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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "21cc6738",
+   "metadata": {},
+   "source": [
+    "# A 1-D example of Chemical Explosive Mode Analysis (CEMA) for $CH_4$ ignition"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c984d88d",
+   "metadata": {},
+   "source": [
+    "Let us implement an sample CEMA for methane combustion case.To do so we first generate the 1-D methane combustion data using Arrhenius.jl"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "9989f948",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "PyObject <module 'cantera' from '/usr/lib/python3/dist-packages/cantera/__init__.py'>"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "using Arrhenius\n",
+    "using ForwardDiff\n",
+    "using DifferentialEquations\n",
+    "using LinearAlgebra\n",
+    "using Plots\n",
+    "using PyCall\n",
+    "ct = pyimport(\"cantera\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "10536c49",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Arrhenius.Solution(53, 325, [2.016, 1.008, 15.999, 31.998, 17.007, 18.015, 33.006, 34.014, 12.011, 13.018999999999998  …  43.025, 43.025, 43.025, 42.016999999999996, 28.014, 39.95, 43.089, 44.097, 43.045, 44.053], [\"H2\", \"H\", \"O\", \"O2\", \"OH\", \"H2O\", \"HO2\", \"H2O2\", \"C\", \"CH\"  …  \"HCNO\", \"HOCN\", \"HNCO\", \"NCO\", \"N2\", \"AR\", \"C3H7\", \"C3H8\", \"CH2CHO\", \"CH3CHO\"], [\"O\", \"H\", \"C\", \"N\", \"Ar\"], [0.0 0.0 … 1.0 1.0; 2.0 1.0 … 3.0 4.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0], Arrhenius.IdealGasThermo([2.34433112 0.00798052075 … -917.935173 0.683010238; 2.5 7.05332819e-13 … 25473.6599 -0.446682853; … ; 3.409062 0.010738574 … 1521.4766 9.55829; 4.7294595 -0.0031932858 … -21572.878 4.1030159], [3.3372792 -4.94024731e-5 … -950.158922 -3.20502331; 2.50000001 -2.30842973e-11 … 25473.6599 -0.446682914; … ; 5.97567 0.008130591 … 490.3218 -5.045251; 5.4041108 0.011723059 … -22593.122 -3.4807917], [200.0 1000.0 3500.0; 200.0 1000.0 3500.0; … ; 300.0 1000.0 5000.0; 200.0 1000.0 6000.0], false), Arrhenius.Transport(7, [-2.366640508787296e-10 -1.2299135731442892e-9 … 8.693592178691943e-8 8.794793602234737e-8; 2.1192301406670526e-8 6.668825847991541e-8 … -3.271068954511353e-6 -3.309147211220777e-6; … ; 5.7448300967228756e-5 0.0001298839327866297 … -0.0049138820317118 -0.00497108414638971; -6.404100632763197e-5 -0.00013572755355049403 … 0.0051027272253661325 0.005162127672107992], [-0.0016279891577270268 7.206309409434899e-5 … -0.00011924922331411204 -0.00015757657556546702; 0.06178549433963653 -0.0021239618978405594 … 0.004243684109832842 0.005587634198577984; … ; 96.65522170663003 -2.0666287999557595 … 5.836219561810777 7.641767878764346; -102.92280467937866 1.8536466578657944 … -5.97664208790569 -7.812749492011645], [3.348485446466861e-5 5.169493728638671e-5 … 3.2144250898910778e-6 3.195770414642371e-6; -0.0012612725705465348 -0.0019481129061467706 … -0.00012272386962752324 -0.00012201165084214482; … ; -2.0626595269712937 -3.1895799986447386 … -0.20978320877300424 -0.20856574763356261; 2.2014391913047877 3.404772178136124 … 0.22590225587600743 0.2245912490539997]), Arrhenius.Reaction(\n",
+       "⢛⣛⢺⣻⠚⡛⣍⢙⡋⢉⡟⣿⡓⢓⢶⡍⣙⣼⣧⣋⣓⢛⠛⠛⠛⢛⢛⣁⣙⢌⠙⡫⣚⣋⣊⡋⣺⣟⢓⡐\n",
