5_Opensystems-Copy1.ipynb

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    "# Open Systems and Nonlinear Interactions\n",
    "\n",
    "\n",
    "## 1. Mass Action Law of Open Systems\n",
    "\n",
    "Despite being developed in the context of closed chemical systems, the __Mass Action Law__ is also applicable to open systems, i.e., systems where there is no consevarion of mass. This is a special case in chemistry (open chemical reactors), but extremely relevanty for biological systems, since biological species are are often part of  open systems that exchanges mass, energy or information with the environment. Let's explore how the Mass action takes into account these type of open systems. The first and most simple example of open system is when a given reactant is degraded or flows away. This is illustrated as the following interaction scheme:\n",
    "\n",
    "$$   X_1  \\overset{k_1}{\\longrightarrow} 0 \\tag{1} $$ \n",
    "\n",
    "with initial concentration $X_1(0)$ and stoichometric matrices\n",
    "\n",
    "$$\n",
    "A=\\begin{bmatrix}\n",
    " 1\\end{bmatrix} ;\n",
    "B=\\begin{bmatrix}\n",
    "0   \n",
    "\\end{bmatrix} ; (B-A)^T= \\begin{bmatrix}\n",
    " - 1 \\end{bmatrix} \\tag{2}\n",
    "$$\n",
    " \n",
    "in this particular case\n",
    "\n",
    "$$\n",
    "K=\\begin{pmatrix}\n",
    " k_1\\\\ \n",
    "\\end{pmatrix} \\tag{3}\n",
    "$$\n",
    "\n",
    "and \n",
    "\n",
    "$$X^A=\\begin{pmatrix}\n",
    "X_1^1\n",
    "\\end{pmatrix} \\tag{4}\n",
    "$$\n",
    "\n",
    "so \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t} \\end{bmatrix}= \\begin{bmatrix}\n",
    " - 1 \\end{bmatrix} \\begin{bmatrix}\n",
    "k_1 \\end{bmatrix}\\begin{pmatrix}\n",
    "X_1\n",
    "\\end{pmatrix} = \\begin{bmatrix}\n",
    " - 1 \\end{bmatrix} \\begin{pmatrix}\n",
    "k_1  X_1\n",
    "\\end{pmatrix} = \\begin{bmatrix}\n",
    " - k_1 X_1 \\end{bmatrix}  \\tag{5}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "So the ODE is simply.\n",
    "\n",
    "$$\\frac{\\mathrm{d} X_1}{\\mathrm{d} t} =  - k_1 X_1 \\tag{6}$$ \n",
    "\n",
    "with a solution that is equivalent to the irreversible first order reaction.\n",
    " \n",
    " $$ X_1 (t)= X_1 (0) e^{- k_1 t}  \\tag{7}$$\n",
    " "
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  {
   "cell_type": "code",
   "execution_count": 45,
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    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using DifferentialEquations\n",
    "using ParameterizedFunctions\n",
    "using Plots \n",
    "gr()"
   ]
  },
  {
   "cell_type": "code",
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    "X₀ = [0.1,0.0]\n",
    "t₀=0\n",
    "t_final=10\n",
    "t=collect(t₀:0.1:t_final) # time\n",
    "k_1 = 1.\n",
    "plot(t,t->X₀[1]*exp(-k_1*t),label=\"\\\\ X_1\")\n",
    "title!(\"Degradation first order\")\n",
    "xaxis!(\"Time\")\n",
    "yaxis!(\"Concentration\")"
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    "Another simple example is the generation of a given reactant in the system, such as:\n",
    "$$   0  \\overset{k_1}{\\longrightarrow} X_1 \\tag{8} $$ \n",
    "\n",
    "with initial concentration $X_1(0)$ and stoichometric matrices\n",
    "\n",
    "$$\n",
    "A=\\begin{bmatrix}\n",
    " 0\\end{bmatrix} ;\n",
    "B=\\begin{bmatrix}\n",
    "1   \n",
    "\\end{bmatrix} ; (B-A)^T= \\begin{bmatrix}\n",
    " 1 \\end{bmatrix} \\tag{9}\n",
    "$$\n",
    " \n",
    "in this particular case\n",
    "\n",
    "$$\n",
    "K=\\begin{pmatrix}\n",
    " k_1\\\\ \n",
    "\\end{pmatrix} \\tag{10}\n",
    "$$\n",
    "\n",
    "and \n",
    "\n",
    "$$X^A=\\begin{pmatrix}\n",
    "X_1^0\n",
    "\\end{pmatrix}=\\begin{pmatrix}\n",
    "1\n",
    "\\end{pmatrix} \\tag{11}\n",
    "$$\n",
    "\n",
    "so \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t} \\end{bmatrix}= \\begin{bmatrix}\n",
    " 1 \\end{bmatrix} \\begin{bmatrix}\n",
    "k_1 \\end{bmatrix}\\begin{pmatrix}\n",
    "1\n",
    "\\end{pmatrix}  \\tag{12}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "So, sor this particular case, the corresponding differential equation is simply:\n",
    "\n",
    " $$\\frac{\\mathrm{d} X_1}{\\mathrm{d} t} =  k_1 \\tag{13}$$ \n",
    " \n",
    " which when we integrate we obtain the following solution:\n",
    " \n",
    "$$\\begin{align}\n",
    "\\int \\mathrm{d} X_1 &= \\int k_1  \\mathrm{d} t  \\tag{14}\\\\\n",
    " X_1(t) &=  k_1  \\cdot t + C \\tag{15}\n",
    "\\end{align} $$\n",
    "\n",
    "again, to calculate the constant $C$ we use the initial condition for $t=0$, so $ X_1(0)=C$. Therefore:\n",
    "$$ X_1(t) =  k_1  \\cdot t + X_1(0) \\tag{16}$$ \n",
    "which shows that the amount of $X_1$ grows linearly with $t$."
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   "source": [
    "plot(t,t->X₀[1]+k_1*t,label=\"\\\\ X_1\")\n",
    "title!(\"Production zero order\")\n",
    "xaxis!(\"Time\")\n",
    "yaxis!(\"Concentration\")"
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   "source": [
    "Let's now explore some special cases of open systems that look in principle very similar to the previous one:\n",
    "\n",
    "$$   X_1  \\overset{k_1}{\\longrightarrow} 2 X_1 \\tag{17} $$ \n",
    "\n",
    "and this one\n",
    "\n",
    "$$   2X_1  \\overset{k_1}{\\longrightarrow} 3 X_1 \\tag{18} $$ \n",
    "\n",
    "These two reaction schemes seem very similar to the previous case of linear producton, since for every reaction you end up with a new molecule of your reactant $X_1$, so we expect a similar dynamics. On the contrary, when we write the differential equations for these two cases we have:\n",
    "\n",
    "\n",
    "$$\n",
    "A=\\begin{bmatrix}\n",
    " 1\\end{bmatrix} ;\n",
    "B=\\begin{bmatrix}\n",
    "2   \n",
    "\\end{bmatrix} ; (B-A)^T= \\begin{bmatrix}\n",
    " 1 \\end{bmatrix} \\tag{19}\n",
    "$$\n",
    " \n",
    "and for the second order production case\n",
    "$$\n",
    "A=\\begin{bmatrix}\n",
    " 2\\end{bmatrix} ;\n",
    "B=\\begin{bmatrix}\n",
    "3   \n",
    "\\end{bmatrix} ; (B-A)^T= \\begin{bmatrix}\n",
    " 1 \\end{bmatrix} \\tag{20}\n",
    "$$\n",
    "\n",
    "In fact, $(B-A)^T$ has the same value in the three cases, but $X^A$ does not. For the first order production\n",
    "\n",
    "$$X^A=\\begin{pmatrix}\n",
    "X_1^1\n",
    "\\end{pmatrix} \\tag{21}\n",
    "$$\n",
    "\n",
    "and for the second order\n",
    "\n",
    "$$X^A=\\begin{pmatrix}\n",
    "X_1^2\n",
    "\\end{pmatrix} \\tag{22}\n",
    "$$\n",
    "\n",
    "so, following the Mass acton general form, for the first order productionreaction:\n",
    "\n",
    " $$\\frac{\\mathrm{d} X_1}{\\mathrm{d} t} =  k_1 \\cdot X_1 \\tag{23}$$  \n",
    " \n",
    " and for the second order production \n",
    " \n",
    " $$\\frac{\\mathrm{d} X_1}{\\mathrm{d} t} = k_1 \\cdot X_1^2 \\tag{24}$$ \n",
    " \n",
    " \n",
    " We know the solution for the first order production:\n",
    " \n",
    "  $$ X_1 (t)= X_1 (0) \\cdot e^{k_1 \\cdot t}  \\tag{25}$$\n",
    " \n",
    "withe the second one is :\n",
    "\n",
    "$$\\begin{align}\n",
    "\\int \\frac{\\mathrm{d} X_1}{X_1^2} &=  \\int k_1 \\cdot \\mathrm{d}t \\tag{26} \\\\\n",
    "-X_1^{-1} &=  k_1 \\cdot t + C \\tag{27}\\\\\n",
    "X_1(t) &=\\frac{-1}{k_1 \\cdot t + C}  \\tag{28}\n",
    "\\end{align}$$\n",
    "\n",
    "using the intial condition of $X_1(t=0)$ we obtain the value of the integration constant. \n",
    "$$\\begin{align}\n",
    "X_1(0) &=\\frac{-1}{k_1 \\cdot 0 + C} = \\frac{-1}{C}  \\tag{29}\\\\\n",
    "C &= \\frac{-1}{X_1(0)}  \\tag{30}\\\\\n",
    "\\end{align}$$\n",
    "\n",
    "so the final equation of is\n",
    "$$\\begin{align}\n",
    "X_1(t) &=\\frac{-1}{k_1  t - \\frac{1}{X_1(0)} } \\tag{31}\n",
    "\\end{align}$$\n",
    "so, multiplying numerator and denominator of the right side of the equation by $-X_1(0)$ we obtain\n",
    "$$\\begin{align}\n",
    "X_1(t) &=\\frac{(-1)\\cdot(-X_1(0))}{(-X_1(0))(k_1 \\cdot t  - \\frac{1}{X_1(0)}) }= \\frac{X_1(0)}{(1-X_1(0)k_1\\cdot t)}\\tag{32}\n",
    "\\end{align}$$"
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     "execution_count": 48,
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   "source": [
    "X₀ = [0.5,0.0]\n",
    "k_1 = 0.1\n",
    "plot(t,t->X₀[1]+k_1*t,label=\"\\\\ zero order reaction \\\\ X_1 \")\n",
    "k_1 = 0.1\n",
    "plot!(t,t->X₀[1]*exp(k_1*t),label=\" first order reaction \\\\ X_1 \")\n",
    "k_1 = 0.1\n",
    "plot!(t,t->X₀[1]/(1-k_1*t*X₀[1]),label=\" Second order reaction \\\\ X_1 \")\n",
    "title!(\"Production  first order\")\n",
    "xaxis!(\"Time\")\n",
    "yaxis!(\"Concentration\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that, despite being similar in terms of the production of molecules of $X_1$ per reaction, the dynamics of zero, first and second order production of a given species are completely different, even when using identical parameter values. Also, the amount of $X-1$ produced for the first and second order reactions depends strongly on the initial condition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 780.245, 1479.62)\" x=\"780.245\" y=\"1479.62\">2.5</text>\n",
