:-1] = (X[3:] - X[:-3]) * X[1:-2] - X[2:-1] + self.f\n return Xdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.LorenzBounded-Tuple{}","page":"Home","title":"ChaoticDynamicalSystemsLibrary.LorenzBounded","text":" LorenzBounded()\n\nThe Lorenz attractor in the presence of a confining potential.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0704391073439155\nLyapunov spectrum (estimated): Any[0.8497533717540197, 0.001413707285345669, -12.120634680260636]\nMaximum Lyapunov exponent (estimated): 0.7124135270827626\nMultiscale Entropy: 1.1983193915488475\nNon-autonomous: false\nPeriod: 1.3716\nPesin entropy: 0.8513198508911037\n\nCitation\n\nSprott & Xiong (2015). Chaos.\n\nOriginal Python Code from the dysts package\n\nclass LorenzBounded(DynSys):\n @staticjit\n def _rhs(x, y, z, t, beta, r, rho, sigma):\n xdot = sigma * y - sigma * x - sigma/r**2 * y * x ** 2 - sigma/r**2 * y ** 3 - sigma/r**2 * y * z ** 2 + sigma/r**2 * x ** 3 + sigma/r**2 * x * y ** 2 + sigma/r**2 * x * z ** 2\n ydot = rho * x - x * z - y - rho/r**2 * x ** 3 - rho/r**2 * x * y ** 2 - rho/r**2 * x * z ** 2 + 1/r**2 * z * x ** 3 + 1/r**2 * x * z * y ** 2 + 1/r**2 * x * z ** 3 + 1/r**2 * y * x ** 2 + 1/r**2 * y ** 3 + 1/r**2 * y * z ** 2\n zdot = x * y - beta * z - 1/r**2 * y * x ** 3 - 1/r**2 * x * y ** 3 - 1/r**2 * x * y * z ** 2 + beta/r**2 * z * x ** 2 + beta/r**2 * z * y ** 2 + beta/r**2 * z ** 3\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"method"},{"location":"#ChaoticDynamicalSystemsLibrary.LorenzCoupled-Tuple{}","page":"Home","title":"ChaoticDynamicalSystemsLibrary.LorenzCoupled","text":" LorenzCoupled()\n\nTwo coupled Lorenz attractors.\n\nStats\n\nEmbedding dimension: 6\nHamiltonian: false\nKaplan-Yorke dimension: 3.0408069075490425\nLyapunov spectrum (estimated): Any[1.2218762961515068, 0.1222254715421154, -1.265209618630438, -1.9333037412103384, -13.618111101381306, -15.073328947851422]\nMaximum Lyapunov exponent (estimated): 1.0284398414904556\nMultiscale Entropy: 1.1717493753138832\nNon-autonomous: false\nPeriod: 1.6207\nPesin entropy: 1.3441017676936224\n\nCitation\n\nLorenz, Edward N. Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): 130-141.\n\nOriginal Python Code from the dysts package\n\nclass LorenzCoupled(DynSys):\n @staticjit\n def _rhs(x1, y1, z1, x2, y2, z2, t, beta, eps, rho, rho1, rho2, sigma):\n x1dot = sigma * y1 - sigma * x1\n y1dot = rho1 * x1 - x1 * z1 - y1\n z1dot = x1 * y1 - beta * z1\n x2dot = sigma * y2 - sigma * x2 + eps * x1 - eps * x2\n y2dot = rho2 * x2 - x2 * z2 - y2\n z2dot = x2 * y2 - beta * z2\n return x1dot, y1dot, z1dot, x2dot, y2dot, z2dot\n\n\n\n\n\n","category":"method"},{"location":"#ChaoticDynamicalSystemsLibrary.LorenzStenflo","page":"Home","title":"ChaoticDynamicalSystemsLibrary.LorenzStenflo","text":" LorenzStenflo()\n\nAtmospheric acoustic-gravity waves.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.1738121741011174\nLyapunov spectrum (estimated): Any[0.44944459925783387, 0.026077542739630446, -2.7358399678801906, -3.4213273148340293]\nMaximum Lyapunov exponent (estimated): 0.3807742646421621\nMultiscale Entropy: 1.0299236916146275\nNon-autonomous: false\nPeriod: 3.8877142857142855\nPesin entropy: 0.47552214199746434\n\nCitation\n\nLetellier & Rossler (2007). Hyperchaos. Scholarpedia 2.8 (2007): 1936.\n\nOriginal Python Code from the dysts package\n\nclass LorenzStenflo(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + d * w\n ydot = c * x - x * z - y\n zdot = x * y - b * z\n wdot = -x - a * w\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.LuChen","page":"Home","title":"ChaoticDynamicalSystemsLibrary.LuChen","text":" LuChen()\n\nA system that switches shapes between the Lorenz and Chen attractors as the parameter c is varied from 12.7 to 28.5. Also called the Chen Celikovsky attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0358980309317456\nLyapunov spectrum (estimated): Any[0.6651845739025707, 0.1450338199952451, -22.56999542505058]\nMaximum Lyapunov exponent (estimated): 0.3266868574498627\nMultiscale Entropy: 0.4483903807918106\nNon-autonomous: false\nPeriod: 2.3385\nPesin entropy: 0.8102183938978158\n\nCitation\n\nLu, Chen. Int J Bifurcat Chaos. 2002: 659-661.\n\nOriginal Python Code from the dysts package\n\nclass LuChen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * y - a * x\n ydot = -x * z + c * y\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.LuChenCheng","page":"Home","title":"ChaoticDynamicalSystemsLibrary.LuChenCheng","text":" LuChenCheng()\n\nA four scroll attractor that reduces to Lorenz-like dynamics under bifurcations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0293036172873244\nLyapunov spectrum (estimated): Any[0.35106210160478324, -0.012414362698175424, -11.556516575635818]\nMaximum Lyapunov exponent (estimated): 0.2612775880231121\nMultiscale Entropy: 0.6029763775293518\nNon-autonomous: false\nPeriod: 0.92333\nPesin entropy: 0.3510621518701211\n\nCitation\n\nLu, Chen, Cheng. Int J Bifurcat Chaos. 2004: 1507–1537.\n\nOriginal Python Code from the dysts package\n\nclass LuChenCheng(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = -(a * b) / (a + b) * x - y * z + c\n ydot = a * y + x * z\n zdot = b * z + x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.MacArthur","page":"Home","title":"ChaoticDynamicalSystemsLibrary.MacArthur","text":" MacArthur()\n\nPopulation abundances in a plankton community, based on MacArthur's consumer resource model. Growth rates are determined using Liebig's law of the maximum\n\nStats\n\nEmbedding dimension: 10\nHamiltonian: false\nKaplan-Yorke dimension: 3.8961444775216614\nLyapunov spectrum (estimated): Any[0.1934617869736995, 0.004866584152339167, -0.0431132227766039, -0.17665092099562724, -0.30357752906614904, -0.30392441898183636, -0.30474608910196055, -0.3051840587615213, -0.3058908399877519, -17.37453227551654]\nMaximum Lyapunov exponent (estimated): 0.2494638510848618\nMultiscale Entropy: 0.7844902548776685\nNon-autonomous: false\nPeriod: 43.406\nPesin entropy: 0.19832837112603868\n\nCitation\n\nMacArthur, R. 1969. Species packing, and what competition minimizes. PNAS 64:1369. Parameter values taken from a model of phytoplankton in Huisman & Weissing. Nature 1999.