+       "⠈⠉⠱⠥⠁⠀⠈⠛⠹⠦⠅⠙⠉⡶⠍⠻⠷⠆⠮⠭⠎⠗⠀⣀⢀⡀⢀⣀⣀⣀⡁⠈⣀⣀⡀⠙⠂⠩⠁⠭\n",
+       "⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢈⣉⠙⣒⣑⢑⢛⡲⡲⡶⣞⣂⠶⡛⠀⠀⠀⣀⠀\n",
+       "⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠉⠉, \n",
+       "⠹⣉⠉⢙⠛⡻⣍⠉⠉⠉⡳⢖⡒⠒⢖⣷⣼⣜⣂⣆⡒⠓⠛⠑⠚⢛⠛⠓⠙⢥⢤⡙⠙⢉⣋⣷⠛⡛⣳⢓\n",
+       "⠀⠈⠓⠦⠁⠀⠈⠉⠓⠦⡄⠀⠙⠲⠌⠀⠅⠄⠠⠘⠝⢦⡀⣀⣀⠀⠀⢀⠀⡀⠀⠀⠀⢠⣠⠐⠂⠄⠤⠀\n",
+       "⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠚⠁⠉⠉⠒⣊⡓⣊⠉⠥⠤⠌⢋⠀⠀⠀⢀⣀\n",
+       "⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀, \n",
+       "⠹⣉⠉⢙⠛⡻⣍⠉⠉⠉⡳⢖⡒⠒⢖⣷⣼⣜⣂⣆⡒⠓⠛⠑⠚⢛⠛⠓⠙⢥⢤⡙⠙⢉⣋⣷⠛⡛⣳⢓\n",
+       "⠀⠈⠓⠦⠁⠀⠈⠉⠓⠦⡄⠀⠙⠲⠌⠀⠅⠄⠠⠘⠝⢦⡀⣀⣀⠀⠀⢀⠀⡀⠀⠀⠀⢠⣠⠐⠂⠄⠤⠀\n",
+       "⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠚⠁⠉⠉⠒⣊⡓⣊⠉⠥⠤⠌⢋⠀⠀⠀⢀⣀\n",
+       "⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀, Bool[1, 1, 1, 1, 1, 1, 1, 1, 1, 1  …  1, 1, 1, 1, 1, 1, 1, 1, 0, 1], [1.2e11 -1.0 0.0; 5.0e11 -1.0 0.0; … ; 2.41e10 0.0 0.0; 1.927e10 -0.32 0.0], [6.02e8 0.0 3000.0; 1.04e20 -2.76 1600.0; … ; 3.0000000000000003e57 -14.6 18170.0; 4.42e55 -13.545 11357.0], [0.562 5836.0 8552.0 91.0; 0.783 2941.0 6964.0 74.0; … ; 0.1894 8748.0 7891.0 277.0; 0.315 3285.0 6667.0 369.0], [1, 2, 33, 39, 43, 167, 187, 205, 212, 227, 230, 269], [12, 50, 52, 54, 56, 57, 59, 63, 70, 71  …  158, 174, 185, 237, 241, 289, 304, 312, 318, 320], [-1, 1, 2, 3, 4, 5, 6, 7, 8, 9  …  19, 20, -1, -1, 21, 22, 23, 24, 25, 26], \n",
+       "⡇⢸⠀⠀⣿⡇⣿⣿⢸⡇⡇⢸⠀⠀⠀⠀⡇⡇⡇⡇⡇⡇⢸⠀⠀⣿⠀⢸⡇⣿⠀⠀⢸⠀⠀⡇⠀⡇⣿⡇\n",
+       "⡇⢸⠀⠀⣿⡇⣿⣿⢸⡇⡇⢸⠀⠀⠀⠀⡇⡇⡇⡇⡇⡇⢸⠀⠀⣿⠀⢸⡇⣿⠀⠀⢸⠀⠀⡇⠀⡇⣿⡇\n",
+       "⡇⢸⠀⠀⣿⡇⣿⣿⢸⡇⡇⢸⠀⠀⠀⠀⡇⡇⡇⡇⡇⡇⢸⠀⠀⣿⠀⢸⡇⣿⠀⠀⢸⠀⠀⡇⠀⡇⣿⡇\n",
+       "⠁⠈⠀⠀⠉⠁⠉⠉⠈⠁⠁⠈⠀⠀⠀⠀⠁⠁⠁⠁⠁⠁⠈⠀⠀⠉⠀⠈⠁⠉⠀⠀⠈⠀⠀⠁⠀⠁⠉⠁, [[3], [2, 3], [1, 3], [3, 7], [3, 8], [3, 10], [3, 11], [3, 12], [3, 12], [3, 13]  …  [8, 50], [13, 51], [13, 25], [3, 50], [2, 50], [2, 50], [5, 50], [7, 50], [7, 50], [13, 50]], [[4], [5], [2, 5], [4, 5], [5, 7], [2, 15], [2, 17], [1, 15], [2, 17], [2, 18]  …  [7, 51], [14, 50], [50], [18, 26], [51], [13, 26], [19, 26], [4, 51], [5, 18, 26], [26]], 325, \n",
+       "⢻⣛⢻⣻⠛⠻⣍⢙⡋⢉⡿⣿⡓⢓⢶⣿⣽⣼⣧⣏⣑⢛⠛⠛⠛⢛⢛⣓⣙⢭⢽⡻⣛⣋⣋⣿⣻⣟⣳⣓\n",
+       "⠈⠉⠳⠧⠁⠀⠈⠛⠻⠦⡅⠙⠙⡶⠍⠻⠷⠆⠮⠽⠟⢷⡀⣀⣀⡀⢀⣀⣀⣀⡁⠈⣀⣠⣠⠙⠂⠭⠥⠭\n",
+       "⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⠙⣛⣙⢓⣛⡳⣺⡿⣿⣦⠾⣛⠀⠀⠀⣀⣀\n",
+       "⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉, [-1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, -1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0]))"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "gas_ct = ct.Solution(\"gri30.xml\")\n",
+    "gas = CreateSolution(\"../mechanism/gri30.yaml\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "b59c86ef",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "53\n",
+      "325"
+     ]
+    }
+   ],
+   "source": [
+    "ns = gas.n_species\n",
+    "print(ns);\n",
+    "nr = gas.n_reactions\n",
+    "print(\"\\n\",nr);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "bb5a6252",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "53-element Vector{String}:\n",
+       " \"H2\"\n",
+       " \"H\"\n",
+       " \"O\"\n",
+       " \"O2\"\n",
+       " \"OH\"\n",
+       " \"H2O\"\n",
+       " \"HO2\"\n",
+       " \"H2O2\"\n",
+       " \"C\"\n",
+       " \"CH\"\n",
+       " \"CH2\"\n",
+       " \"CH2(S)\"\n",
+       " \"CH3\"\n",
+       " ⋮\n",
+       " \"H2CN\"\n",
+       " \"HCNN\"\n",
+       " \"HCNO\"\n",
+       " \"HOCN\"\n",