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       "<g clip-path=\"url(#clip6500)\">\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 1284.25, 1559.48)\" x=\"1284.25\" y=\"1559.48\">Time</text>\n",
       "</g>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 89.2861, 773.647)\" x=\"89.2861\" y=\"773.647\">Concentration</text>\n",
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       "  679.443,1279.37 699.604,1273.9 719.764,1268.43 739.924,1262.97 760.085,1257.5 780.245,1252.03 800.406,1246.57 820.566,1241.1 840.726,1235.63 860.887,1230.16 \n",
       "  881.047,1224.7 901.207,1219.23 921.368,1213.76 941.528,1208.29 961.689,1202.83 981.849,1197.36 1002.01,1191.89 1022.17,1186.43 1042.33,1180.96 1062.49,1175.49 \n",
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       "  1687.46,1006.01 1707.62,1000.54 1727.78,995.071 1747.94,989.604 1768.1,984.137 1788.26,978.669 1808.43,973.202 1828.59,967.735 1848.75,962.268 1868.91,956.8 \n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1654.35, 273.581)\" x=\"1654.35\" y=\"273.581\"> zero order reaction  X</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 2130.54, 287.21)\" x=\"2130.54\" y=\"287.21\">1 </text>\n",
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     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X₀ = [0.75,0.0]\n",
    "k_1 = 0.1\n",
    "plot(t,t->X₀[1]+k_1*t,label=\"\\\\ zero order reaction \\\\ X_1 \")\n",
    "k_1 = 0.1\n",
    "plot!(t,t->X₀[1]*exp(k_1*t),label=\" first order reaction \\\\ X_1 \")\n",
    "k_1 = 0.1\n",
    "plot!(t,t->X₀[1]/(1-k_1*t*X₀[1]),label=\" Second order reaction \\\\ X_1 \")\n",
    "title!(\"Production  first order\")\n",
    "xaxis!(\"Time\")\n",
    "yaxis!(\"Concentration\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This first and second order production are basic examples of a nonlinear system. In this case, these type of system are defined as autocatalytic, where the production of a given species depends on the amount of the same species. This again is quite difficult to picture in the context of a chemical reaction. On the other hand, nature is full of systems where autocalysis is a main feature or, systems that are open in the sense that they can exchange mass, energy or information with their surroundings (for instance, a cell). From the Physics point of view, this constant supply or removal of a species in a system of interacting entities can push it away from its thermodinamic equilibrium, driving very important features that are essential for live itself (as we will see later in the course).  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example of Open System: The Brusselator\n",
    "\n",
    "The Brusselator is a simplified model for a very famous chemical reaction, the __Belousov-Zhabotinsky__ reaction. This reactions has presented a very impornat challenge in the fields of Chemistry and Physics, because it was thought that it broke the second principle of Thermodynamics. You can find more History about the BZ system here. The reaction scheme is as follows:\n",
    "$$\n",
    "\\begin{align}\n",
    "0  &\\overset{k_1}{\\underset{k_2}{\\longleftrightarrow}} x \\tag{33} \\\\\n",
    "x &\\overset{k_3}\\longrightarrow y \\tag{34}\\\\\n",
    "2x + y &\\overset{k_4}\\longrightarrow 3x \\tag{35}\\\\\n",
    "\\end{align}\n",
    " $$\n",
    "\n",
    "where `x, b, y, d` are the reactants species. We will use the general formulation of the mass action in differential form to derive the differentiatl equations that govern the dynamics of the system. The state vector can be defined as $X= [x,b,y,d]$.\n",
    "$$\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d} X}{\\mathrm{d} t}&= (B-A)^T \\cdot K \\cdot X^A \\tag{36}\\\\\n",
    "\\end{align}\n",
    "$$\n",
    " \n",
    "First, we write the matrices of stoichiometric coefficients:\n",
    "\n",
    "$$\n",
    "A=\\begin{bmatrix}\n",
    "  0 & 0  \\\\  \n",
    "  1 & 0   \\\\\n",
    "  1 & 0   \\\\\n",
    "  2 & 1    \\end{bmatrix} ;\n",
    "B=\\begin{bmatrix}\n",
    "  1 & 0   \\\\ \n",
    "  0 & 0   \\\\\n",
    "  0 & 1   \\\\\n",
    "  3 & 0     \\end{bmatrix}; \\tag{37}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The stoichiometric matrix is [1 -1 -1 1; 0 0 1 -1] \n"
     ]
    }
   ],
   "source": [
    "A = [0 0;1 0;1 0 ;2 1]\n",
    "B = [1 0;0 0;0 1;3 0]\n",
    "stoichiometric_matrix= (B-A)'\n",
    "println(\"The stoichiometric matrix is $stoichiometric_matrix \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "in this particular case\n",
    "\n",
    "$$\n",
    "K=\\begin{pmatrix}\n",
    " k_1 & 0 & 0 & 0  \\tag{38}\\\\ \n",
    " 0 &  k_2 & 0 & 0 \\\\ \n",
    " 0 &  0 & k_3 & 0 \\\\ \n",
    " 0 &  0 & 0 &  k_4 \n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "and \n",
    "\n",
    "\n",
    "$$X^A=\\begin{pmatrix}\n",
    "X_1^0\\cdot X_2^0  \\\\\n",
    "X_1^1\\cdot X_2^0  \\\\\n",
    "X_1^1\\cdot X_2^0  \\\\\n",
    "X_1^2\\cdot X_2^1  \n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "1 \\\\\n",
    " X_1 \\\\\n",
    " X_1 \\\\\n",
    " X_1^2 \\cdot X_2\n",
    "\\end{pmatrix} \\tag{39}\n",
    "$$\n",
    "\n",
    "so, the equations that define the system are\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\end{bmatrix}& \n",
    "=  \\begin{bmatrix} 1 & -1 & - 1 & 1  \\\\ 0 & 0 &  1 & -1  \\end{bmatrix}\n",
    "\\begin{pmatrix}\n",
    " k_1 & 0 & 0 & 0  \\\\ \n",
    " 0 &  k_2 & 0 & 0 \\\\ \n",
    " 0 &  0 & k_3 & 0 \\\\ \n",
    " 0 &  0 & 0 &  k_4 \n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "1 \\\\\n",
    " X_1 \\\\\n",
    " X_1 \\\\\n",
    " X_1^2 \\cdot X_2\n",
    "\\end{pmatrix} \n",
    "\\end{align} \\tag{40}\n",
    "$$\n",
    "and after multiplying the matrices\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\end{bmatrix}& \n",
    "=  \\begin{bmatrix} 1 & -1 & - 1 & 1  \\\\ 0 & 0 &  1 & -1  \\end{bmatrix}\n",
    "\\begin{pmatrix}\n",
    "k_1 \\\\\n",
    " k_2 \\cdot X_1 \\\\\n",
    " k_3 \\cdot X_1 \\\\\n",
    " k_4 \\cdot X_1^2 \\cdot X_2\n",
    "\\end{pmatrix}  \\tag{41}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "\n",
    "so finally, \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \n",
    "\\end{bmatrix}&= \n",
    " \\begin{pmatrix}\n",
    " k_1  - k_2 \\cdot X_1 - k_3 \\cdot X_1 +  k_4 \\cdot X_1^2 \\cdot X_2  \\\\\n",
    " k_3 \\cdot X_1 -  k_4 \\cdot X_1^2 \\cdot X_2 \n",
    "\\end{pmatrix} \\tag{42}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Therefore, the  equations for the evolution of `[x]` and `[y]` are as follows:\n",
    "\n",
    "$$\\begin{align}       \n",
    "            \\frac{ d[x] }{dt} &= k_1  - (k_2+k_3) \\cdot x + k_4 \\cdot x^2 \\cdot y \\tag{43} \\\\ \n",
    "            \\frac{ d[y] }{dt} &= k_3 \\cdot x -  k_4 \\cdot x^2 \\cdot y  \\tag{44}   \n",
    "            \\end{align}            $$\n",
    "\n",
    "We assume as initial conditions:\n",
    "\n",
    "$$\\begin{align}       \n",
    "            [x (0)] &= 0 \\tag{45} \\\\ \n",
    "            [y (0)] &= 0   \\tag{46}   \n",
    "            \\end{align}            \n",
    "            $$\n",
    "           \n",
    "To calculate the equilibrium, we just set teh two equations to zero and solve for `X` and `Y`. \n",
    "$$\\begin{align}       \n",
    "            [x]_{eq} &= \\frac{k_1}{k_4}\\tag{47} \\\\ \n",
    "            [y]_{eq} &= \\frac{k_4  k_2}{k_3  k_1} \\tag{48}   \n",
    "            \\end{align}            $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using DifferentialEquations\n",
    "using ParameterizedFunctions\n",
    "using Plots;  gr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::BR{var\"#83#87\",var\"#84#88\",var\"#85#89\",Nothing,Nothing,var\"#86#90\",Expr,Expr}) (generic function with 2 methods)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "brusselator3! = @ode_def BR begin\n",
    "    dX = k_1  - (k_2+k_3) * X  + k_4 * X^2 * Y \n",
    "    dY = k_3 * X - k_4 * X^2 * Y\n",
    "    end k_1 k_2 k_3 k_4 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "tspan = (0.0,50.0)\n",
    "k_1=1.\n",
    "k_2=1.\n",
    "k_3=3.\n",
    "k_4=1.\n",
    "B=3.\n",
    "D=1.\n",
    "u₀=[0.0,0.0]\n",
    "p=[k_1,k_2,k_3,k_4];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
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       "  502.416,1255.34 503.972,1253.74 505.528,1252.07 507.084,1250.33 508.641,1248.51 510.197,1246.6 511.753,1244.58 513.309,1242.44 514.866,1240.18 516.422,1237.77 \n",
       "  517.978,1235.19 519.534,1232.44 521.09,1229.49 522.647,1226.32 524.203,1222.91 525.759,1219.24 527.315,1215.29 528.871,1211.02 530.428,1206.38 531.984,1201.19 \n",
       "  533.54,1195.49 535.096,1189.29 536.653,1182.56 538.209,1175.2 539.765,1167.09 541.321,1158 542.877,1147.7 544.434,1135.86 545.99,1122.14 547.546,1106.1 \n",
       "  549.102,1087.27 550.658,1065.13 552.215,1039.04 553.771,1007.82 555.327,970.182 556.883,924.521 558.44,869.209 559.996,803.898 561.552,729.731 563.108,649.368 \n",
       "  564.664,569.633 566.221,500.707 567.777,450.204 569.333,420.782 570.889,410.169 572.445,413.98 574.002,426.854 575.558,445.634 577.114,467.917 578.67,491.979 \n",
       "  580.227,516.941 581.783,542.198 583.339,567.381 584.895,592.276 586.451,616.745 588.008,640.706 589.564,664.119 591.12,686.959 592.676,709.217 594.233,730.889 \n",