\n\nOriginal Python Code from the dysts package\n\nclass MacArthur(DynSys):\n def growth_rate(self, rr):\n u0 = rr / (self.k.T + rr)\n u = self.r * u0.T\n return np.min(u.T, axis=1)\n\n def rhs(self, X, t):\n nn, rr = X[:5], X[5:]\n mu = self.growth_rate(rr)\n nndot = nn * (mu - self.m)\n rrdot = self.d * (self.s - rr) - np.matmul(self.c, (mu * nn))\n return np.hstack([nndot, rrdot])\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.MackeyGlass","page":"Home","title":"ChaoticDynamicalSystemsLibrary.MackeyGlass","text":" MackeyGlass()\n\nA physiological circuit with time-delayed feedback. A standard delay dimension of three is used.\n\nStats\n\nEmbedding dimension: 10\nHamiltonian: false\nKaplan-Yorke dimension: 8.012917999650943\nLyapunov spectrum (estimated): Any[0.21081023801544096, 0.16332731438425996, -0.03580777756738295, -0.041336844485590406, -0.04889078450243534, -0.059836905162829496, -0.07714200039885702, -0.10872244658769878, -0.18584871959889662, -2.2412562118189827]\nMaximum Lyapunov exponent (estimated): 0.07287527454565443\nNon-autonomous: false\nPeriod: 5.9612230962292045\nPesin entropy: 0.3741375523997009\n\nCitation\n\nGlass, L. and Mackey, M. C. (1979). Pathological physiological conditions resulting from instabilities in physiological control systems. Ann. NY. Acad. Sci.\n\nOriginal Python Code from the dysts package\n\nclass MackeyGlass(DynSysDelay):\n @staticjit\n def _rhs(x, xt, t, beta, gamma, n, tau):\n xdot = beta * (xt / (1 + xt ** n)) - gamma * x\n return xdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.MooreSpiegel","page":"Home","title":"ChaoticDynamicalSystemsLibrary.MooreSpiegel","text":" MooreSpiegel()\n\nA thermo-mechanical oscillator.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.3217713120052443\nLyapunov spectrum (estimated): Any[0.4074353399833356, -0.022079955878999143, -1.1975751283606624]\nMaximum Lyapunov exponent (estimated): 0.6976623003008144\nMultiscale Entropy: 0.5401952179209213\nNon-autonomous: false\nPeriod: 1.2887\nPesin entropy: 0.4076334003995097\n\nCitation\n\nMoore, Spiegel. A Thermally Excited Nonlinear Oscillator, Ap. J., 143, 871 (1966)\n\nOriginal Python Code from the dysts package\n\nclass MooreSpiegel(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, eps):\n xdot = y\n ydot = a * z\n zdot = -z + eps * y - y * x ** 2 - b * x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.MultiChua","page":"Home","title":"ChaoticDynamicalSystemsLibrary.MultiChua","text":" MultiChua()\n\nMultiple interacting Chua electronic circuits.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0710214570482988\nLyapunov spectrum (estimated): Any[0.2210571234274344, -0.00140279701621115, -3.10264254580399]\nMaximum Lyapunov exponent (estimated): 1.462134784614238\nMultiscale Entropy: 0.8118881305702679\nNon-autonomous: false\nPeriod: 2.2586\nPesin entropy: 0.22151080996231726\n\nCitation\n\nMufcstak E. Yalcin, Johan A. K. Suykens, Joos Vandewalle. Cellular Neural Networks, Multi-Scroll Chaos And Synchronization. \n\nOriginal Python Code from the dysts package\n\nclass MultiChua(DynSys):\n def diode(self, x):\n m, c = self.m, self.c\n total = m[-1] * x\n for i in range(1, 6):\n total += 0.5 * (m[i - 1] - m[i]) * (np.abs(x + c[i]) - np.abs(x - c[i]))\n return total\n\n def rhs(self, X, t):\n x, y, z = X\n xdot = self.a * (y - self.diode(x))\n ydot = x - y + z\n zdot = -self.b * y\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.NewtonLiepnik","page":"Home","title":"ChaoticDynamicalSystemsLibrary.NewtonLiepnik","text":" NewtonLiepnik()\n\nEuler's equations for a rigid body, augmented with a linear feedback term.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2206092778719664\nLyapunov spectrum (estimated): Any[0.1723564207468104, -0.00036976661148194504, -0.7795789311467557]\nMaximum Lyapunov exponent (estimated): 0.14945341784639976\nMultiscale Entropy: 0.6322918700854281\nNon-autonomous: false\nPeriod: 8.5095\nPesin entropy: 0.17242608410350868\n\nCitation\n\nLeipnik, R. B., and T. A. Newton (1981). Double strange attractors in rigid body motion with linear feedback control. Physics Letters A 86.2 (1981): 63-67.\n\nOriginal Python Code from the dysts package\n\nclass NewtonLiepnik(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x + y + 10 * y * z\n ydot = -x - 0.4 * y + 5 * x * z\n zdot = b * z - 5 * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.NoseHoover","page":"Home","title":"ChaoticDynamicalSystemsLibrary.NoseHoover","text":" NoseHoover()\n\nFixed temperature molecular dynamics for a strained flow under a reduced order model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 6.213204677134776\nLyapunov spectrum (estimated): Any[0.04757010871740569, 0.0013844486958287268, -0.011653762286922777]\nMaximum Lyapunov exponent (estimated): 0.010464383779400952\nMultiscale Entropy: 0.5656074689683536\nNon-autonomous: false\nPeriod: 5.8673\nPesin entropy: 0.04895455741323442\n\nCitation\n\nNose, S (1985). A unified formulation of the constant temperature molecular-dynamics methods. Journal of Chemical Physics (1984). Hoover, William G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A. Hoover, William G., Anthony JC Ladd, and Bill Moran (1982). High-strain-rate plastic flow studied via nonequilibrium molecular dynamics. Physical Review Letters.