+       " \"HNCO\"\n",
+       " \"NCO\"\n",
+       " \"N2\"\n",
+       " \"AR\"\n",
+       " \"C3H7\"\n",
+       " \"C3H8\"\n",
+       " \"CH2CHO\"\n",
+       " \"CH3CHO\""
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "species_arr = gas.species_names"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "2205b286",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#get equivalence ratio\n",
+    "gas_ct.TP = 900, ct.one_atm\n",
+    "phi = 1\n",
+    "gas_ct.set_equivalence_ratio(phi, \"CH4\",\"O2:1, N2:3.76\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "91ef853c",
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "LoadError",
+     "evalue": "UndefVarError: find not defined",
+     "output_type": "error",
+     "traceback": [
+      "UndefVarError: find not defined",
+      "",
+      "Stacktrace:",
+      " [1] top-level scope",
+      "   @ In[21]:2",
+      " [2] eval",
+      "   @ ./boot.jl:360 [inlined]",
+      " [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+      "   @ Base ./loading.jl:1094"
+     ]
+    }
+   ],
+   "source": [
+    "specs = gas_ct.X\n",
+    "find(x->(x>0),specs)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 46,
+   "id": "c8b39202",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Y0 = zeros(ns)\n",
+    "Y0[species_index(gas, \"CH4\")] = 0.05\n",
+    "Y0[species_index(gas, \"O2\")] = 0.22\n",
+    "Y0[species_index(gas,\"N2\")] = 0.73\n",
+    "T0 = 1300  #K\n",
+    "P = one_atm\n",
+    "u0 = vcat(Y0, T0);"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 47,
+   "id": "9458bb24",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "dudt! (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 47,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "@inbounds function dudt!(du, u, p, t)\n",
+    "    T = u[end]\n",
+    "    Y = @view(u[1:ns])\n",
+    "    mean_MW = 1. / dot(Y, 1 ./ gas.MW)\n",
+    "    ρ_mass = P / R / T * mean_MW\n",
+    "    X = Y2X(gas, Y, mean_MW)\n",
+    "    C = Y2C(gas, Y, ρ_mass)\n",
+    "    cp_mole, cp_mass = get_cp(gas, T, X, mean_MW)\n",
+    "    h_mole = get_H(gas, T, Y, X)\n",
+    "    S0 = get_S(gas, T, P, X)\n",
+    "    wdot = wdot_func(gas.reaction, T, C, S0, h_mole)\n",
+    "    Ydot = wdot / ρ_mass .* gas.MW\n",
+    "    Tdot = -dot(h_mole, wdot) / ρ_mass / cp_mass\n",
+    "    du .= vcat(Ydot, Tdot)\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 48,
+   "id": "a10b1e0d",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[0.0007372047769254919, 0.00015783813441258581, 0.0024684764564995667, 0.030240894023154722, 0.009514030316890703, 0.09925295177222965, 5.999200322512603e-6, 2.745199996984539e-7, 1.1655762192849003e-14, 1.1195793281653313e-15, 9.169500260625093e-16, 7.417468071139761e-17, 1.1526296348853825e-15, 1.6559665608689453e-16, 0.02639335196179212, 0.09569054288713183, 1.3926994094453778e-8, 1.1347285657166752e-10, 1.3074647243442918e-15, 2.6681749783590557e-17, 3.8453520809023416e-17, 5.6420499565209635e-21, 1.4785698813585826e-19, 3.2237756053507546e-24, 7.281778600979616e-25, 1.6873468039224525e-29, 3.6881881095716126e-31, 1.2623809037968665e-17, 4.2550975151592306e-18, 1.2943563840775742e-20, 3.8472657287692335e-7, 4.0801441084301696e-8, 7.14521858643824e-9, 4.924599456330184e-9, 1.0282015957786918e-8, 0.010380621594693025, 4.3757920969141466e-6, 8.035663360454821e-7, 3.6552718100736904e-7, 7.382181252022507e-12, 2.950729500815938e-10, 3.6453197064588955e-16, 4.785953872629456e-19, 8.234820493660357e-15, 1.9286533429595644e-11, 2.8854025065189156e-9, 5.447650914165151e-10, 0.7251518037980842, 0.0, 1.2457688091893588e-42, 3.1329280487968883e-43, 4.898003839005981e-23, 7.896716066200371e-25, 2625.321240165389]"
