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Expand Up @@ -110,17 +110,28 @@ <h2>Strange Attractors</h2>

<h3 id="intro">Introduction</h3>

<p>The term <strong>&apos;Strange Attractor&apos;</strong> is used to
describe an attractor (a region or shape to which points
are &apos;pulled&apos; as the result of a certain process) that displays
sensitive dependence on initial conditions (that is, points
which are initially close on the attractor become
exponentially separated with time).
<p>In dynamical systems, a <strong>&apos;Strange Attractor&apos;</strong>
is a type of attractor (a region or shape to which points are 'pulled'
as the result of a certain process) that arises in certain non-linear systems
and is characterized by its fractal structure. Unlike regular attractors,
which maybe points, close loops, or more complex but still regular shapes,
a strange attractor, is highly sensitive to small changes in the initial
conditions, leading to chaotic behaivor within the system
(<a href="#schuster">Schuster 1989, pp. 105-106</a>;
<a href="#strogatz">Strogatz 2018, chapter 9</a>). The name
was instroduced in the early 1970s by
David Ruelle and Floris Takens
in a paper in which they proposed that fluid turbulence is an example
of what we now call chaos (<a href="#lorenz1993">Lorenz 1993, p. 48</a>; <a href="#ruelle1971">Ruelle &amp; Takens 1971</a>).
</p>

<p>
The most famous strange attractor is undoubtedly
the Lorenz attractor - a three dimensional object whose
body plan resembles a butterfly or a mask. The Lorenz
attractor, named for its discoverer Edward N. Lorenz,
arose from a mathematical model of the atmosphere [<a href="#lorenz1963">5</a>].</p>
arose from a mathematical model of the atmosphere (<a href="#lorenz1963">Lorenz 1963</a>).
</p>

<p>Imagine a rectangular slice of air heated from below and
cooled from above by edges kept at constant
Expand Down Expand Up @@ -503,8 +514,9 @@ <h3 id="references">References</h3>
<li>Dadras, S., Momeni, H.R. (2009). A novel three-dimensional autonomous chaotic system
generating two, three and four-scroll attractors. <i>Physics Letters A.</i> Volume 373,
Issue 40. pp. 3637-3642. <a href="https://doi.org/10.1016/j.physleta.2009.07.088" target="_blank"
title="Open link"><i class="fas fa-external-link-alt"></i></a></li>
<li id="lorenz1963">Langford, W. F. (1984). Numerical studies of torus bifurcations,
title="Open link"><i class="fas fa-external-link-alt"></i></a>
</li>
<li id="kangford1984">Langford, W. F. (1984). Numerical studies of torus bifurcations,
Numerical methods for bifurcation problems (Dortmund, 1983),
<em>Internat. Schriftenreihe Numer. Math.</em>, vol. 70, Birkhäuser, Basel, pp. 285-295.
<a href="https://www.researchgate.net/publication/238282172_Numerical_Studies_of_Torus_Bifurcations"
Expand All @@ -514,9 +526,9 @@ <h3 id="references">References</h3>
Atmospheric Sciences.</i> <b>20</b>(2): 130&#x2013;141. <a
href="http://journals.ametsoc.org/doi/abs/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2"
target="_blank" title="Open link"><i class="fas fa-external-link-alt"></i></a></li>
<li id="lorenz1963">Lucas, S. K., Sander, E. &amp; Tallman, L. (2020). Modeling Dynamical Systems for 3D
Printing.
Notices of the AMS. 67(11). pp. 1692-1702.
<li id="lorenz1993">Lorenz E. N. (1993). <i>The Essence of Chaos</i> University of Washington Press.</li>
<li id="lucas2020">Lucas, S. K., Sander, E. &amp; Tallman, L. (2020). Modeling Dynamical Systems for 3D
Printing. <i>Notices of the AMS.</i> 67(11). pp. 1692-1702.
<a href="" target="_blank" title="Open link"><i class="fas fa-external-link-alt"></i></a>
</li>
<li>Pan, L., Zhou, W., Fang,J., Li, D. (2010). A new three-scroll unified chaotic system coined.
Expand All @@ -527,15 +539,18 @@ <h3 id="references">References</h3>
<li>R&#xF6;ssler, O. E. (1976). An Equation for Continuous Chaos. <i>Physics Letters,</i> 57<b>A</b>
(5): 397&#x2013;398. <a href="https://doi.org/10.1016/0375-9601(76)90101-8" target="_blank"
title="Open link"><i class="fas fa-external-link-alt"></i></a></li>
<li id="ruelle1971">Ruelle, D. &amp; Takens, F. (1971). On the Natrue of Turbulence. <i>Commun. math. Phys. 20. 167-192.</i></li>
<li id="schuster1989">Schuster, H. G. (1989). <i>Deterministic Chaos.</i> VCH.</li>
<li>Sol&#xED;s P&#xE9;rez, J. E., G&#xF3;mez-Aguilar, J. F., Baleanu, D., Tchier, F. (2018). Chaotic
Attractors
with Fractional Conformable Derivatives in the Liouville&#x2013;Caputo Sense and Its Dynamical
Behaviors.<i> Entropy.</i> 2018, 20(5), 384. <a href="https://doi.org/10.3390/e20050384" target="_blank"
title="Open link"><i class="fas fa-external-link-alt"></i></a></li>
<li>Sprott. J. C. (2014). A dynamical system with a strange attractor and invariant tori
<li>Sprott. J. C. (2014). A dynamical system with a strange attractor and invariant tori.
<i>Physic Letters A,</i> 378 1361-1363. <a href="http://sprott.physics.wisc.edu/pubs/paper423.pdf"
target="_blank" title="Open link"><i class="fas fa-external-link-alt"></i></a>
</li>
<li id="strogatz2018">Strogatz, S. H. (2018). <i>Nonlinear Dynamics and Chaos.</i> CRC Press.</li>
<li>Thomas, Ren&#xE9;. (1999). Deterministic chaos seen in terms of feedback circuits: Analysis,
synthesis, &#x2018;labyrinth chaos&#x2019;. <i>Int. J. Bifurcation and Chaos.</i> 9 (10):
1889&#x2013;1905. <a href="https://doi.org/10.1142/S0218127499001383" target="_blank"
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