       "  595.789,751.983 597.345,772.512 598.901,792.486 600.457,811.916 602.014,830.814 603.57,849.195 605.126,867.071 606.682,884.456 608.238,901.364 609.795,917.806 \n",
       "  611.351,933.791 612.907,949.331 614.463,964.435 616.02,979.112 617.576,993.37 619.132,1007.22 620.688,1020.66 622.244,1033.7 623.801,1046.34 625.357,1058.59 \n",
       "  626.913,1070.46 628.469,1081.95 630.025,1093.05 631.582,1103.78 633.138,1114.14 634.694,1124.12 636.25,1133.73 637.807,1142.98 639.363,1151.86 640.919,1160.38 \n",
       "  642.475,1168.55 644.031,1176.36 645.588,1183.82 647.144,1190.95 648.7,1197.73 650.256,1204.19 651.812,1210.32 653.369,1216.13 654.925,1221.63 656.481,1226.84 \n",
       "  658.037,1231.76 659.594,1236.39 661.15,1240.75 662.706,1244.85 664.262,1248.7 665.818,1252.3 667.375,1255.66 668.931,1258.81 670.487,1261.74 672.043,1264.47 \n",
       "  673.6,1267.01 675.156,1269.37 676.712,1271.56 678.268,1273.6 679.824,1275.48 681.381,1277.22 682.937,1278.82 684.493,1280.3 686.049,1281.65 687.605,1282.89 \n",
       "  689.162,1284.03 690.718,1285.07 692.274,1286.02 693.83,1286.88 695.387,1287.66 696.943,1288.36 698.499,1289 700.055,1289.57 701.611,1290.07 703.168,1290.52 \n",
       "  704.724,1290.91 706.28,1291.25 707.836,1291.54 709.392,1291.79 710.949,1292 712.505,1292.16 714.061,1292.29 715.617,1292.38 717.174,1292.44 718.73,1292.47 \n",
       "  720.286,1292.47 721.842,1292.45 723.398,1292.39 724.955,1292.31 726.511,1292.21 728.067,1292.08 729.623,1291.93 731.179,1291.77 732.736,1291.58 734.292,1291.37 \n",
       "  735.848,1291.14 737.404,1290.9 738.961,1290.63 740.517,1290.36 742.073,1290.06 743.629,1289.75 745.185,1289.42 746.742,1289.07 748.298,1288.71 749.854,1288.34 \n",
       "  751.41,1287.95 752.966,1287.54 754.523,1287.12 756.079,1286.68 757.635,1286.23 759.191,1285.77 760.748,1285.29 762.304,1284.79 763.86,1284.28 765.416,1283.75 \n",
       "  766.972,1283.21 768.529,1282.65 770.085,1282.07 771.641,1281.47 773.197,1280.86 774.754,1280.22 776.31,1279.57 777.866,1278.9 779.422,1278.2 780.978,1277.49 \n",
       "  782.535,1276.75 784.091,1275.99 785.647,1275.2 787.203,1274.39 788.759,1273.56 790.316,1272.7 791.872,1271.81 793.428,1270.89 794.984,1269.94 796.541,1268.95 \n",
       "  798.097,1267.93 799.653,1266.88 801.209,1265.78 802.765,1264.64 804.322,1263.46 805.878,1262.23 807.434,1260.95 808.99,1259.61 810.546,1258.22 812.103,1256.77 \n",
       "  813.659,1255.22 815.215,1253.56 816.771,1251.81 818.328,1249.98 819.884,1248.07 821.44,1246.09 822.996,1244.03 824.552,1241.88 826.109,1239.64 827.665,1237.29 \n",
       "  829.221,1234.81 830.777,1232.2 832.333,1229.41 833.89,1226.43 835.446,1223.22 837.002,1219.75 838.558,1215.98 840.115,1211.86 841.671,1207.36 843.227,1202.42 \n",
       "  844.783,1196.99 846.339,1191.02 847.896,1184.43 849.452,1177.18 851.008,1169.06 852.564,1159.83 854.12,1149.48 855.677,1137.88 857.233,1124.77 858.789,1109.73 \n",
       "  860.345,1092.22 861.902,1071.55 863.458,1046.9 865.014,1017.3 866.57,981.66 868.126,938.739 869.683,887.167 871.239,825.711 872.795,754.425 874.351,675.522 \n",
       "  875.908,595.805 877.464,523.857 879.02,467.508 880.576,432.093 882.132,416.145 883.689,415.806 885.245,425.932 886.801,442.931 888.357,464.199 889.913,487.622 \n",
       "  891.47,512.208 893.026,537.266 894.582,562.363 896.138,587.246 897.695,611.75 899.251,635.775 900.807,659.271 902.363,682.207 903.919,704.568 905.476,726.347 \n",
       "  907.032,747.551 908.588,768.191 910.144,788.274 911.7,807.811 913.257,826.816 914.813,845.3 916.369,863.278 917.925,880.764 919.482,897.771 921.038,914.309 \n",
       "  922.594,930.389 924.15,946.022 925.706,961.217 927.263,975.984 928.819,990.33 930.375,1004.26 931.931,1017.79 933.487,1030.91 935.044,1043.64 936.6,1055.98 \n",
       "  938.156,1067.93 939.712,1079.5 941.269,1090.68 942.825,1101.49 944.381,1111.93 945.937,1121.99 947.493,1131.68 949.05,1141.01 950.606,1149.97 952.162,1158.56 \n",
       "  953.718,1166.81 955.274,1174.69 956.831,1182.23 958.387,1189.43 959.943,1196.29 961.499,1202.81 963.056,1209.01 964.612,1214.89 966.168,1220.47 967.724,1225.74 \n",
       "  969.28,1230.72 970.837,1235.41 972.393,1239.83 973.949,1243.98 975.505,1247.88 977.062,1251.53 978.618,1254.95 980.174,1258.14 981.73,1261.11 983.286,1263.88 \n",
       "  984.843,1266.47 986.399,1268.87 987.955,1271.1 989.511,1273.17 991.067,1275.08 992.624,1276.85 994.18,1278.48 995.736,1279.98 997.292,1281.36 998.849,1282.63 \n",
       "  1000.4,1283.79 1001.96,1284.85 1003.52,1285.82 1005.07,1286.7 1006.63,1287.49 1008.19,1288.22 1009.74,1288.87 1011.3,1289.45 1012.85,1289.97 1014.41,1290.42 \n",
       "  1015.97,1290.83 1017.52,1291.18 1019.08,1291.48 1020.64,1291.74 1022.19,1291.95 1023.75,1292.13 1025.3,1292.26 1026.86,1292.36 1028.42,1292.43 1029.97,1292.47 \n",
       "  1031.53,1292.47 1033.09,1292.45 1034.64,1292.4 1036.2,1292.33 1037.75,1292.23 1039.31,1292.11 1040.87,1291.97 1042.42,1291.8 1043.98,1291.62 1045.53,1291.41 \n",
       "  1047.09,1291.19 1048.65,1290.95 1050.2,1290.69 1051.76,1290.42 1053.32,1290.12 1054.87,1289.81 1056.43,1289.49 1057.98,1289.15 1059.54,1288.79 1061.1,1288.42 \n",
       "  1062.65,1288.03 1064.21,1287.63 1065.77,1287.21 1067.32,1286.78 1068.88,1286.33 1070.43,1285.87 1071.99,1285.39 1073.55,1284.9 1075.1,1284.39 1076.66,1283.87 \n",
       "  1078.22,1283.33 1079.77,1282.77 1081.33,1282.19 1082.88,1281.6 1084.44,1280.99 1086,1280.36 1087.55,1279.71 1089.11,1279.04 1090.67,1278.35 1092.22,1277.64 \n",
       "  1093.78,1276.91 1095.33,1276.15 1096.89,1275.37 1098.45,1274.57 1100,1273.74 1101.56,1272.89 1103.11,1272 1104.67,1271.09 1106.23,1270.15 1107.78,1269.17 \n",
       "  1109.34,1268.16 1110.9,1267.11 1112.45,1266.02 1114.01,1264.89 1115.56,1263.71 1117.12,1262.49 1118.68,1261.22 1120.23,1259.89 1121.79,1258.46 1123.35,1256.93 \n",
       "  1124.9,1255.32 1126.46,1253.64 1128.01,1251.9 1129.57,1250.09 1131.13,1248.23 1132.68,1246.3 1134.24,1244.3 1135.8,1242.23 1137.35,1240.06 1138.91,1237.79 \n",
       "  1140.46,1235.39 1142.02,1232.84 1143.58,1230.12 1145.13,1227.21 1146.69,1224.06 1148.25,1220.65 1149.8,1216.94 1151.36,1212.88 1152.91,1208.44 1154.47,1203.57 \n",
       "  1156.03,1198.22 1157.58,1192.34 1159.14,1185.86 1160.69,1178.74 1162.25,1170.85 1163.81,1161.89 1165.36,1151.84 1166.92,1140.57 1168.48,1127.81 1170.03,1113.18 \n",
       "  1171.59,1096.18 1173.14,1076.17 1174.7,1052.36 1176.26,1023.88 1177.81,989.693 1179.37,948.6 1180.93,899.049 1182.48,839.714 1184.04,770.072 1185.59,692.702 \n",
       "  1187.15,612.834 1188.71,538.239 1190.26,478.21 1191.82,438.257 1193.38,418.238 1194.93,414.691 1196.49,423.332 1198.04,439.175 1199.6,459.56 1201.16,482.571 \n",
       "  1202.71,506.947 1204.27,531.937 1205.83,557.047 1207.38,581.984 1208.94,606.574 1210.49,630.704 1212.05,654.313 1213.61,677.368 1215.16,699.85 1216.72,721.754 \n",
       "  1218.27,743.078 1219.83,763.833 1221.39,784.032 1222.94,803.684 1224.5,822.802 1226.06,841.397 1227.61,859.483 1229.17,877.073 1230.72,894.18 1232.28,910.817 \n",
       "  1233.84,926.994 1235.39,942.721 1236.95,958.009 1238.51,972.866 1240.06,987.301 1241.62,1001.32 1243.17,1014.93 1244.73,1028.14 1246.29,1040.96 1247.84,1053.38 \n",
       "  1249.4,1065.41 1250.96,1077.06 1252.51,1088.33 1254.07,1099.22 1255.62,1109.73 1257.18,1119.87 1258.74,1129.64 1260.29,1139.04 1261.85,1148.08 1263.41,1156.76 \n",
       "  1264.96,1165.08 1266.52,1173.04 1268.07,1180.65 1269.63,1187.92 1271.19,1194.85 1272.74,1201.45 1274.3,1207.71 1275.85,1213.66 1277.41,1219.3 1278.97,1224.63 \n",
       "  1280.52,1229.67 1282.08,1234.43 1283.64,1238.9 1285.19,1243.11 1286.75,1247.07 1288.3,1250.77 1289.86,1254.24 1291.42,1257.48 1292.97,1260.5 1294.53,1263.31 \n",
       "  1296.09,1265.93 1297.64,1268.37 1299.2,1270.64 1300.75,1272.74 1302.31,1274.69 1303.87,1276.48 1305.42,1278.14 1306.98,1279.67 1308.54,1281.08 1310.09,1282.37 \n",
       "  1311.65,1283.55 1313.2,1284.63 1314.76,1285.62 1316.32,1286.52 1317.87,1287.33 1319.43,1288.07 1320.99,1288.73 1322.54,1289.33 1324.1,1289.86 1325.65,1290.33 \n",
       "  1327.21,1290.75 1328.77,1291.11 1330.32,1291.42 1331.88,1291.69 1333.43,1291.91 1334.99,1292.09 1336.55,1292.24 1338.1,1292.35 1339.66,1292.42 1341.22,1292.46 \n",
       "  1342.77,1292.48 1344.33,1292.46 1345.88,1292.42 1347.44,1292.35 1349,1292.25 1350.55,1292.14 1352.11,1292 1353.67,1291.84 1355.22,1291.66 1356.78,1291.46 \n",
       "  1358.33,1291.24 1359.89,1291 1361.45,1290.75 1363,1290.48 1364.56,1290.19 1366.12,1289.88 1367.67,1289.56 1369.23,1289.22 1370.78,1288.87 1372.34,1288.5 \n",
       "  1373.9,1288.11 1375.45,1287.72 1377.01,1287.3 1378.57,1286.87 1380.12,1286.43 1381.68,1285.97 1383.23,1285.49 1384.79,1285.01 1386.35,1284.5 1387.9,1283.98 \n",
       "  1389.46,1283.44 1391.01,1282.89 1392.57,1282.32 1394.13,1281.73 1395.68,1281.12 1397.24,1280.5 1398.8,1279.85 1400.35,1279.19 1401.91,1278.5 1403.46,1277.8 \n",
       "  1405.02,1277.07 1406.58,1276.32 1408.13,1275.54 1409.69,1274.74 1411.25,1273.92 1412.8,1273.07 1414.36,1272.19 1415.91,1271.28 1417.47,1270.35 1419.03,1269.38 \n",
       "  1420.58,1268.37 1422.14,1267.33 1423.7,1266.25 1425.25,1265.13 1426.81,1263.97 1428.36,1262.76 1429.92,1261.5 1431.48,1260.19 1433.03,1258.83 1434.59,1257.4 \n",