\n\nOriginal Python Code from the dysts package\n\nclass NoseHoover(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y\n ydot = -x + y * z\n zdot = a - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.NuclearQuadrupole","page":"Home","title":"ChaoticDynamicalSystemsLibrary.NuclearQuadrupole","text":" NuclearQuadrupole()\n\nA quadrupole boson Hamiltonian that produces chaos via a resonance overlap mechanism\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: true\nKaplan-Yorke dimension: 4.280996083113027\nLyapunov spectrum (estimated): Any[0.27858365099437626, 0.06629535725881477, -0.00449888101008936, -0.2658004759134054]\nMaximum Lyapunov exponent (estimated): 0.29554197767663787\nMultiscale Entropy: 0.8209529097052416\nNon-autonomous: false\nPeriod: 6.023076923076923\nPesin entropy: 0.3448827969293456\n\nCitation\n\nBaran V. and Raduta A. A. (1998), International Journal of Modern Physics E\n\nOriginal Python Code from the dysts package\n\nclass NuclearQuadrupole(DynSys):\n @staticjit\n def _rhs(q1, q2, p1, p2, t, a, b, d):\n q1dot = a * p1\n q2dot = a * p2\n p1dot = - a * q1 + 3 / np.sqrt(2) * b * q1 ** 2 - 3 / np.sqrt(2) * b * q2 ** 2 - d * q1 ** 3 - d * q1 * q2 ** 2\n p2dot = -a * q2 - 3 * np.sqrt(2) * b * q1 * q2 - d * q2 * q1 ** 2 - d * q2 ** 3\n return q1dot, q2dot, p1dot, p2dot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.OscillatingFlow","page":"Home","title":"ChaoticDynamicalSystemsLibrary.OscillatingFlow","text":" OscillatingFlow()\n\nA model fluid flow that produces KAM tori. Original parameters have been rescaled by k\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.962629189215532\nLyapunov spectrum (estimated): Any[0.4274159768365289, -0.005029615144529367, -0.43888500460322705]\nMaximum Lyapunov exponent (estimated): 0.13470179997834794\nMultiscale Entropy: 0.6177682255070645\nNon-autonomous: true\nPeriod: 23.857\nPesin entropy: 0.4274159768365288\n\nCitation\n\nT. H. Solomon and J. P. Gollub, Phys. Rev. A 38, 6280 (1988).\n\nOriginal Python Code from the dysts package\n\nclass OscillatingFlow(DynSys):\n @staticjit\n def _rhs(x, y, z, t, b, k, omega, u):\n f = x + b * np.sin(z)\n dx = u * np.cos(k * y) * np.sin(k * f)\n dy = -u * np.sin(k * y) * np.cos(k * f)\n dz = omega\n return dx, dy, dz\n\n def _postprocessing(self, x, y, z):\n return np.cos(self.k * x), y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.PehlivanWei","page":"Home","title":"ChaoticDynamicalSystemsLibrary.PehlivanWei","text":" PehlivanWei()\n\nA system with quadratic nonlinearity, which undergoes Hopf bifurcations of codimension 1 and 2.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1563635563177295\nLyapunov spectrum (estimated): Any[0.09512327528959069, 0.0015323304198873254, -0.6168618108533457]\nMaximum Lyapunov exponent (estimated): 0.10415076085878888\nMultiscale Entropy: 0.6547493000137818\nNon-autonomous: false\nPeriod: 3.7487330316742087\nPesin entropy: 0.09665562724246173\n\nCitation\n\nPehlivan, Ihsan, and Wei Zhouchao (2012). Analysis, nonlinear control, and chaos generator circuit of another strange chaotic system.\n\nOriginal Python Code from the dysts package\n\nclass PehlivanWei(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y - y * z\n ydot = y + y * z - 2 * x\n zdot = 2 - x * y - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Qi","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Qi","text":" Qi()\n\nA hyperchaotic system with a wide power spectrum.\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: false\nKaplan-Yorke dimension: 2.498645226832379\nLyapunov spectrum (estimated): Any[2.0360429519545677, 0.1320125824983059, -4.347891883424509, -44.356041824978305]\nMaximum Lyapunov exponent (estimated): 2.1916060472989556\nMultiscale Entropy: 0.8855652233700573\nNon-autonomous: false\nPeriod: 0.40820999999999996\nPesin entropy: 2.168055534452874\n\nCitation\n\nG. Qi, M. A. van Wyk, B. J. van Wyk, and G. Chen, (2008). On a new hyperchaotic system. Physics Letters A.\n\nOriginal Python Code from the dysts package\n\nclass Qi(DynSys):\n @staticjit\n def _rhs(x, y, z, w, t, a, b, c, d):\n xdot = a * y - a * x + y * z * w\n ydot = b * x + b * y - x * z * w\n zdot = -c * z + x * y * w\n wdot = -d * w + x * y * z\n return xdot, ydot, zdot, wdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.QiChen","page":"Home","title":"ChaoticDynamicalSystemsLibrary.QiChen","text":" QiChen()\n\nA double-wing chaotic attractor that arises from two bistable attractors. The system has quadratic nonlinearity.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0917466081236316\nLyapunov spectrum (estimated): Any[3.8880075026301206, 0.13379585631170307, -43.8359895934497]\nMaximum Lyapunov exponent (estimated): 4.0275703123027435\nMultiscale Entropy: 0.846453868214798\nNon-autonomous: false\nPeriod: 0.39179\nPesin entropy: 4.021803358941824\n\nCitation\n\nQi et al. Chaos, Solitons & Fractals 2008.\n\nOriginal Python Code from the dysts package\n\nclass QiChen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = a * y - a * x + y * z\n ydot = c * x + y - x * z\n zdot = x * y - b * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.RabinovichFabrikant","page":"Home","title":"ChaoticDynamicalSystemsLibrary.RabinovichFabrikant","text":" RabinovichFabrikant()\n\nA reduced-order model of propagating waves in monequilibrium media.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.36076141745315\nLyapunov spectrum (estimated): Any[0.24233580922074083, 1.7068504600119996e-5, -0.6717710576786404]\nMaximum Lyapunov exponent (estimated): 0.2541443669572464\nMultiscale Entropy: 0.9189339042839536\nNon-autonomous: false\nPeriod: 2.3356\nPesin entropy: 0.24239107527154555\n\nCitation\n\nRabinovich, Mikhail I.; Fabrikant, A. L. (1979). Stochastic Self-Modulation of Waves in Nonequilibrium Media. Sov. Phys. JETP.\n\nOriginal Python Code from the dysts package\n\nclass RabinovichFabrikant(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, g):\n xdot = y * z - y + y * x ** 2 + g * x\n ydot = 3 * x * z + x - x ** 3 + g * y\n zdot = -2 * a * z - 2 * x * y * z\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.RikitakeDynamo","page":"Home","title":"ChaoticDynamicalSystemsLibrary.RikitakeDynamo","text":" RikitakeDynamo()\n\nElectric current and magnetic field of two coupled disk dynamos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0672834652602137\nLyapunov spectrum (estimated): Any[0.14571192786308473, 0.0007222098218782572, -2.1743990404663296]\nMaximum Lyapunov exponent (estimated): 0.1318127385277225\nMultiscale Entropy: 0.9664471212975273\nNon-autonomous: false\nPeriod: 5.8975\nPesin entropy: 0.14655639292019562\n\nCitation\n\nRikitake, T., Oscillations of a system of disk dynamos, Mathematical Proceedings of the Cambridge Philosophical Society, 1958.