+     ]
+    }
+   ],
+   "source": [
+    "tspan = [0.0, 0.07];\n",
+    "prob = ODEProblem(dudt!, u0, tspan);\n",
+    "sol = solve(prob, TRBDF2(), reltol=1e-6, abstol=1e-9);\n",
+    "print(sol[end])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 49,
+   "id": "6dcc99bf",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/svg+xml": [
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+      ]
+     },
+     "execution_count": 49,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#plot stuff\n",
+    "plt1 = plot(sol.t, sol[species_index(gas, \"CH4\"), :], lw=2, label=\"CH4 mass fraction\");\n",
+    "plot(plt1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 50,
+   "id": "743f60a0",
+   "metadata": {},
+   "outputs": [
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+       "  578.488,1281.85 579.874,1238.71 580.714,1194.48 581.154,1159.72 581.501,1121.32 581.706,1090.82 581.897,1053.39 582.013,1023.89 582.135,983.395 582.206,952.822 \n",
+       "  582.289,905.642 582.333,873.104 582.362,847.546 582.403,803.15 582.433,763.944 582.463,714.404 582.486,672.952 582.512,627.237 582.53,599.706 582.55,580.609 \n",
+       "  582.585,567.821 582.643,558.177 582.773,535.235 583.165,479.855 583.557,437.852 584.882,334.893 587,236.464 590.127,159.365 594.717,110.927 600.824,92.0311 \n",
+       "  609.919,87.9763 624.799,88.9923 666.962,92.0279 713.038,94.3474 767.684,96.112 832.75,97.2942 916.58,98.0195 1010.94,98.3401 1151.38,98.4839 1315.65,98.5136 \n",
+       "  1639.11,98.5183 2143.95,98.5178 2291.86,98.5178 \n",
+       "  \"/>\n",
+       "<path clip-path=\"url(#clip820)\" d=\"\n",
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+       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
+       "<polyline clip-path=\"url(#clip820)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
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+       "  \"/>\n",
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+       "  1767.94,155.698 1911.39,155.698 \n",
+       "  \"/>\n",
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+      ]
+     },
+     "execution_count": 50,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "plt2 = plot(sol.t, sol[end,:],lw=2, label =\"Temperature\");\n",
+    "plot(plt2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 72,
+   "id": "067127ea",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "  0.230798 seconds (405.00 k allocations: 25.309 MiB, 16.89% gc time, 99.58% compilation time)\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "54×54 Matrix{Float64}:\n",
+       "    -1.52695e6         7.13195e6    …   21485.2               0.824522\n",
+       "     7.63502e5        -4.49871e6       -10742.5              -0.584406\n",