       "  1436.15,1255.87 1437.7,1254.24 1439.26,1252.53 1440.81,1250.73 1442.37,1248.87 1443.93,1246.92 1445.48,1244.91 1447.04,1242.81 1448.59,1240.62 1450.15,1238.33 \n",
       "  1451.71,1235.92 1453.26,1233.37 1454.82,1230.66 1456.38,1227.76 1457.93,1224.65 1459.49,1221.29 1461.04,1217.65 1462.6,1213.68 1464.16,1209.34 1465.71,1204.58 \n",
       "  1467.27,1199.36 1468.83,1193.63 1470.38,1187.31 1471.94,1180.36 1473.49,1172.72 1475.05,1164.3 1476.61,1154.94 1478.16,1144.37 1479.72,1132.25 1481.28,1118.14 \n",
       "  1482.83,1101.52 1484.39,1081.77 1485.94,1058.19 1487.5,1029.98 1489.06,996.271 1490.61,956.091 1492.17,908.382 1493.73,851.999 1495.28,785.708 1496.84,709.658 \n",
       "  1498.39,629.291 1499.95,552.462 1501.51,487.987 1503.06,443.335 1504.62,420.385 1506.17,414.783 1507.73,420.874 1509.29,435.372 1510.84,454.974 1512.4,477.548 \n",
       "  1513.96,501.724 1515.51,526.627 1517.07,551.74 1518.62,576.731 1520.18,601.403 1521.74,625.637 1523.29,649.362 1524.85,672.533 1526.41,695.136 1527.96,717.163 \n",
       "  1529.52,738.613 1531.07,759.491 1532.63,779.805 1534.19,799.566 1535.74,818.791 1537.3,837.496 1538.86,855.69 1540.41,873.386 1541.97,890.595 1543.52,907.331 \n",
       "  1545.08,923.604 1546.64,939.426 1548.19,954.806 1549.75,969.754 1551.3,984.277 1552.86,998.385 1554.42,1012.08 1555.97,1025.38 1557.53,1038.27 1559.09,1050.78 \n",
       "  1560.64,1062.89 1562.2,1074.62 1563.75,1085.97 1565.31,1096.94 1566.87,1107.53 1568.42,1117.75 1569.98,1127.6 1571.54,1137.08 1573.09,1146.2 1574.65,1154.95 \n",
       "  1576.2,1163.35 1577.76,1171.38 1579.32,1179.07 1580.87,1186.41 1582.43,1193.41 1583.99,1200.08 1585.54,1206.42 1587.1,1212.43 1588.65,1218.13 1590.21,1223.53 \n",
       "  1591.77,1228.62 1593.32,1233.43 1594.88,1237.96 1596.44,1242.22 1597.99,1246.23 1599.55,1249.98 1601.1,1253.5 1602.66,1256.79 1604.22,1259.87 1605.77,1262.73 \n",
       "  1607.33,1265.4 1608.88,1267.88 1610.44,1270.18 1612,1272.31 1613.55,1274.29 1615.11,1276.11 1616.67,1277.8 1618.22,1279.36 1619.78,1280.79 1621.33,1282.1 \n",
       "  1622.89,1283.31 1624.45,1284.41 1626,1285.42 1627.56,1286.34 1629.12,1287.17 1630.67,1287.92 1632.23,1288.6 1633.78,1289.2 1635.34,1289.75 1636.9,1290.23 \n",
       "  1638.45,1290.66 1640.01,1291.03 1641.57,1291.36 1643.12,1291.63 1644.68,1291.87 1646.23,1292.06 1647.79,1292.21 1649.35,1292.33 1650.9,1292.41 1652.46,1292.46 \n",
       "  1654.02,1292.48 1655.57,1292.47 1657.13,1292.43 1658.68,1292.36 1660.24,1292.28 1661.8,1292.17 1663.35,1292.03 1664.91,1291.88 1666.46,1291.7 1668.02,1291.5 \n",
       "  1669.58,1291.29 1671.13,1291.05 1672.69,1290.8 1674.25,1290.53 1675.8,1290.25 1677.36,1289.95 1678.91,1289.63 1680.47,1289.3 1682.03,1288.95 1683.58,1288.58 \n",
       "  1685.14,1288.2 1686.7,1287.81 1688.25,1287.4 1689.81,1286.97 1691.36,1286.53 1692.92,1286.07 1694.48,1285.59 1696.03,1285.11 1697.59,1284.6 1699.15,1284.08 \n",
       "  1700.7,1283.55 1702.26,1282.99 1703.81,1282.43 1705.37,1281.84 1706.93,1281.24 1708.48,1280.62 1710.04,1279.98 1711.6,1279.32 1713.15,1278.64 1714.71,1277.94 \n",
       "  1716.26,1277.22 1717.82,1276.48 1719.38,1275.71 1720.93,1274.92 1722.49,1274.11 1724.04,1273.26 1725.6,1272.39 1727.16,1271.49 1728.71,1270.56 1730.27,1269.59 \n",
       "  1731.83,1268.59 1733.38,1267.55 1734.94,1266.47 1736.49,1265.35 1738.05,1264.19 1739.61,1262.99 1741.16,1261.74 1742.72,1260.45 1744.28,1259.1 1745.83,1257.69 \n",
       "  1747.39,1256.23 1748.94,1254.7 1750.5,1253.09 1752.06,1251.41 1753.61,1249.64 1755.17,1247.78 1756.73,1245.82 1758.28,1243.75 1759.84,1241.56 1761.39,1239.25 \n",
       "  1762.95,1236.79 1764.51,1234.15 1766.06,1231.21 1767.62,1228.01 1769.18,1224.61 1770.73,1221.02 1772.29,1217.26 1773.84,1213.3 1775.4,1209.12 1776.96,1204.66 \n",
       "  1778.51,1199.85 1780.07,1194.62 1781.62,1188.83 1783.18,1182.38 1784.74,1175.11 1786.29,1166.86 1787.85,1157.45 1789.41,1146.66 1790.96,1134.29 1792.52,1120.08 \n",
       "  1794.07,1103.78 1795.63,1084.97 1797.19,1062.85 1798.74,1036.57 1800.3,1005.15 1801.86,967.223 1803.41,921.217 1804.97,866.093 1806.52,801.35 1808.08,727.127 \n",
       "  1809.64,647.095 1811.19,568.989 1812.75,501.54 1814.31,452.971 1815.86,424.665 1817.42,414.426 1818.97,418.387 1820.53,431.393 1822.09,450.059 1823.64,472.272 \n",
       "  1825.2,496.249 1826.76,521.066 1828.31,546.179 1829.87,571.223 1831.42,595.986 1832.98,620.33 1834.54,644.171 1836.09,667.471 1837.65,690.204 1839.2,712.359 \n",
       "  1840.76,733.935 1842.32,754.937 1843.87,775.377 1845.43,795.265 1846.99,814.612 1848.54,833.431 1850.1,851.735 1851.65,869.537 1853.21,886.852 1854.77,903.691 \n",
       "  1856.32,920.065 1857.88,935.985 1859.44,951.462 1860.99,966.504 1862.55,981.121 1864.1,995.319 1865.66,1009.11 1867.22,1022.49 1868.77,1035.47 1870.33,1048.06 \n",
       "  1871.89,1060.26 1873.44,1072.08 1875,1083.51 1876.55,1094.56 1878.11,1105.24 1879.67,1115.54 1881.22,1125.47 1882.78,1135.03 1884.34,1144.23 1885.89,1153.06 \n",
       "  1887.45,1161.53 1889,1169.65 1890.56,1177.41 1892.12,1184.83 1893.67,1191.9 1895.23,1198.64 1896.78,1205.05 1898.34,1211.13 1899.9,1216.9 1901.45,1222.37 \n",
       "  1903.01,1227.54 1904.57,1232.41 1906.12,1237.01 1907.68,1241.33 1909.23,1245.39 1910.79,1249.2 1912.35,1252.77 1913.9,1256.1 1915.46,1259.22 1917.02,1262.12 \n",
       "  1918.57,1264.82 1920.13,1267.34 1921.68,1269.68 1923.24,1271.85 1924.8,1273.86 1926.35,1275.73 1927.91,1277.44 1929.47,1279.03 1931.02,1280.49 1932.58,1281.83 \n",
       "  1934.13,1283.05 1935.69,1284.18 1937.25,1285.2 1938.8,1286.14 1940.36,1286.99 1941.92,1287.76 1943.47,1288.45 1945.03,1289.08 1946.58,1289.64 1948.14,1290.13 \n",
       "  1949.7,1290.57 1951.25,1290.96 1952.81,1291.29 1954.36,1291.58 1955.92,1291.82 1957.48,1292.02 1959.03,1292.18 1960.59,1292.3 1962.15,1292.39 1963.7,1292.45 \n",
       "  1965.26,1292.47 1966.81,1292.47 1968.37,1292.44 1969.93,1292.38 1971.48,1292.3 1973.04,1292.19 1974.6,1292.06 1976.15,1291.91 1977.71,1291.74 1979.26,1291.55 \n",
       "  1980.82,1291.34 1982.38,1291.11 1983.93,1290.86 1985.49,1290.6 1987.05,1290.31 1988.6,1290.02 1990.16,1289.7 1991.71,1289.37 1993.27,1289.02 1994.83,1288.66 \n",
       "  1996.38,1288.28 1997.94,1287.89 1999.5,1287.48 2001.05,1287.06 2002.61,1286.62 2004.16,1286.17 2005.72,1285.7 2007.28,1285.22 2008.83,1284.72 2010.39,1284.21 \n",
       "  2011.94,1283.68 2013.5,1283.13 2015.06,1282.57 2016.61,1281.99 2018.17,1281.39 2019.73,1280.77 2021.28,1280.13 2022.84,1279.48 2024.39,1278.8 2025.95,1278.1 \n",
       "  2027.51,1277.38 2029.06,1276.64 2030.62,1275.88 2032.18,1275.09 2033.73,1274.28 2035.29,1273.44 2036.84,1272.58 2038.4,1271.68 2039.96,1270.76 2041.51,1269.8 \n",
       "  2043.07,1268.81 2044.63,1267.79 2046.18,1266.72 2047.74,1265.62 2049.29,1264.48 2050.85,1263.29 2052.41,1262.05 2053.96,1260.76 2055.52,1259.41 2057.07,1257.94 \n",
       "  2058.63,1256.38 2060.19,1254.74 2061.74,1253.03 2063.3,1251.25 2064.86,1249.41 2066.41,1247.51 2067.97,1245.55 2069.52,1243.52 2071.08,1241.4 2072.64,1239.2 \n",
       "  2074.19,1236.88 2075.75,1234.44 2077.31,1231.84 2078.86,1229.07 2080.42,1226.09 2081.97,1222.87 2083.53,1219.37 2085.09,1215.56 2086.64,1211.39 2088.2,1206.82 \n",
       "  2089.76,1201.79 2091.31,1196.26 2092.87,1190.16 2094.42,1183.44 2095.98,1176.04 2097.54,1167.84 2099.09,1158.55 2100.65,1148.1 2102.21,1136.32 2103.76,1122.91 \n",
       "  2105.32,1107.46 2106.87,1089.41 2108.43,1068.11 2109.99,1042.75 2111.54,1012.41 2113.1,976.021 2114.65,932.113 2116.21,879.221 2117.77,816.135 2119.32,743.52 \n",
       "  2120.88,664.616 2122.44,585.315 2123.99,514.906 2125.55,461.619 2127.1,429.019 2128.66,415.442 2130.22,416.859 2131.77,428.186 2133.33,445.903 2134.89,467.443 \n",
       "  2136.44,491.056 2138,515.736 2139.55,540.813 2141.11,565.89 2142.67,590.725 2144.22,615.162 2145.78,639.115 2147.34,662.532 2148.89,685.387 2150.45,707.663 \n",
       "  2152,729.36 2153.56,750.485 2155.12,771.046 2156.67,791.051 2158.23,810.512 2159.79,829.441 2161.34,847.853 2162.9,865.762 2164.45,883.181 2166.01,900.121 \n",
       "  2167.57,916.594 2169.12,932.61 2170.68,948.181 2172.23,963.315 2173.79,978.022 2175.35,992.31 2176.9,1006.18 2178.46,1019.65 2180.02,1032.72 2181.57,1045.4 \n",
       "  2183.13,1057.68 2184.68,1069.58 2186.24,1081.09 2187.8,1092.22 2189.35,1102.98 2190.91,1113.36 2192.47,1123.37 2194.02,1133.01 2195.58,1142.28 2197.13,1151.19 \n",
       "  2198.69,1159.74 2200.25,1167.93 2201.8,1175.77 2203.36,1183.27 2204.92,1190.41 2206.47,1197.22 2208.03,1203.7 2209.58,1209.86 2211.14,1215.7 2212.7,1221.23 \n",
       "  2214.25,1226.46 2215.81,1231.4 2217.37,1236.05 2218.92,1240.43 2220.48,1244.54 2222.03,1248.4 2223.59,1252.01 2225.15,1255.39 2226.7,1258.55 2228.26,1261.5 \n",
       "  2229.81,1264.25 2231.37,1266.81 2232.93,1269.19 2234.48,1271.4 2236.04,1273.45 2237.6,1275.34 2239.15,1277.09 2240.71,1278.7 2242.26,1280.18 2243.82,1281.55 \n",
       "  2245.38,1282.8 2246.93,1283.94 2248.49,1284.99 2250.05,1285.95 2251.6,1286.82 2253.16,1287.6 2254.71,1288.31 2256.27,1288.95 2257.83,1289.52 2259.38,1290.03 \n",