\n\nOriginal Python Code from the dysts package\n\nclass RikitakeDynamo(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, mu):\n xdot = -mu * x + y * z\n ydot = -mu * y - a * x + x * z\n zdot = 1 - x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Rossler","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Rossler","text":" Rossler()\n\nSpiral-type chaos in a simple oscillator model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0146095059018845\nLyapunov spectrum (estimated): Any[0.0763301393780693, 0.007522476799147231, -5.741024338716747]\nMaximum Lyapunov exponent (estimated): 0.15059688939547888\nMultiscale Entropy: 0.605878349292794\nNon-autonomous: false\nPeriod: 5.9086\nPesin entropy: 0.08385261617721652\n\nCitation\n\nRossler, O. E. (1976), An Equation for Continuous Chaos, Physics Letters, 57A (5): 3972013398,\n\nOriginal Python Code from the dysts package\n\nclass Rossler(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c):\n xdot = -y - z\n ydot = x + a * y\n zdot = b + z * x - c * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Sakarya","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Sakarya","text":" Sakarya()\n\nAn attractor that arises due to merging of two disjoint bistable attractors.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.245095314885476\nLyapunov spectrum (estimated): Any[0.3080921246264039, 0.00035292037656675234, -1.2584697568737606]\nMaximum Lyapunov exponent (estimated): 0.26501036485491514\nMultiscale Entropy: 0.6082767822064958\nNon-autonomous: false\nPeriod: 4.9844\nPesin entropy: 0.3084450450029706\n\nCitation\n\nLi, Chunbiao, et al (2015). A novel four-wing strange attractor born in bistability. IEICE Electronics Express 12.4.\n\nOriginal Python Code from the dysts package\n\nclass Sakarya(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, h, p, q, r, s):\n xdot = a * x + h * y + s * y * z\n ydot = -b * y - p * x + q * x * z\n zdot = c * z - r * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SaltonSea","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SaltonSea","text":" SaltonSea()\n\nAn eco-epidemiological model of bird and fish abundances in the Salton sea of southern California.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0843843542474496\nLyapunov spectrum (estimated): Any[0.3914445015077359, -0.0037578848522063194, -4.592867667006148]\nMaximum Lyapunov exponent (estimated): 1.0702552017260383\nMultiscale Entropy: 0.9335269725171643\nNon-autonomous: false\nPeriod: 0.96738\nPesin entropy: 0.39160860195935765\n\nCitation\n\nUpadhyay, Bairagi, Kundu, Chattopadhyay (2007).Chaos in eco-epidemiological problem of the Salton Sea and its possible control. Applied mathematics and computation.\n\nOriginal Python Code from the dysts package\n\nclass SaltonSea(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, d, k, lam, m, mu, r, th):\n xdot = r * x * (1 - (x + y) / k) - lam * x * y\n ydot = lam * x * y - m * y * z / (y + a) - mu * y\n zdot = th * y * z / (y + a) - d * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SanUmSrisuchinwong","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SanUmSrisuchinwong","text":" SanUmSrisuchinwong()\n\nA two-scroll attractor arising from dynamical equations with piecewise nonlinearities. Equivalent to the Wimol-Banlue system\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.194219372064799\nLyapunov spectrum (estimated): Any[0.23913890446639047, -0.0002916165416126154, -1.2296759041605134]\nMaximum Lyapunov exponent (estimated): 0.2606135207962152\nMultiscale Entropy: 0.9947542017597873\nNon-autonomous: false\nPeriod: 7.4654\nPesin entropy: 0.23946267672516214\n\nCitation\n\nSan-Um, Srisuchinwong. J. Comp 2012\n\nOriginal Python Code from the dysts package\n\nclass SanUmSrisuchinwong(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y - x\n ydot = -z * np.tanh(x)\n zdot = -a + x * y + np.abs(y)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ShimizuMorioka","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ShimizuMorioka","text":" ShimizuMorioka()\n\nA system that bifurcates from a symmetric limit cycle to an asymmetric one, similar to the Lorenz attractor.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0312187919639926\nLyapunov spectrum (estimated): Any[0.044032234819090736, -1.3942528300804223e-5, -1.409991553164768]\nMaximum Lyapunov exponent (estimated): 0.03577425334463406\nMultiscale Entropy: 0.7156563118943259\nNon-autonomous: false\nPeriod: 12.038\nPesin entropy: 0.044239420179671494\n\nCitation\n\nShimizu, Morioka. Phys Lett A. 1980: 201-204\n\nOriginal Python Code from the dysts package\n\nclass ShimizuMorioka(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = y\n ydot = x - a * y - x * z\n zdot = -b * z + x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottA","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottA","text":" SprottA()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 60.66528307527488\nLyapunov spectrum (estimated): Any[0.029259920155059096, 0.004990159457687409, -0.0015561317158397701]\nMaximum Lyapunov exponent (estimated): 0.012993052652429771\nMultiscale Entropy: 0.5748267225365974\nNon-autonomous: false\nPeriod: 6.3728461538461545\nPesin entropy: 0.0343008088722448\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottA(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y\n ydot = -x + y * z\n zdot = 1 - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottB","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottB","text":" SprottB()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1852994894688953\nLyapunov spectrum (estimated): Any[0.21987503866971636, 0.005187437926969091, -1.2145826242115807]\nMaximum Lyapunov exponent (estimated): 0.20674275256006883\nMultiscale Entropy: 0.8826029903839796\nNon-autonomous: false\nPeriod: 11.050142857142857\nPesin entropy: 0.2250624765966855\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottB(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y * z\n ydot = x - y\n zdot = 1 - x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottC","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottC","text":" SprottC()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1529236029587073\nLyapunov spectrum (estimated): Any[0.17898394736414464, -0.0007304674803429964, -1.1644550651924115]\nMaximum Lyapunov exponent (estimated): 0.15806924766944033\nMultiscale Entropy: 1.14839462464542\nNon-autonomous: false\nPeriod: 8.8852\nPesin entropy: 0.17933456833596748\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottC(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y * z\n ydot = x - y\n zdot = 1 - x ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottD","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottD","text":" SprottD()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0807565309303526\nLyapunov spectrum (estimated): Any[0.11415315682442645, 0.001787998513315278, -1.4601870646976078]\nMaximum Lyapunov exponent (estimated): 0.10170941886932346\nMultiscale Entropy: 1.0030259093693934\nNon-autonomous: false\nPeriod: 4.8333\nPesin entropy: 0.11594526886491094\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottD(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = -y\n ydot = x + z\n zdot = x * z + 3 * y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottE","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottE","text":" SprottE()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0894640654250147\nLyapunov spectrum (estimated): Any[0.09834470642927143, -0.00031069798260046597, -1.0953799295090543]\nMaximum Lyapunov exponent (estimated): 0.09016067590084897\nMultiscale Entropy: 0.8387888716672988\nNon-autonomous: false\nPeriod: 7.6275\nPesin entropy: 0.09866310814686535\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottE(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y * z\n ydot = x ** 2 - y\n zdot = 1 - 4 * x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottF","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottF","text":" SprottF()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2516153966506316\nLyapunov spectrum (estimated): Any[0.13246006583273318, 0.0192716855937824, -0.603011686855317]\nMaximum Lyapunov exponent (estimated): 0.1183090101484132\nMultiscale Entropy: 0.7674306147320477\nNon-autonomous: false\nPeriod: 19.244\nPesin entropy: 0.1517317514265156\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottF(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y + z\n ydot = -x + a * y\n zdot = x ** 2 - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottG","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottG","text":" SprottG()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0519020031307744\nLyapunov spectrum (estimated): Any[0.03349131570257798, -0.0012749369106980868, -0.6182204265060449]\nMaximum Lyapunov exponent (estimated): 0.09067832253388744\nMultiscale Entropy: 0.6185493944257671\nNon-autonomous: false\nPeriod: 8.7097\nPesin entropy: 0.03403484153064641\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottG(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = a * x + z\n ydot = x * z - y\n zdot = -x + y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottH","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottH","text":" SprottH()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2414973487697853\nLyapunov spectrum (estimated): Any[0.13871794302385684, 0.003501431527456669, -0.5889065665901029]\nMaximum Lyapunov exponent (estimated): 0.13100182470690253\nMultiscale Entropy: 0.6325364503922819\nNon-autonomous: false\nPeriod: 20.229\nPesin entropy: 0.14221937455131348\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottH(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = -y + z ** 2\n ydot = x + a * y\n zdot = x - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottI","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottI","text":" SprottI()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.019844528960972\nLyapunov spectrum (estimated): Any[0.020459651934364873, 9.478591964272237e-6, -1.031514712617076]\nMaximum Lyapunov exponent (estimated): 0.023218704376506836\nMultiscale Entropy: 0.5893805046352655\nNon-autonomous: false\nPeriod: 10.93\nPesin entropy: 0.02077411702709478\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottI(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = -a * y\n ydot = x + z\n zdot = x + y ** 2 - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottJ","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottJ","text":" SprottJ()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0221468017281006\nLyapunov spectrum (estimated): Any[0.04451808076432957, 0.0010621228852282804, -2.055842238064512]\nMaximum Lyapunov exponent (estimated): 0.12290789099642041\nMultiscale Entropy: 0.6352974298685767\nNon-autonomous: false\nPeriod: 4.836\nPesin entropy: 0.04588958821017871\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottJ(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = 2 * z\n ydot = -2 * y + z\n zdot = -x + y + y ** 2\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottJerk","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottJerk","text":" SprottJerk()\n\nAn algebraidally simple flow depending on a third time derivative. Numerical integration of this attractor is extremely sensitive to the initial conditions\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.031190248069004\nLyapunov spectrum (estimated): Any[0.06738258231535933, -0.0007831911114099333, -2.1347047232043663]\nMaximum Lyapunov exponent (estimated): 0.08785190282162628\nMultiscale Entropy: 0.5314887786603959\nNon-autonomous: false\nPeriod: 11.921\nPesin entropy: 0.06776939078007471\n\nCitation\n\nSprott, J. C. Simplest dissipative chaotic flow. Physics Letters A (1997).