+       "    -3.05046e6         2.3922e7        -28991.5              -5.04239\n",
+       " -5439.7              -1.94202e7        -1416.48             -3.77348\n",
+       "    -6.39097e6         4.54834e7           -2.15116e5        24.7637\n",
+       "     1.0205e7         -4.76365e7    …       2.60533e5       -15.4171\n",
+       "  5391.86          19308.6               1455.83              0.150413\n",
+       "   -36.2059        -1744.06                -6.56466           0.210893\n",
+       "    -4.62871e-6        0.000280186         -1.68825e-18       1.31532e-9\n",
+       "     8.09969e-7        5.60372e-5           5.85536e-12       1.36516e-10\n",
+       "     4.41289e-6        2.61133e-6   …       2.77507e-12       2.2412e-11\n",
+       "    -2.26432e-7        5.40157e-6          -1.70388e-10       9.57e-12\n",
+       "     2.85661e-7        1.08957e-5           3.11854e5         7.702e-12\n",
+       "     ⋮                              ⋱                    \n",
+       "    -2.08883e-10      -2.43301e-9          -9.57743e-12      -2.04416e-14\n",
+       "    -2.88522e-17      -2.72726e-6          -1.32036e-18       4.8977e-12\n",
+       "    -1.95895e-13      -0.154448            -8.96475e-15       8.09177e-8\n",
+       "     0.222192         -3.58519      …      -1.56636e-5        4.71834e-7\n",
+       "    -0.219206         -2.72015             -8.62846e-5        2.93628e-6\n",
+       "    55.0283       -31106.4                  3.12279          -0.00718459\n",
+       "     0.0               0.0                  0.0               0.0\n",
+       "    -7.04006e-35      -9.53325e-33          2.72048e-36       6.30171e-39\n",
+       "     1.58833e-34      -6.73867e-33  …       1.72678e-36      -3.24201e-39\n",
+       "    -7.55654e-15       1.09565e-11          2.68386e5        -4.46754e-18\n",
+       "     1.08791e-15      -2.34898e-15         -1.18843e6         7.88067e-22\n",
+       "     2.09748e10       -1.38905e11           1.24381e9    -48623.4"
+      ]
+     },
+     "execution_count": 72,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#compute the jacobian \n",
+    "function dudt(u)\n",
+    "    T = u[end]\n",
+    "    Y = @view(u[1:ns])\n",
+    "    mean_MW = 1. / dot(Y, 1 ./ gas.MW)\n",
+    "    ρ_mass = P / R / T * mean_MW\n",
+    "    X = Y2X(gas, Y, mean_MW)\n",
+    "    C = Y2C(gas, Y, ρ_mass)\n",
+    "    cp_mole, cp_mass = get_cp(gas, T, X, mean_MW)\n",
+    "    h_mole = get_H(gas, T, Y, X)\n",
+    "    S0 = get_S(gas, T, P, X)\n",
+    "    wdot = wdot_func(gas.reaction, T, C, S0, h_mole)\n",
+    "    Ydot = wdot / ρ_mass .* gas.MW\n",
+    "    Tdot = -dot(h_mole, wdot) / ρ_mass / cp_mass\n",
+    "    du = vcat(Ydot, Tdot)\n",
+    "end\n",
+    "@time J_w = ForwardDiff.jacobian(dudt,sol[end])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 73,
+   "id": "33e12125",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "54-element Vector{Float64}:\n",
+       "      -5.1402940277116776e8\n",
+       "      -5.13152406186228e8\n",
+       "      -1.047488154189249e8\n",
+       "      -4.016960521435924e7\n",
+       "      -3.712423735420916e7\n",
+       "      -3.656225824242137e7\n",
+       "      -3.465295356192211e7\n",
+       "      -3.458235019079888e7\n",