       "  2260.94,1290.48 2262.5,1290.88 2264.05,1291.22 2265.61,1291.52 2267.16,1291.77 2268.72,1291.98 2270.28,1292.15 2271.83,1292.28 2273.39,1292.38 2274.95,1292.44 \n",
       "  2276.5,1292.47 2278.06,1292.47 2279.61,1292.45 2281.17,1292.4 2282.73,1292.32 2284.28,1292.22 2285.84,1292.09 2287.39,1291.95 2288.95,1291.78 2290.51,1291.59 \n",
       "  2292.06,1291.38 2293.62,1291.16 2295.18,1290.91 2296.73,1290.65 2298.29,1290.37 2299.84,1290.08 2301.4,1289.77 2302.96,1289.44 2304.51,1289.1 2306.07,1288.74 \n",
       "  2307.63,1288.37 2309.18,1287.98 2310.74,1287.57 2312.29,1287.15 2313.85,1286.72 2315.41,1286.27 2316.96,1285.81 2318.52,1285.33 2320.08,1284.83 2321.63,1284.32 \n",
       "  2323.19,1283.8 2324.74,1283.25 2326.3,1282.69 2327.86,1282.11 2329.41,1281.52 2330.97,1280.9 2332.53,1280.27 2334.08,1279.62 2335.64,1278.95 2337.19,1278.25 \n",
       "  2338.75,1277.54 2340.31,1276.81 2341.86,1276.05 2343.42,1275.27 2344.97,1274.46 2346.53,1273.63 2348.09,1272.77 2349.64,1271.88 2351.2,1270.96 2352.76,1270.01 \n",
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       "  191.173,1356.23 192.729,1350.86 194.285,1345.28 195.841,1339.54 197.398,1333.65 198.954,1327.64 200.51,1321.51 202.066,1315.3 203.622,1309 205.179,1302.63 \n",
       "  206.735,1296.21 208.291,1289.74 209.847,1283.22 211.404,1276.67 212.96,1270.09 214.516,1263.48 216.072,1256.85 217.628,1250.2 219.185,1243.54 220.741,1236.86 \n",
       "  222.297,1230.18 223.853,1223.49 225.409,1216.79 226.966,1210.08 228.522,1203.38 230.078,1196.67 231.634,1189.96 233.191,1183.24 234.747,1176.53 236.303,1169.82 \n",
       "  237.859,1163.11 239.415,1156.41 240.972,1149.7 242.528,1143 244.084,1136.3 245.64,1129.61 247.196,1122.92 248.753,1116.23 250.309,1109.55 251.865,1102.87 \n",
       "  253.421,1096.2 254.978,1089.53 256.534,1082.86 258.09,1076.21 259.646,1069.55 261.202,1062.9 262.759,1056.26 264.315,1049.62 265.871,1042.99 267.427,1036.37 \n",
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       "  471.291,259.306 472.848,254.802 474.404,250.35 475.96,245.952 477.516,241.611 479.073,237.329 480.629,233.108 482.185,228.949 483.741,224.857 485.297,220.832 \n",
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       "  1000.4,598.699 1001.96,592.087 1003.52,585.532 1005.07,579.032 1006.63,572.584 1008.19,566.185 1009.74,559.834 1011.3,553.528 1012.85,547.266 1014.41,541.046 \n",
       "  1015.97,534.865 1017.52,528.723 1019.08,522.617 1020.64,516.548 1022.19,510.512 1023.75,504.51 1025.3,498.54 1026.86,492.602 1028.42,486.694 1029.97,480.815 \n",
       "  1031.53,474.966 1033.09,469.146 1034.64,463.353 1036.2,457.588 1037.75,451.85 1039.31,446.139 1040.87,440.455 1042.42,434.797 1043.98,429.166 1045.53,423.561 \n",
       "  1047.09,417.983 1048.65,412.431 1050.2,406.906 1051.76,401.407 1053.32,395.936 1054.87,390.492 1056.43,385.075 1057.98,379.686 1059.54,374.325 1061.1,368.992 \n",
       "  1062.65,363.687 1064.21,358.412 1065.77,353.166 1067.32,347.951 1068.88,342.766 1070.43,337.613 1071.99,332.492 1073.55,327.404 1075.1,322.349 1076.66,317.329 \n",
       "  1078.22,312.345 1079.77,307.397 1081.33,302.486 1082.88,297.614 1084.44,292.782 1086,287.991 1087.55,283.243 1089.11,278.538 1090.67,273.878 1092.22,269.263 \n",
       "  1093.78,264.696 1095.33,260.178 1096.89,255.71 1098.45,251.295 1100,246.934 1101.56,242.631 1103.11,238.386 1104.67,234.204 1106.23,230.086 1107.78,226.036 \n",
       "  1109.34,222.057 1110.9,218.151 1112.45,214.324 1114.01,210.577 1115.56,206.916 1117.12,203.343 1118.68,199.864 1120.23,196.496 1121.79,193.285 1123.35,190.221 \n",
       "  1124.9,187.29 1126.46,184.485 1128.01,181.798 1129.57,179.228 1131.13,176.778 1132.68,174.453 1134.24,172.263 1135.8,170.222 1137.35,168.345 1138.91,166.655 \n",
       "  1140.46,165.176 1142.02,163.937 1143.58,162.969 1145.13,162.31 1146.69,161.998 1148.25,162.078 1149.8,162.597 1151.36,163.606 1152.91,165.16 1154.47,167.317 \n",
       "  1156.03,170.142 1157.58,173.698 1159.14,178.058 1160.69,183.294 1162.25,189.552 1163.81,197.169 1165.36,206.21 1166.92,216.84 1168.48,229.368 1170.03,244.25 \n",
       "  1171.59,262.088 1173.14,283.627 1174.7,309.761 1176.26,341.527 1177.81,380.108 1179.37,426.897 1180.93,483.754 1182.48,552.325 1184.04,633.485 1185.59,725.057 \n",
       "  1187.15,821.979 1188.71,916.395 1190.26,998.665 1191.82,1062.65 1193.38,1107.75 1194.93,1136.79 1196.49,1153.58 1198.04,1162.7 1199.6,1166.61 1201.16,1167.12 \n",
       "  1202.71,1165.41 1204.27,1162.21 1205.83,1157.99 1207.38,1153.05 1208.94,1147.57 1210.49,1141.67 1212.05,1135.45 1213.61,1128.95 1215.16,1122.2 1216.72,1115.24 \n",
       "  1218.27,1108.09 1219.83,1100.76 1221.39,1093.24 1222.94,1085.57 1224.5,1077.73 1226.06,1069.75 1227.61,1061.62 1229.17,1053.34 1230.72,1044.93 1232.28,1036.39 \n",
       "  1233.84,1027.72 1235.39,1018.93 1236.95,1010.02 1238.51,1001.01 1240.06,991.894 1241.62,982.686 1243.17,973.392 1244.73,964.021 1246.29,954.583 1247.84,945.086 \n",
       "  1249.4,935.539 1250.96,925.954 1252.51,916.339 1254.07,906.707 1255.62,897.068 1257.18,887.432 1258.74,877.812 1260.29,868.216 1261.85,858.656 1263.41,849.142 \n",
       "  1264.96,839.682 1266.52,830.286 1268.07,820.961 1269.63,811.715 1271.19,802.555 1272.74,793.487 1274.3,784.516 1275.85,775.648 1277.41,766.885 1278.97,758.23 \n",
       "  1280.52,749.684 1282.08,741.249 1283.64,732.924 1285.19,724.712 1286.75,716.61 1288.3,708.618 1289.86,700.735 1291.42,692.96 1292.97,685.288 1294.53,677.718 \n",
       "  1296.09,670.246 1297.64,662.867 1299.2,655.579 1300.75,648.377 1302.31,641.259 1303.87,634.221 1305.42,627.26 1306.98,620.373 1308.54,613.555 1310.09,606.805 \n",
       "  1311.65,600.119 1313.2,593.494 1314.76,586.927 1316.32,580.416 1317.87,573.957 1319.43,567.548 1320.99,561.187 1322.54,554.872 1324.1,548.601 1325.65,542.372 \n",
       "  1327.21,536.183 1328.77,530.032 1330.32,523.919 1331.88,517.842 1333.43,511.799 1334.99,505.79 1336.55,499.813 1338.1,493.868 1339.66,487.953 1341.22,482.069 \n",
       "  1342.77,476.213 1344.33,470.387 1345.88,464.588 1347.44,458.817 1349,453.074 1350.55,447.357 1352.11,441.668 1353.67,436.004 1355.22,430.367 1356.78,424.757 \n",
       "  1358.33,419.173 1359.89,413.615 1361.45,408.084 1363,402.58 1364.56,397.102 1366.12,391.652 1367.67,386.23 1369.23,380.835 1370.78,375.467 1372.34,370.129 \n",
       "  1373.9,364.818 1375.45,359.537 1377.01,354.285 1378.57,349.063 1380.12,343.872 1381.68,338.712 1383.23,333.584 1384.79,328.489 1386.35,323.427 1387.9,318.4 \n",
       "  1389.46,313.408 1391.01,308.452 1392.57,303.533 1394.13,298.652 1395.68,293.811 1397.24,289.01 1398.8,284.252 1400.35,279.536 1401.91,274.866 1403.46,270.242 \n",
       "  1405.02,265.666 1406.58,261.138 1408.13,256.661 1409.69,252.236 1411.25,247.865 1412.8,243.549 1414.36,239.292 1415.91,235.096 1417.47,230.964 1419.03,226.898 \n",
       "  1420.58,222.902 1422.14,218.98 1423.7,215.134 1425.25,211.368 1426.81,207.687 1428.36,204.095 1429.92,200.596 1431.48,197.195 1433.03,193.896 1434.59,190.709 \n",
       "  1436.15,187.679 1437.7,184.807 1439.26,182.081 1440.81,179.492 1442.37,177.039 1443.93,174.721 1445.48,172.543 1447.04,170.516 1448.59,168.651 1450.15,166.967 \n",
       "  1451.71,165.486 1453.26,164.232 1454.82,163.235 1456.38,162.531 1457.93,162.157 1459.49,162.155 1461.04,162.573 1462.6,163.46 1464.16,164.872 1465.71,166.868 \n",
       "  1467.27,169.51 1468.83,172.866 1470.38,177.008 1471.94,182.012 1473.49,187.959 1475.05,194.963 1476.61,203.203 1478.16,212.969 1479.72,224.663 1481.28,238.796 \n",
       "  1482.83,255.99 1484.39,276.979 1485.94,302.609 1487.5,333.834 1489.06,371.721 1490.61,417.448 1492.17,472.303 1493.73,537.686 1495.28,615.107 1496.84,704.764 \n",
       "  1498.39,801.553 1499.95,897.64 1501.51,983.923 1503.06,1052.31 1504.62,1100.24 1506.17,1131.34 1507.73,1150.69 1509.29,1161.26 1510.84,1166.11 1512.4,1167.23 \n",
       "  1513.96,1165.91 1515.51,1162.98 1517.07,1158.95 1518.62,1154.14 1520.18,1148.76 1521.74,1142.95 1523.29,1136.79 1524.85,1130.35 1526.41,1123.65 1527.96,1116.73 \n",
       "  1529.52,1109.62 1531.07,1102.32 1532.63,1094.84 1534.19,1087.21 1535.74,1079.41 1537.3,1071.46 1538.86,1063.35 1540.41,1055.11 1541.97,1046.73 1543.52,1038.21 \n",
       "  1545.08,1029.57 1546.64,1020.8 1548.19,1011.92 1549.75,1002.93 1551.3,993.835 1552.86,984.646 1554.42,975.37 1555.97,966.014 1557.53,956.588 1559.09,947.102 \n",
       "  1560.64,937.564 1562.2,927.986 1563.75,918.378 1565.31,908.749 1566.87,899.112 1568.42,889.475 1569.98,879.851 1571.54,870.249 1573.09,860.679 1574.65,851.153 \n",
       "  1576.2,841.681 1577.76,832.27 1579.32,822.929 1580.87,813.666 1582.43,804.488 1583.99,795.4 1585.54,786.409 1587.1,777.518 1588.65,768.732 1590.21,760.053 \n",
       "  1591.77,751.485 1593.32,743.027 1594.88,734.682 1596.44,726.449 1597.99,718.328 1599.55,710.315 1601.1,702.411 1602.66,694.612 1604.22,686.917 1605.77,679.323 \n",
       "  1607.33,671.827 1608.88,664.426 1610.44,657.117 1612,649.898 1613.55,642.763 1615.11,635.71 1616.67,628.734 1618.22,621.831 1619.78,615 1621.33,608.236 \n",
       "  1622.89,601.536 1624.45,594.898 1626,588.318 1627.56,581.795 1629.12,575.324 1630.67,568.905 1632.23,562.534 1633.78,556.21 1635.34,549.93 1636.9,543.692 \n",