\n\nOriginal Python Code from the dysts package\n\nclass SprottJerk(DynSys):\n @staticjit\n def _rhs(x, y, z, t, mu):\n xdot = y\n ydot = z\n zdot = -x + y ** 2 - mu * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottK","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottK","text":" SprottK()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0417036451446204\nLyapunov spectrum (estimated): Any[0.03778259119323125, -0.0005109461395399151, -0.8934722146671759]\nMaximum Lyapunov exponent (estimated): 0.05962593786799159\nMultiscale Entropy: 0.6562812206790845\nNon-autonomous: false\nPeriod: 7.006\nPesin entropy: 0.03788160511007809\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottK(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = x * y - z\n ydot = x - y\n zdot = x + a * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottL","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottL","text":" SprottL()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0329815956000377\nLyapunov spectrum (estimated): Any[0.03407149258791976, -0.000391471796353918, -1.012962157697122]\nMaximum Lyapunov exponent (estimated): 0.09216197583296963\nMultiscale Entropy: 0.5605870039438788\nNon-autonomous: false\nPeriod: 6.7262\nPesin entropy: 0.0343958205328805\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottL(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = y + b * z\n ydot = a * x ** 2 - y\n zdot = 1 - x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottM","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottM","text":" SprottM()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.04996202872584\nLyapunov spectrum (estimated): Any[0.05230117379497565, -0.00033063791832836577, -1.0391447706177515]\nMaximum Lyapunov exponent (estimated): 0.08422043130700084\nMultiscale Entropy: 0.7481178461456197\nNon-autonomous: false\nPeriod: 5.3644\nPesin entropy: 0.05234518101225785\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottM(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = -z\n ydot = -x ** 2 - y\n zdot = a + a * x + y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottMore","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottMore","text":" SprottMore()\n\nA multifractal system with a nearly 3D attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.999952495039482\nLyapunov spectrum (estimated): Any[0.32499834596046606, -0.0001641583999746131, -0.31918544080126926]\nMaximum Lyapunov exponent (estimated): 1.0946909441433958\nMultiscale Entropy: 0.6370154977546665\nNon-autonomous: false\nPeriod: 6.62095890410959\nPesin entropy: 0.32515558132375566\n\nCitation\n\nSprott, J. C. (2020). Do We Need More Chaos Examples? Chaos Theory and Applications 2(2),1-3, 2020.\n\nOriginal Python Code from the dysts package\n\nclass SprottMore(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y\n ydot = -x - np.sign(z) * y\n zdot = y ** 2 - np.exp(-(x ** 2))\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottN","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottN","text":" SprottN()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.028951405806632\nLyapunov spectrum (estimated): Any[0.06095861449194353, 6.560209716837452e-5, -2.101648044064128]\nMaximum Lyapunov exponent (estimated): 0.11869612209728901\nMultiscale Entropy: 0.6385218945868028\nNon-autonomous: false\nPeriod: 4.8135\nPesin entropy: 0.061478272616693266\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottN(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = -2 * y\n ydot = x + z ** 2\n zdot = 1 + y - 2 * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottO","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottO","text":" SprottO()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2474645346215256\nLyapunov spectrum (estimated): Any[0.06685684927644238, 0.011533284505120292, -0.3165440349833235]\nMaximum Lyapunov exponent (estimated): 0.09141077854590315\nMultiscale Entropy: 0.4161154943097029\nNon-autonomous: false\nPeriod: 9.755\nPesin entropy: 0.07839013378156268\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottO(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = y\n ydot = x - z\n zdot = x + x * z + a * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottP","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottP","text":" SprottP()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2122245136720187\nLyapunov spectrum (estimated): Any[0.1026165705645786, 0.00025136328640969387, -0.4843551814513015]\nMaximum Lyapunov exponent (estimated): 0.1064676723437159\nMultiscale Entropy: 0.8486396469370624\nNon-autonomous: false\nPeriod: 5.3414\nPesin entropy: 0.10329419803150507\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottP(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a):\n xdot = a * y + z\n ydot = -x + y ** 2\n zdot = x + y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottQ","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottQ","text":" SprottQ()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.2246852931286063\nLyapunov spectrum (estimated): Any[0.13698250147166127, 0.0010386716745898255, -0.6140516336902575]\nMaximum Lyapunov exponent (estimated): 0.17169784338063596\nMultiscale Entropy: 0.8105158145249107\nNon-autonomous: false\nPeriod: 4.8443\nPesin entropy: 0.13852213114060488\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottQ(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -z\n ydot = x - y\n zdot = a * x + y ** 2 + b * z\n return (xdot, ydot, zdot)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottR","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottR","text":" SprottR()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.071744469374985\nLyapunov spectrum (estimated): Any[0.07608481335025485, 0.0013697096816499383, -1.0784000269941107]\nMaximum Lyapunov exponent (estimated): 0.09661415277799693\nMultiscale Entropy: 0.5877166245477188\nNon-autonomous: false\nPeriod: 7.0133\nPesin entropy: 0.07840404228136309\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottR(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = a - y\n ydot = b + z\n zdot = x * y - z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottS","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottS","text":" SprottS()\n\nA member of the Sprott family of algebraically-simple chaotic sytems.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.182390107009358\nLyapunov spectrum (estimated): Any[0.2149312025337423, 0.0030194370306177213, -1.1945344436340624]\nMaximum Lyapunov exponent (estimated): 0.2737629235389973\nMultiscale Entropy: 0.6690400524585065\nNon-autonomous: false\nPeriod: 3.3505\nPesin entropy: 0.21876911249150574\n\nCitation\n\nSprott (1994). Some simple chaotic flows. Physical Review E (1994).