+       "      -3.222971058255094e7\n",
+       "      -3.218474998638545e7\n",
+       "      -2.1424526246316772e7\n",
+       "      -1.6560164920316156e7\n",
+       "      -1.5350392042396849e7\n",
+       "       ⋮\n",
+       " -376970.8910503711\n",
+       " -263703.93432325067\n",
+       " -159718.90651112387\n",
+       " -123941.40921549303\n",
+       "   -6630.883336015616\n",
+       "    -304.2359055369181\n",
+       "      -3.951849578020171e-8\n",
+       "      -5.965913016854841e-9\n",
+       "      -1.1194473550679741e-10\n",
+       "       0.0\n",
+       "       2.555719020746241e-8\n",
+       "       1.1448487553316728e-7"
+      ]
+     },
+     "execution_count": 73,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "eig_vals = eigvals(J_w)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 78,
+   "id": "488bb2c0",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "2-element Vector{Int64}:\n",
+       " 53\n",
+       " 54"
+      ]
+     },
+     "execution_count": 78,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#now extract all the positive eigen values: \n",
+    "findall(x->(x > 0), eig_vals) "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 79,
+   "id": "19874e03",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "54×54 Matrix{Float64}:\n",
+       "  2.70714e-8   -7.1509e-7    -2.51146e-6   …   3.68468e-6    2.64177e-6\n",
+       "  1.72087e-9    7.05283e-5    2.57949e-6       1.02975e-6    9.2112e-7\n",
+       " -2.65337e-7   -7.3539e-6     3.41717e-6       9.43655e-6    1.23394e-5\n",
+       "  1.77707e-5    1.80682e-6   -3.41017e-5      -4.11591e-5    2.88098e-5\n",
+       "  0.000175692   9.22861e-6    8.56997e-5       2.33229e-5    2.75997e-5\n",
+       " -0.000274375  -1.75474e-5   -8.70614e-5   …  -1.07149e-5   -3.82003e-5\n",
+       " -0.000537388  -5.1194e-5    -4.49305e-7       7.61265e-9    1.7135e-8\n",
+       "  0.000618398   6.21162e-5   -2.38254e-8       3.1115e-10    5.47463e-10\n",
+       "  4.26755e-18  -4.03337e-17   6.37716e-7       2.2017e-16    2.0083e-16\n",
+       " -8.61843e-18  -8.09984e-18  -1.97865e-5       2.15168e-17   1.88548e-17\n",
+       " -6.28377e-19  -1.0581e-19    0.000285173  …   1.68573e-17   1.3881e-17\n",
+       " -2.15674e-19  -1.08432e-18  -0.000372372      1.40081e-18   1.16429e-18\n",
+       " -4.37923e-19  -2.01582e-18   7.11077e-5       1.87427e-17   1.41786e-17\n",
+       "  ⋮                                        ⋱                \n",
+       " -4.35165e-22   5.23838e-22  -1.60644e-10      7.95851e-21   6.79772e-21\n",
+       "  5.54854e-21   3.90167e-19   2.17009e-7       9.22833e-17   8.15808e-17\n",
+       " -7.22635e-16   2.17072e-14  -7.47314e-10      1.49925e-13   1.25346e-13\n",
+       "  1.67162e-14  -1.21096e-11   5.10093e-7   …   1.74148e-11   1.37376e-11\n",
+       "  2.14649e-14  -7.77908e-11   4.40618e-7       4.34532e-12   4.04681e-12\n",
+       " -6.00393e-7    0.00195975   -9.3441e-8       -1.7715e-6    -1.00397e-5\n",
+       "  0.0           0.0           0.0              0.0           0.0\n",
+       " -1.60856e-46   1.27799e-45  -1.33364e-34      0.0           9.10949e-22\n",
+       "  1.67265e-47   9.5001e-46   -2.28985e-36  …   2.48673e-21   2.04963e-21\n",