       "  1638.45,537.495 1640.01,531.337 1641.57,525.216 1643.12,519.132 1644.68,513.082 1646.23,507.066 1647.79,501.082 1649.35,495.13 1650.9,489.209 1652.46,483.318 \n",
       "  1654.02,477.456 1655.57,471.623 1657.13,465.818 1658.68,460.041 1660.24,454.292 1661.8,448.57 1663.35,442.875 1664.91,437.206 1666.46,431.564 1668.02,425.948 \n",
       "  1669.58,420.359 1671.13,414.796 1672.69,409.259 1674.25,403.749 1675.8,398.266 1677.36,392.81 1678.91,387.38 1680.47,381.978 1682.03,376.604 1683.58,371.258 \n",
       "  1685.14,365.941 1686.7,360.654 1688.25,355.396 1689.81,350.169 1691.36,344.973 1692.92,339.808 1694.48,334.675 1696.03,329.575 1697.59,324.507 1699.15,319.474 \n",
       "  1700.7,314.475 1702.26,309.511 1703.81,304.584 1705.37,299.695 1706.93,294.845 1708.48,290.034 1710.04,285.265 1711.6,280.538 1713.15,275.855 1714.71,271.219 \n",
       "  1716.26,266.629 1717.82,262.088 1719.38,257.598 1720.93,253.161 1722.49,248.778 1724.04,244.452 1725.6,240.184 1727.16,235.977 1728.71,231.834 1730.27,227.757 \n",
       "  1731.83,223.754 1733.38,219.825 1734.94,215.972 1736.49,212.197 1738.05,208.503 1739.61,204.893 1741.16,201.372 1742.72,197.944 1744.28,194.615 1745.83,191.392 \n",
       "  1747.39,188.282 1748.94,185.292 1750.5,182.43 1752.06,179.707 1753.61,177.131 1755.17,174.714 1756.73,172.466 1758.28,170.399 1759.84,168.527 1761.39,166.862 \n",
       "  1762.95,165.419 1764.51,164.258 1766.06,163.507 1767.62,163.115 1769.18,163.038 1770.73,163.254 1772.29,163.763 1773.84,164.587 1775.4,165.771 1776.96,167.38 \n",
       "  1778.51,169.502 1780.07,172.248 1781.62,175.749 1783.18,180.16 1784.74,185.657 1786.29,192.437 1787.85,200.721 1789.41,210.752 1790.96,222.792 1792.52,237.129 \n",
       "  1794.07,254.071 1795.63,274.107 1797.19,298.192 1798.74,327.289 1800.3,362.538 1801.86,405.522 1803.41,458.1 1804.97,521.625 1806.52,596.916 1808.08,684.154 \n",
       "  1809.64,779.975 1811.19,876.719 1812.75,965.428 1814.31,1037.36 1815.86,1090.38 1817.42,1126.02 1818.97,1147.57 1820.53,1159.74 1822.09,1165.68 1823.64,1167.35 \n",
       "  1825.2,1166.42 1826.76,1163.78 1828.31,1159.94 1829.87,1155.28 1831.42,1150.01 1832.98,1144.28 1834.54,1138.19 1836.09,1131.8 1837.65,1125.15 1839.2,1118.28 \n",
       "  1840.76,1111.21 1842.32,1103.95 1843.87,1096.51 1845.43,1088.9 1846.99,1081.14 1848.54,1073.22 1850.1,1065.15 1851.65,1056.93 1853.21,1048.58 1854.77,1040.09 \n",
       "  1856.32,1031.48 1857.88,1022.74 1859.44,1013.88 1860.99,1004.91 1862.55,995.842 1864.1,986.673 1865.66,977.415 1867.22,968.077 1868.77,958.667 1870.33,949.193 \n",
       "  1871.89,939.667 1873.44,930.097 1875,920.493 1876.55,910.867 1878.11,901.229 1879.67,891.591 1881.22,881.963 1882.78,872.355 1884.34,862.779 1885.89,853.244 \n",
       "  1887.45,843.76 1889,834.335 1890.56,824.978 1892.12,815.697 1893.67,806.499 1895.23,797.39 1896.78,788.376 1898.34,779.464 1899.9,770.655 1901.45,761.953 \n",
       "  1903.01,753.359 1904.57,744.875 1906.12,736.503 1907.68,728.242 1909.23,720.093 1910.79,712.054 1912.35,704.125 1913.9,696.304 1915.46,688.588 1917.02,680.975 \n",
       "  1918.57,673.461 1920.13,666.043 1921.68,658.715 1923.24,651.476 1924.8,644.322 1926.35,637.25 1927.91,630.256 1929.47,623.337 1931.02,616.49 1932.58,609.712 \n",
       "  1934.13,602.998 1935.69,596.347 1937.25,589.756 1938.8,583.221 1940.36,576.739 1941.92,570.309 1943.47,563.928 1945.03,557.593 1946.58,551.303 1948.14,545.056 \n",
       "  1949.7,538.85 1951.25,532.683 1952.81,526.554 1954.36,520.461 1955.92,514.404 1957.48,508.38 1959.03,502.389 1960.59,496.431 1962.15,490.503 1963.7,484.606 \n",
       "  1965.26,478.738 1966.81,472.899 1968.37,467.088 1969.93,461.306 1971.48,455.55 1973.04,449.822 1974.6,444.121 1976.15,438.446 1977.71,432.798 1979.26,427.176 \n",
       "  1980.82,421.58 1982.38,416.011 1983.93,410.469 1985.49,404.953 1987.05,399.464 1988.6,394.002 1990.16,388.568 1991.71,383.161 1993.27,377.782 1994.83,372.43 \n",
       "  1996.38,367.107 1997.94,361.813 1999.5,356.549 2001.05,351.314 2002.61,346.109 2004.16,340.935 2005.72,335.793 2007.28,330.684 2008.83,325.608 2010.39,320.565 \n",
       "  2011.94,315.558 2013.5,310.586 2015.06,305.651 2016.61,300.754 2018.17,295.896 2019.73,291.078 2021.28,286.302 2022.84,281.569 2024.39,276.88 2025.95,272.236 \n",
       "  2027.51,267.638 2029.06,263.088 2030.62,258.588 2032.18,254.139 2033.73,249.743 2035.29,245.402 2036.84,241.119 2038.4,236.897 2039.96,232.737 2041.51,228.642 \n",
       "  2043.07,224.617 2044.63,220.663 2046.18,216.785 2047.74,212.985 2049.29,209.269 2050.85,205.638 2052.41,202.098 2053.96,198.654 2055.52,195.325 2057.07,192.161 \n",
       "  2058.63,189.148 2060.19,186.27 2061.74,183.52 2063.3,180.889 2064.86,178.377 2066.41,175.985 2067.97,173.72 2069.52,171.59 2071.08,169.611 2072.64,167.801 \n",
       "  2074.19,166.18 2075.75,164.776 2077.31,163.619 2078.86,162.742 2080.42,162.184 2081.97,161.986 2083.53,162.196 2085.09,162.863 2086.64,164.041 2088.2,165.789 \n",
       "  2089.76,168.168 2091.31,171.246 2092.87,175.091 2094.42,179.779 2095.98,185.388 2097.54,192.058 2099.09,200.121 2100.65,209.691 2102.21,220.975 2103.76,234.328 \n",
       "  2105.32,250.245 2106.87,269.364 2108.43,292.464 2109.99,320.47 2111.54,354.447 2113.1,395.643 2114.65,445.778 2116.21,506.614 2117.77,579.693 2119.32,664.771 \n",
       "  2120.88,758.876 2122.44,856.208 2123.99,947.349 2125.55,1023.59 2127.1,1080.68 2128.66,1119.56 2130.22,1143.67 2131.77,1157.61 2133.33,1164.64 2134.89,1167.14 \n",
       "  2136.44,1166.75 2138,1164.43 2139.55,1160.83 2141.11,1156.32 2142.67,1151.17 2144.22,1145.53 2145.78,1139.51 2147.34,1133.18 2148.89,1126.59 2150.45,1119.77 \n",
       "  2152,1112.74 2153.56,1105.52 2155.12,1098.12 2156.67,1090.54 2158.23,1082.81 2159.79,1074.92 2161.34,1066.89 2162.9,1058.71 2164.45,1050.38 2166.01,1041.92 \n",
       "  2167.57,1033.33 2169.12,1024.62 2170.68,1015.79 2172.23,1006.84 2173.79,997.795 2175.35,988.646 2176.9,979.407 2178.46,970.086 2180.02,960.69 2181.57,951.229 \n",
       "  2183.13,941.713 2184.68,932.151 2186.24,922.553 2187.8,912.931 2189.35,903.295 2190.91,893.657 2192.47,884.026 2194.02,874.414 2195.58,864.83 2197.13,855.286 \n",
       "  2198.69,845.789 2200.25,836.35 2201.8,826.977 2203.36,817.678 2204.92,808.462 2206.47,799.334 2208.03,790.301 2209.58,781.366 2211.14,772.534 2212.7,763.807 \n",
       "  2214.25,755.189 2215.81,746.682 2217.37,738.285 2218.92,730.001 2220.48,721.829 2222.03,713.769 2223.59,705.818 2225.15,697.976 2226.7,690.239 2228.26,682.604 \n",
       "  2229.81,675.068 2231.37,667.627 2232.93,660.28 2234.48,653.021 2236.04,645.849 2237.6,638.759 2239.15,631.749 2240.71,624.814 2242.26,617.952 2243.82,611.16 \n",
       "  2245.38,604.433 2246.93,597.769 2248.49,591.165 2250.05,584.618 2251.6,578.125 2253.16,571.683 2254.71,565.292 2256.27,558.947 2257.83,552.647 2259.38,546.391 \n",
       "  2260.94,540.177 2262.5,534.002 2264.05,527.865 2265.61,521.764 2267.16,515.7 2268.72,509.669 2270.28,503.672 2271.83,497.706 2273.39,491.772 2274.95,485.869 \n",
       "  2276.5,479.994 2278.06,474.149 2279.61,468.333 2281.17,462.544 2282.73,456.782 2284.28,451.048 2285.84,445.341 2287.39,439.66 2288.95,434.006 2290.51,428.379 \n",
       "  2292.06,422.778 2293.62,417.203 2295.18,411.656 2296.73,406.134 2298.29,400.64 2299.84,395.173 2301.4,389.732 2302.96,384.319 2304.51,378.933 2306.07,373.576 \n",
       "  2307.63,368.246 2309.18,362.945 2310.74,357.674 2312.29,352.432 2313.85,347.221 2315.41,342.04 2316.96,336.891 2318.52,331.775 2320.08,326.691 2321.63,321.642 \n",
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     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob1 = ODEProblem(brusselator3!,u₀,tspan,p)\n",
    "sol1 = solve(prob1)\n",
    "plot(sol1,label=[\"X\",\"Y\"])\n",
    "title!(\"Brusselator\")\n",
    "xlabel!(\"Time [s]\")\n",
    "ylabel!(\"Concentration [a.u]\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A very convenient way to plot these type of oscillatory systems is the phase plot, where we plot one variable versus the other.This is a direct way of looking at the limit cycle. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
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       "  446.911,186.392 450.278,183.574 453.83,180.906 457.55,178.38 461.428,175.994 465.461,173.748 469.654,171.647 474.019,169.7 478.575,167.923 483.349,166.332 \n",
       "  488.376,164.951 493.696,163.808 499.358,162.933 505.418,162.362 511.939,162.136 518.993,162.3 526.658,162.901 535.018,163.995 544.167,165.638 554.204,167.894 \n",
       "  565.238,170.827 577.382,174.51 590.76,179.018 605.499,184.43 622.004,190.977 640.768,198.941 661.798,208.344 685.361,219.355 712.009,232.304 742.573,247.678 \n",
       "  778.163,266.121 820.172,288.439 870.274,315.593 930.421,348.704 1002.85,389.053 1090.08,438.077 1194.88,497.372 1319.78,568.538 1464.65,651.91 1625,745.608 \n",
       "  1787.01,842.979 1933.22,935.293 2047.74,1014.32 2119.71,1074.06 2152.12,1115.25 2152.81,1141.12 2132.23,1156.32 2097.69,1164.16 2054.46,1167.05 2006.86,1166.97 \n",
       "  1956.9,1164.87 1905.97,1161.41 1854.97,1157.01 1804.4,1151.94 1754.6,1146.36 1705.78,1140.39 1658.03,1134.1 1611.41,1127.54 1565.97,1120.74 1521.71,1113.74 \n",