\n\nOriginal Python Code from the dysts package\n\nclass SprottS(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = -x - 4 * y\n ydot = x + z ** 2\n zdot = 1 + x\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SprottTorus","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SprottTorus","text":" SprottTorus()\n\nA multiattractor system that goes to a torus or a complex attractor depending on the initial conditions.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.759693065535405\nLyapunov spectrum (estimated): Any[0.29656066211909027, 0.0005468818242604863, -0.3910853761095065]\nMaximum Lyapunov exponent (estimated): 0.1056964344650892\nMultiscale Entropy: 0.9562901894463037\nNon-autonomous: false\nPeriod: 13.607\nPesin entropy: 0.29710937347161515\n\nCitation\n\nSprott Physics Letters A 2014\n\nOriginal Python Code from the dysts package\n\nclass SprottTorus(DynSys):\n @staticjit\n def _rhs(x, y, z, t):\n xdot = y + 2 * x * y + x * z\n ydot = 1 - 2 * x ** 2 + y * z\n zdot = x - x ** 2 - y ** 2\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.StickSlipOscillator","page":"Home","title":"ChaoticDynamicalSystemsLibrary.StickSlipOscillator","text":" StickSlipOscillator()\n\nA weakly forced (quasiautonomous) oscillator with dynamics similar to stick-slip friction. The parameter gamma controls the transition to chaos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 5.682155291949877\nLyapunov spectrum (estimated): Any[0.07073329226101453, 3.356854711815694e-5, -0.02131392884593247]\nMaximum Lyapunov exponent (estimated): 0.06406131604734494\nMultiscale Entropy: 0.9213029740441252\nNon-autonomous: true\nPeriod: 1.6129\nPesin entropy: 0.07108101456222432\n\nCitation\n\nAwrejcewicz, Jan, and M. M. Holicke (1999). Int J of Bifurcation and Chaos.\n\nOriginal Python Code from the dysts package\n\nclass StickSlipOscillator(DynSys):\n def _t(self, v):\n return self.t0 * np.sign(v) - self.alpha * v + self.beta * v ** 3\n\n @staticjit\n def _rhs(x, v, th, t, a, alpha, b, beta, eps, gamma, t0, vs, w):\n tq = t0 * np.sign(v - vs) - alpha * v + beta * (v - vs) ** 3\n xdot = v\n vdot = eps * (gamma * np.cos(th) - tq) + a * x - b * x ** 3\n thdot = w\n return xdot, vdot, thdot\n\n @staticjit\n def _postprocessing(x, v, th):\n return x, v, np.cos(th)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.SwingingAtwood","page":"Home","title":"ChaoticDynamicalSystemsLibrary.SwingingAtwood","text":" SwingingAtwood()\n\nA mechanical system consisting of two swinging weights connected by ropes and pulleys. This is only chaotic when m2 is sufficiently larger than m1, and there are nonzero initial momenta\n\nStats\n\nEmbedding dimension: 4\nHamiltonian: true\nKaplan-Yorke dimension: 4.0198847313975445\nLyapunov spectrum (estimated): Any[7.735006558021491, 3.0380875684439133, -3.002075505122432, -7.664246463289054]\nMaximum Lyapunov exponent (estimated): 1.0111284961954359\nNon-autonomous: false\nPeriod: 0.3417647058823529\nPesin entropy: 10.773094126465404\n\nCitation\n\nTufillaro, Nicholas B.; Abbott, Tyler A.; Griffiths, David J. (1984). Swinging Atwood's Machine. American Journal of Physics. 52 (10): 895–903.\n\nOriginal Python Code from the dysts package\n\nclass SwingingAtwood(DynSys):\n @staticjit\n def _rhs(r, th, pr, pth, t, m1, m2):\n g = 9.82\n rdot = pr / (m1 + m2)\n thdot = pth / (m1 * r ** 2)\n prdot = pth ** 2 / (m1 * r ** 3) - m2 * g + m1 * g * np.cos(th)\n pthdot = -m1 * g * r * np.sin(th)\n return rdot, thdot, prdot, pthdot\n\n @staticjit\n def _postprocessing(r, th, pr, pth):\n return r, np.sin(th), pr, pth\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Thomas","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Thomas","text":" Thomas()\n\nA cyclically-symmetric attractor correspondng to a frictionally-damped particle traversing a 3D lattice of forces\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1271136075295507\nLyapunov spectrum (estimated): Any[0.7589685797630259, 0.057355308262307496, -6.421950974462397]\nMaximum Lyapunov exponent (estimated): 0.6324054000339867\nMultiscale Entropy: 0.9921951854111759\nNon-autonomous: false\nPeriod: 4.969\nPesin entropy: 0.8163238880253334\n\nCitation\n\nThomas, Rene (1999). Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, labyrinth chaos. Int. J. Bifurc. Chaos.\n\nOriginal Python Code from the dysts package\n\nclass Thomas(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b):\n xdot = -a * x + b * np.sin(y)\n ydot = -a * y + b * np.sin(z)\n zdot = -a * z + b * np.sin(x)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.Torus","page":"Home","title":"ChaoticDynamicalSystemsLibrary.Torus","text":" Torus()\n\nA minimal quasiperiodic flow on a torus. All lyapunov exponents and related quantities are zero\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0\nLyapunov spectrum (estimated): Any[0.0, 0.0, 0.0]\nMaximum Lyapunov exponent (estimated): 0.00817284128815307\nMultiscale Entropy: 0.47330913201518593\nNon-autonomous: false\nPeriod: 6.36\nPesin entropy: 0.0\n\nCitation\n\nSee, for example, Strogatz (1994). Nonlinear Dynamics and Chaos.\n\nOriginal Python Code from the dysts package\n\nclass Torus(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, n, r):\n xdot = (-a * n * np.sin(n * t)) * np.cos(t) - (r + a * np.cos(n * t)) * np.sin(\n t\n )\n ydot = (-a * n * np.sin(n * t)) * np.sin(t) + (r + a * np.cos(n * t)) * np.cos(\n t\n )\n zdot = a * n * np.cos(n * t)\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.TurchinHanski","page":"Home","title":"ChaoticDynamicalSystemsLibrary.TurchinHanski","text":" TurchinHanski()\n\nA chaotic three species food web. The species growth rate has been increased from the default parameters in the original paper, in order to ensure that the system exhibits sustained chaotic oscillations.