+       " -1.70227e-25  -1.63193e-24   5.22568e-14      1.61638e-21   5.76934e-21\n",
+       "  2.00756e-29   3.84882e-28  -9.66608e-18     -8.08188e-22   4.40292e-21\n",
+       " -1.0           0.999996      1.0              1.0           1.0"
+      ]
+     },
+     "execution_count": 79,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "eigvecs(J_w)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 82,
+   "id": "927e9dc8",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "54×54 Diagonal{Float64, Vector{Float64}}:\n",
+       " -5.14029e8    ⋅           ⋅         …   ⋅    ⋅           ⋅ \n",
+       "   ⋅         -5.13152e8    ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅         -1.04749e8      ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅         …   ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅         …   ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "  ⋮                                  ⋱                   \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅         …   ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅         …   ⋅    ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅            0.0   ⋅           ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅   2.55572e-8   ⋅ \n",
+       "   ⋅           ⋅           ⋅             ⋅    ⋅          1.14485e-7"
+      ]
+     },
+     "execution_count": 82,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "F = eigen(J_w)\n",
+    "Q = eigvecs(F) # right eigenvectors \n",
+    "QL = inv(eigvecs(F)) # left eigenvectors \n",
+    "Λ = Diagonal(eigvals(F))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 86,
+   "id": "1a49e810",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "54.000000000000014"
+      ]
+     },
+     "execution_count": 86,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "EP = Diagonal(Q*QL)\n",
+    "sum(EP)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 87,
+   "id": "6db2f983",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "54×54 Diagonal{Float64, Vector{Float64}}:\n",
+       " 0.0185185   ⋅          ⋅         …   ⋅          ⋅          ⋅ \n",
+       "  ⋅         0.0185185   ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅         0.0185185      ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅         …   ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅         …   ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       " ⋮                                ⋱                        \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅         …   ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅         …   ⋅          ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅            0.0185185   ⋅          ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅         0.0185185   ⋅ \n",
+       "  ⋅          ⋅          ⋅             ⋅          ⋅         0.0185185"
+      ]
+     },
+     "execution_count": 87,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "EI = Diagonal(Q*QL)/sum(EP)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.6.0",
+   "language": "julia",
+   "name": "julia-1.6"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.6.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}