       "  1478.62,1106.55 1436.67,1099.17 1395.86,1091.62 1356.16,1083.91 1317.53,1076.04 1279.97,1068.03 1243.43,1059.86 1207.9,1051.56 1173.33,1043.12 1139.72,1034.55 \n",
       "  1107.05,1025.85 1075.27,1017.04 1044.39,1008.11 1014.38,999.072 985.23,989.936 956.916,980.709 929.428,971.399 902.756,962.013 876.888,952.561 851.816,943.052 \n",
       "  827.529,933.496 804.021,923.903 781.286,914.284 759.318,904.649 738.113,895.01 717.665,885.377 697.968,875.761 679.019,866.172 660.809,856.622 643.334,847.118 \n",
       "  626.585,837.672 610.553,828.289 595.231,818.98 580.606,809.752 566.67,800.61 553.41,791.562 540.813,782.614 528.859,773.768 517.533,765.027 506.819,756.394 \n",
       "  496.701,747.871 487.16,739.459 478.179,731.158 469.739,722.969 461.819,714.891 454.4,706.924 447.461,699.065 440.979,691.313 434.932,683.664 429.297,676.115 \n",
       "  424.049,668.663 419.165,661.303 414.631,654.033 410.429,646.85 406.539,639.749 402.946,632.727 399.631,625.782 396.578,618.91 393.773,612.108 391.199,605.372 \n",
       "  388.842,598.699 386.688,592.087 384.724,585.532 382.937,579.032 381.315,572.584 379.849,566.185 378.53,559.834 377.347,553.528 376.294,547.266 375.361,541.046 \n",
       "  374.542,534.865 373.828,528.723 373.213,522.617 372.691,516.548 372.256,510.512 371.902,504.51 371.624,498.54 371.419,492.602 371.281,486.694 371.207,480.815 \n",
       "  371.194,474.966 371.239,469.146 371.338,463.353 371.489,457.588 371.689,451.85 371.935,446.139 372.227,440.455 372.561,434.797 372.936,429.166 373.35,423.561 \n",
       "  373.803,417.983 374.292,412.431 374.818,406.906 375.379,401.407 375.974,395.936 376.603,390.492 377.264,385.075 377.958,379.686 378.683,374.325 379.439,368.992 \n",
       "  380.225,363.687 381.042,358.412 381.89,353.166 382.768,347.951 383.676,342.766 384.615,337.613 385.585,332.492 386.587,327.404 387.62,322.349 388.686,317.329 \n",
       "  389.785,312.345 390.918,307.397 392.086,302.486 393.291,297.614 394.533,292.782 395.814,287.991 397.134,283.243 398.496,278.538 399.899,273.878 401.344,269.263 \n",
       "  402.834,264.696 404.369,260.178 405.951,255.71 407.582,251.295 409.266,246.934 411.004,242.631 412.799,238.386 414.654,234.204 416.573,230.086 418.56,226.036 \n",
       "  420.617,222.057 422.75,218.151 424.963,214.324 427.259,210.577 429.645,206.916 432.125,203.343 434.705,199.864 437.413,196.496 440.33,193.285 443.435,190.221 \n",
       "  446.703,187.29 450.115,184.485 453.657,181.798 457.322,179.228 461.111,176.778 465.029,174.453 469.089,172.263 473.31,170.222 477.715,168.345 482.337,166.655 \n",
       "  487.214,165.176 492.388,163.937 497.911,162.969 503.838,162.31 510.233,161.998 517.165,162.078 524.709,162.597 532.948,163.606 541.968,165.16 551.865,167.317 \n",
       "  562.74,170.142 574.699,173.698 587.856,178.058 602.331,183.294 618.373,189.552 636.566,197.169 656.993,206.21 679.911,216.84 705.841,229.368 735.56,244.25 \n",
       "  770.11,262.088 810.791,283.627 859.162,309.761 917.046,341.527 986.524,380.108 1070.04,426.897 1170.74,483.754 1291.32,552.325 1432.85,633.485 1590.09,725.057 \n",
       "  1752.4,821.979 1904,916.395 2025.99,998.665 2107.18,1062.65 2147.87,1107.75 2155.08,1136.79 2137.52,1153.58 2105.32,1162.7 2063.89,1166.61 2017.13,1167.12 \n",
       "  1967.59,1165.41 1916.8,1162.21 1865.77,1157.99 1815.09,1153.05 1765.12,1147.57 1716.08,1141.67 1668.1,1135.45 1621.25,1128.95 1575.56,1122.2 1531.05,1115.24 \n",
       "  1487.71,1108.09 1445.53,1100.76 1404.48,1093.24 1364.54,1085.57 1325.69,1077.73 1287.9,1069.75 1251.15,1061.62 1215.4,1053.34 1180.63,1044.93 1146.82,1036.39 \n",
       "  1113.95,1027.72 1081.98,1018.93 1050.91,1010.02 1020.72,1001.01 991.385,991.894 962.892,982.686 935.229,973.392 908.383,964.021 882.345,954.583 857.103,945.086 \n",
       "  832.65,935.539 808.977,925.954 786.078,916.339 763.947,906.707 742.579,897.068 721.969,887.432 702.113,877.812 683.003,868.216 664.636,858.656 647.004,849.142 \n",
       "  630.101,839.682 613.917,830.286 598.445,820.961 583.673,811.715 569.592,802.555 556.188,793.487 543.45,784.516 531.363,775.648 519.909,766.885 509.069,758.23 \n",
       "  498.827,749.684 489.164,741.249 480.064,732.924 471.507,724.712 463.475,716.61 455.947,708.618 448.902,700.735 442.32,692.96 436.18,685.288 430.458,677.718 \n",
       "  425.131,670.246 420.178,662.867 415.573,655.579 411.302,648.377 407.347,641.259 403.692,634.221 400.318,627.26 397.211,620.373 394.353,613.555 391.73,606.805 \n",
       "  389.327,600.119 387.131,593.494 385.128,586.927 383.305,580.416 381.65,573.957 380.152,567.548 378.803,561.187 377.592,554.872 376.511,548.601 375.553,542.372 \n",
       "  374.709,536.183 373.972,530.032 373.337,523.919 372.795,517.842 372.341,511.799 371.97,505.79 371.677,499.813 371.457,493.868 371.305,487.953 371.218,482.069 \n",
       "  371.192,476.213 371.225,470.387 371.312,464.588 371.452,458.817 371.642,453.074 371.879,447.357 372.161,441.668 372.486,436.004 372.853,430.367 373.259,424.757 \n",
       "  373.704,419.173 374.185,413.615 374.703,408.084 375.256,402.58 375.843,397.102 376.465,391.652 377.119,386.23 377.806,380.835 378.524,375.467 379.274,370.129 \n",
       "  380.055,364.818 380.866,359.537 381.708,354.285 382.58,349.063 383.482,343.872 384.415,338.712 385.379,333.584 386.374,328.489 387.401,323.427 388.46,318.4 \n",
       "  389.551,313.408 390.677,308.452 391.837,303.533 393.033,298.652 394.265,293.811 395.535,289.01 396.844,284.252 398.195,279.536 399.588,274.866 401.025,270.242 \n",
       "  402.506,265.666 404.033,261.138 405.608,256.661 407.231,252.236 408.905,247.865 410.632,243.549 412.416,239.292 414.258,235.096 416.163,230.964 418.134,226.898 \n",
       "  420.174,222.902 422.289,218.98 424.482,215.134 426.758,211.368 429.123,207.687 431.582,204.095 434.14,200.596 436.803,197.195 439.579,193.896 442.48,190.709 \n",
       "  445.585,187.679 448.891,184.807 452.376,182.081 456.021,179.492 459.819,177.039 463.766,174.721 467.866,172.543 472.132,170.516 476.581,168.651 481.238,166.967 \n",
       "  486.137,165.486 491.315,164.232 496.82,163.235 502.703,162.531 509.026,162.157 515.855,162.155 523.264,162.573 531.333,163.46 540.151,164.872 549.813,166.868 \n",
       "  560.419,169.51 572.08,172.866 584.909,177.008 599.031,182.012 614.568,187.959 631.679,194.963 650.708,203.203 672.182,212.969 696.81,224.663 725.481,238.796 \n",
       "  759.262,255.99 799.402,276.979 847.329,302.609 904.651,333.834 973.156,371.721 1054.81,417.448 1151.77,472.303 1266.35,537.686 1401.07,615.107 1555.63,704.764 \n",
       "  1718.95,801.553 1875.09,897.64 2006.12,983.923 2096.86,1052.31 2143.51,1100.24 2154.89,1131.34 2142.51,1150.69 2113.05,1161.26 2073.21,1166.11 2027.34,1167.23 \n",
       "  1978.2,1165.91 1927.59,1162.98 1876.56,1158.95 1825.77,1154.14 1775.63,1148.76 1726.38,1142.95 1678.17,1136.79 1631.07,1130.35 1585.14,1123.65 1540.38,1116.73 \n",
       "  1496.78,1109.62 1454.35,1102.32 1413.07,1094.84 1372.91,1087.21 1333.84,1079.41 1295.83,1071.46 1258.85,1063.35 1222.89,1055.11 1187.92,1046.73 1153.91,1038.21 \n",
       "  1120.83,1029.57 1088.68,1020.8 1057.42,1011.92 1027.05,1002.93 997.531,993.835 968.861,984.646 941.023,975.37 914.004,966.014 887.794,956.588 862.382,947.102 \n",
       "  837.761,937.564 813.924,927.986 790.863,918.378 768.572,908.749 747.045,899.112 726.276,889.475 706.26,879.851 686.992,870.249 668.465,860.679 650.676,851.153 \n",
       "  633.618,841.681 617.281,832.27 601.658,822.929 586.738,813.666 572.511,804.488 558.965,795.4 546.087,786.409 533.863,777.518 522.277,768.732 511.313,760.053 \n",
       "  500.955,751.485 491.183,743.027 481.977,734.682 473.318,726.449 465.182,718.328 457.549,710.315 450.399,702.411 443.711,694.612 437.465,686.917 431.642,679.323 \n",
       "  426.22,671.827 421.181,664.426 416.503,657.117 412.168,649.898 408.155,642.763 404.444,635.71 401.016,628.734 397.854,621.831 394.943,615 392.269,608.236 \n",
       "  389.818,601.536 387.576,594.898 385.531,588.318 383.669,581.795 381.98,575.324 380.453,568.905 379.076,562.534 377.84,556.21 376.735,549.93 375.753,543.692 \n",
       "  374.885,537.495 374.126,531.337 373.468,525.216 372.905,519.132 372.432,513.082 372.043,507.066 371.733,501.082 371.497,495.13 371.331,489.209 371.231,483.318 \n",
       "  371.192,477.456 371.213,471.623 371.289,465.818 371.418,460.041 371.597,454.292 371.824,448.57 372.097,442.875 372.413,437.206 372.771,431.564 373.17,425.948 \n",
       "  373.607,420.359 374.082,414.796 374.593,409.259 375.138,403.749 375.718,398.266 376.331,392.81 376.977,387.38 377.654,381.978 378.364,376.604 379.104,371.258 \n",
       "  379.877,365.941 380.68,360.654 381.516,355.396 382.384,350.169 383.284,344.973 384.214,339.808 385.176,334.675 386.17,329.575 387.194,324.507 388.251,319.474 \n",
       "  389.34,314.475 390.461,309.511 391.616,304.584 392.805,299.695 394.03,294.845 395.291,290.034 396.589,285.265 397.927,280.538 399.305,275.855 400.725,271.219 \n",
       "  402.189,266.629 403.699,262.088 405.256,257.598 406.864,253.161 408.523,248.778 410.238,244.452 412.01,240.184 413.841,235.977 415.736,231.834 417.698,227.757 \n",
       "  419.737,223.754 421.852,219.825 424.045,215.972 426.318,212.197 428.673,208.503 431.115,204.893 433.65,201.372 436.283,197.944 439.023,194.615 441.877,191.392 \n",