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 1.2053069221786739\nLyapunov spectrum (estimated): Any[0.03719830596656144, -0.181183885924572, -2.522759887828014]\nMaximum Lyapunov exponent (estimated): 0.12906862565697674\nMultiscale Entropy: 0.4989636261373046\nNon-autonomous: true\nPeriod: 4.40529411764706\nPesin entropy: 0.03719830596656144\n\nCitation\n\nTurchin, Hanski. The American Naturalist 1997. Turchin, Hanski. Ecology 2000\n\nOriginal Python Code from the dysts package\n\nclass TurchinHanski(DynSys):\n @staticjit\n def _rhs(n, p, z, t, a, d, e, g, h, r, s):\n ndot = (\n r * (1 - e * np.sin(z)) * n\n - r * (n ** 2)\n - g * (n ** 2) / (n ** 2 + h ** 2)\n - a * n * p / (n + d)\n )\n pdot = s * (1 - e * np.sin(z)) * p - s * (p ** 2) / n\n zdot = 2 * np.pi\n return ndot, pdot, zdot\n\n @staticjit\n def _postprocessing(x, y, z):\n return x, y, np.sin(z)\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.VallisElNino","page":"Home","title":"ChaoticDynamicalSystemsLibrary.VallisElNino","text":" VallisElNino()\n\nAtmospheric temperature fluctuations with annual forcing.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1205065379188333\nLyapunov spectrum (estimated): Any[0.6561566154431996, 0.022689184568283626, -5.6332694618503885]\nMaximum Lyapunov exponent (estimated): 0.5477586276153029\nMultiscale Entropy: 0.9058707360850027\nNon-autonomous: false\nPeriod: 2.2087\nPesin entropy: 0.6788458000114831\n\nCitation\n\nVallis GK. Conceptual models of El Nio and the southern oscillation. J Geophys Res 1988\n\nOriginal Python Code from the dysts package\n\nclass VallisElNino(DynSys):\n @staticmethod\n def _rhs(x, y, z, t, b, c, p):\n xdot = b * y - c * x - c * p\n ydot = -y + x * z\n zdot = -z - x * y + 1\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.WangSun","page":"Home","title":"ChaoticDynamicalSystemsLibrary.WangSun","text":" WangSun()\n\nA four-scroll attractor\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0595560098996604\nLyapunov spectrum (estimated): Any[0.07990874520682271, -0.001043918775422804, -1.3240808432405]\nMaximum Lyapunov exponent (estimated): 0.06735463002480017\nMultiscale Entropy: 0.9045422097501858\nNon-autonomous: false\nPeriod: 26.958\nPesin entropy: 0.07994326076367073\n\nCitation\n\nWang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A (2009). A 3-D four-wing attractor and its analysis. Brazilian J. Phys. 39.\n\nOriginal Python Code from the dysts package\n\nclass WangSun(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, d, e, f, q):\n xdot = a * x + q * y * z\n ydot = b * x + d * y - x * z\n zdot = e * z + f * x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.WindmiReduced","page":"Home","title":"ChaoticDynamicalSystemsLibrary.WindmiReduced","text":" WindmiReduced()\n\nEnergy transfer into the ionosphere and magnetosphere by the solar wind. The parameter vsw controls the onset of chaos.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0947817139683895\nLyapunov spectrum (estimated): Any[0.15595238514031914, 9.695760899839663e-5, -1.6450480911820564]\nMaximum Lyapunov exponent (estimated): 0.403521000622694\nMultiscale Entropy: 0.5896070173085494\nNon-autonomous: false\nPeriod: 4.6733\nPesin entropy: 0.15635149449218705\n\nCitation\n\nSmith, Thiffeault, Horton. J Geophys Res. 2000. Horton, Weigel, Sprott. Chaos 2001.\n\nOriginal Python Code from the dysts package\n\nclass WindmiReduced(DynSys):\n @staticjit\n def _rhs(i, v, p, t, a1, b1, b2, b3, d1, vsw):\n idot = a1 * (vsw - v)\n vdot = b1 * i - b2 * p ** 1 / 2 - b3 * v\n pdot = (\n vsw ** 2 - p ** (5 / 4) * vsw ** (1 / 2) * (1 + np.tanh(d1 * (i - 1))) / 2\n )\n return idot, vdot, pdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.YuWang","page":"Home","title":"ChaoticDynamicalSystemsLibrary.YuWang","text":" YuWang()\n\nA temperature-compensation circuit with an operational amplifier\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.1703721122420614\nLyapunov spectrum (estimated): Any[2.459880656234634, -0.07297917814211805, -14.00993065520751]\nMaximum Lyapunov exponent (estimated): 1.8525516537708415\nMultiscale Entropy: 0.6340382196367295\nNon-autonomous: false\nPeriod: 1.3366\nPesin entropy: 2.459880656234634\n\nCitation\n\nYu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.\n\nOriginal Python Code from the dysts package\n\nclass YuWang(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d):\n xdot = a * (y - x)\n ydot = b * x - c * x * z\n zdot = np.exp(x * y) - d * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.YuWang2","page":"Home","title":"ChaoticDynamicalSystemsLibrary.YuWang2","text":" YuWang2()\n\nAn alternative temperature-compensation circuit with an operational amplifier\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.0619881661890873\nLyapunov spectrum (estimated): Any[0.802715215563475, 0.007966964664720393, -13.078014115524866]\nMaximum Lyapunov exponent (estimated): 1.1576066800869989\nMultiscale Entropy: 0.9335570471768684\nNon-autonomous: false\nPeriod: 0.39265\nPesin entropy: 0.8106827804626555\n\nCitation\n\nYu, Wang (2012). A novel three dimension autonomous chaotic system with a quadratic exponential nonlinear term. Eng Techn & Applied Science Research.\n\nOriginal Python Code from the dysts package\n\nclass YuWang2(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d):\n xdot = a * (y - x)\n ydot = b * x - c * x * z\n zdot = np.cosh(x * y) - d * z\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"},{"location":"#ChaoticDynamicalSystemsLibrary.ZhouChen","page":"Home","title":"ChaoticDynamicalSystemsLibrary.ZhouChen","text":" ZhouChen()\n\nA feedback circuit model.\n\nStats\n\nEmbedding dimension: 3\nHamiltonian: false\nKaplan-Yorke dimension: 2.045920730005392\nLyapunov spectrum (estimated): Any[0.5917259125455694, 0.005203946388490041, -12.999237688229242]\nMaximum Lyapunov exponent (estimated): 0.6828000288089435\nMultiscale Entropy: 0.7337896982114851\nNon-autonomous: false\nPeriod: 0.87518\nPesin entropy: 0.5969734164748293\n\nCitation\n\nZhou, Chen (2004). A simple smooth chaotic system with a 3-layer attractor. Int J Bifurcat Chaos, 2004\n\nOriginal Python Code from the dysts package\n\nclass ZhouChen(DynSys):\n @staticjit\n def _rhs(x, y, z, t, a, b, c, d, e):\n xdot = a * x + b * y + y * z\n ydot = c * y - x * z + d * y * z\n zdot = e * z - x * y\n return xdot, ydot, zdot\n\n\n\n\n\n","category":"function"}] }