       "  444.856,188.282 447.97,185.292 451.23,182.43 454.65,179.707 458.243,177.131 462.024,174.714 466.01,172.466 470.217,170.399 474.663,168.527 479.368,166.862 \n",
       "  484.352,165.419 489.72,164.258 495.701,163.507 502.194,163.115 509.112,163.038 516.404,163.254 524.058,163.763 532.102,164.587 540.602,165.771 549.662,167.38 \n",
       "  559.424,169.502 570.07,172.248 581.821,175.749 594.933,180.16 609.705,185.657 626.472,192.437 645.608,200.721 667.525,210.752 692.675,222.792 721.548,237.129 \n",
       "  754.671,254.071 792.891,274.107 837.859,298.192 891.262,327.289 955.103,362.538 1032.19,405.522 1125.69,458.1 1237.71,521.625 1369.29,596.916 1520.13,684.154 \n",
       "  1682.77,779.975 1841.5,876.719 1978.58,965.428 2077.28,1037.36 2134.81,1090.38 2155.61,1126.02 2147.56,1147.57 2121.13,1159.74 2083.2,1165.68 2038.06,1167.35 \n",
       "  1989.33,1166.42 1938.89,1163.78 1887.86,1159.94 1836.96,1155.28 1786.64,1150.01 1737.17,1144.28 1688.71,1138.19 1641.36,1131.8 1595.16,1125.15 1550.14,1118.28 \n",
       "  1506.29,1111.21 1463.61,1103.95 1422.07,1096.51 1381.65,1088.9 1342.33,1081.14 1304.09,1073.22 1266.89,1065.15 1230.71,1056.93 1195.52,1048.58 1161.3,1040.09 \n",
       "  1128.03,1031.48 1095.67,1022.74 1064.22,1013.88 1033.65,1004.91 1003.95,995.842 975.091,986.673 947.071,977.415 919.873,968.077 893.488,958.667 867.903,949.193 \n",
       "  843.11,939.667 819.101,930.097 795.868,920.493 773.406,910.867 751.709,901.229 730.773,891.591 710.592,881.963 691.161,872.355 672.474,862.779 654.526,853.244 \n",
       "  637.309,843.76 620.816,834.335 605.037,824.978 589.964,815.697 575.585,806.499 561.89,797.39 548.867,788.376 536.501,779.464 524.773,770.655 513.668,761.953 \n",
       "  503.169,753.359 493.258,744.875 483.918,736.503 475.13,728.242 466.876,720.093 459.135,712.054 451.887,704.125 445.111,696.304 438.785,688.588 432.886,680.975 \n",
       "  427.391,673.461 422.276,666.043 417.519,658.715 413.105,651.476 409.015,644.322 405.233,637.25 401.74,630.256 398.52,623.337 395.556,616.49 392.834,609.712 \n",
       "  390.339,602.998 388.055,596.347 385.97,589.756 384.071,583.221 382.344,576.739 380.779,570.309 379.366,563.928 378.096,557.593 376.96,551.303 375.951,545.056 \n",
       "  375.059,538.85 374.277,532.683 373.599,526.554 373.018,520.461 372.527,514.404 372.121,508.38 371.795,502.389 371.543,496.431 371.362,490.503 371.248,484.606 \n",
       "  371.196,478.738 371.204,472.899 371.268,467.088 371.386,461.306 371.554,455.55 371.771,449.822 372.034,444.121 372.341,438.446 372.689,432.798 373.079,427.176 \n",
       "  373.507,421.58 373.972,416.011 374.475,410.469 375.013,404.953 375.586,399.464 376.193,394.002 376.834,388.568 377.506,383.161 378.211,377.782 378.947,372.43 \n",
       "  379.714,367.107 380.512,361.813 381.34,356.549 382.198,351.314 383.087,346.109 384.006,340.935 384.956,335.793 385.937,330.684 386.95,325.608 387.995,320.565 \n",
       "  389.072,315.558 390.183,310.586 391.329,305.651 392.51,300.754 393.727,295.896 394.982,291.078 396.277,286.302 397.612,281.569 398.988,276.88 400.406,272.236 \n",
       "  401.867,267.638 403.372,263.088 404.924,258.588 406.524,254.139 408.174,249.743 409.876,245.402 411.634,241.119 413.45,236.897 415.328,232.737 417.27,228.642 \n",
       "  419.281,224.617 421.365,220.663 423.525,216.785 425.767,212.985 428.094,209.269 430.513,205.638 433.028,202.098 435.644,198.654 438.399,195.325 441.373,192.161 \n",
       "  444.542,189.148 447.878,186.27 451.359,183.52 454.971,180.889 458.708,178.377 462.569,175.985 466.56,173.72 470.693,171.59 474.988,169.611 479.471,167.801 \n",
       "  484.177,166.18 489.143,164.776 494.417,163.619 500.052,162.742 506.108,162.184 512.652,161.986 519.756,162.196 527.502,162.863 535.976,164.041 545.271,165.789 \n",
       "  555.488,168.168 566.734,171.246 579.123,175.091 592.775,179.779 607.817,185.388 624.488,192.058 643.36,200.121 664.599,209.691 688.545,220.975 715.795,234.328 \n",
       "  747.2,250.245 783.868,269.364 827.161,292.464 878.696,320.47 940.348,354.447 1014.31,395.643 1103.54,445.778 1211.03,506.614 1339.24,579.693 1486.81,664.771 \n",
       "  1647.17,758.876 1808.32,856.208 1951.41,947.349 2059.71,1023.59 2125.96,1080.68 2153.55,1119.56 2150.67,1143.67 2127.65,1157.61 2091.65,1164.64 2047.87,1167.14 \n",
       "  1999.88,1166.75 1949.73,1164.43 1898.76,1160.83 1847.8,1156.32 1797.33,1151.17 1747.67,1145.53 1698.99,1139.51 1651.4,1133.18 1604.95,1126.59 1559.68,1119.77 \n",
       "  1515.59,1112.74 1472.66,1105.52 1430.87,1098.12 1390.22,1090.54 1350.67,1082.81 1312.2,1074.92 1274.78,1066.89 1238.38,1058.71 1202.99,1050.38 1168.56,1041.92 \n",
       "  1135.08,1033.33 1102.53,1024.62 1070.89,1015.79 1040.13,1006.84 1010.24,997.795 981.207,988.646 953.009,979.407 925.638,970.086 899.079,960.69 873.323,951.229 \n",
       "  848.36,941.713 824.182,932.151 800.783,922.553 778.156,912.931 756.297,903.295 735.198,893.657 714.857,884.026 695.267,874.414 676.422,864.83 658.316,855.286 \n",
       "  640.942,845.789 624.293,836.35 608.36,826.977 593.135,817.678 578.608,808.462 564.768,799.334 551.6,790.301 539.09,781.366 527.223,772.534 515.984,763.807 \n",
       "  505.357,755.189 495.324,746.682 485.868,738.285 476.97,730.001 468.611,721.829 460.769,713.769 453.423,705.818 446.551,697.976 440.13,690.239 434.136,682.604 \n",
       "  428.546,675.068 423.343,667.627 418.507,660.28 414.019,653.021 409.861,645.849 406.014,638.759 402.462,631.749 399.187,624.814 396.172,617.952 393.401,611.16 \n",
       "  390.859,604.433 388.531,597.769 386.402,591.165 384.461,584.618 382.696,578.125 381.096,571.683 379.652,565.292 378.353,558.947 377.191,552.647 376.156,546.391 \n",
       "  375.241,540.177 374.437,534.002 373.739,527.865 373.137,521.764 372.628,515.7 372.203,509.669 371.86,503.672 371.592,497.706 371.395,491.772 371.267,485.869 \n",
       "  371.202,479.994 371.197,474.149 371.25,468.333 371.356,462.544 371.514,456.782 371.721,451.048 371.973,445.341 372.271,439.66 372.61,434.006 372.991,428.379 \n",
       "  373.411,422.778 373.87,417.203 374.365,411.656 374.896,406.134 375.463,400.64 376.063,395.173 376.696,389.732 377.361,384.319 378.058,378.933 378.786,373.576 \n",
       "  379.546,368.246 380.335,362.945 381.156,357.674 382.007,352.432 382.888,347.221 383.8,342.04 384.743,336.891 385.717,331.775 386.723,326.691 387.762,321.642 \n",
       "  388.834,316.627 389.94,311.649 391.08,306.707 392.256,301.802 393.467,296.936 394.715,292.11 396.001,287.324 397.325,282.581 398.689,277.88 400.095,273.225 \n",
       "  401.544,268.615 403.037,264.054 404.577,259.542 406.165,255.082 407.805,250.675 409.497,246.323 411.245,242.03 413.051,237.796 414.919,233.624 416.85,229.518 \n",
       "  \n",
       "  \"/>\n",
       "<path clip-path=\"url(#clip7300)\" d=\"\n",
       "M1680.28 326.155 L2280.76 326.155 L2280.76 205.195 L1680.28 205.195  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<polyline clip-path=\"url(#clip7300)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  1680.28,326.155 2280.76,326.155 2280.76,205.195 1680.28,205.195 1680.28,326.155 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip7300)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  1704.28,265.675 1848.28,265.675 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip7300)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1872.28, 283.175)\" x=\"1872.28\" y=\"283.175\">[&quot;limit cycle plot&quot;]</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(sol1,vars=(1,2),label=[\"limit cycle plot\"])\n",
    "title!(\"Brusselator \")\n",
    "xlabel!(\"X [a.u.]\")\n",
    "ylabel!(\"Y [a.u.]\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=red>Computer task:</font> \n",
    "\n",
    "The Lotka-Volterra model, also known as Predator-Prey model describes the inteactions between two populations of species where one feeds into the other.  One can think of rabbits $x$ and foxes $y$, such that rabits multiply where there is no foxes (assuming an infinite amount of food for the rabbits). This is basically a first order production of rabbits, an autocalalitic system similar to the ones studied above that results in exponnetial growth of rabbits.\n",
    "\n",
    "Next, foxes $y$ feed on the rabbits and multiply due to the good food conditions. Then also foxes die at a constant rate (rabbits also die, but the mdoel assumes that their rate of birth is much higher than the rate of death). The scheme of interactions for this very simple system is the following:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " x &\\overset{k_1}{\\longrightarrow} 2 x   \\\\\n",
    " x + y &\\overset{k_2}{\\longrightarrow} 2 y \\\\\n",
    " y &\\overset{k_3}{\\longrightarrow} 0 \n",
    " \\end{align}\n",
    " $$\n",
    " \n",
    " \n",
    "Find the order of the reactions, the units of the kinetic rate constants, write the differential equations that govern the dynamcis of the following system, the steady state values and solve numerically (plot the time evolution and phase plot) for the following values:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x₀=1\n",
    "y₀=1\n",
    "\n",
    "k1=2.0\n",
    "k2=1.0\n",
    "k3=3.0\n",
    "t₀=0.0\n",
    "final_time=10.0;"
   ]
  }
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