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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Intégration numérique avec Julia, une introduction"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Introduction\n",
+ "Pour la documentation voire \n",
+ "[Documentation of DifferentialEquations.jl](https://diffeq.sciml.ai/stable/)\n",
+ "\n",
+ "\n",
+ "Il nous faut tout d'abord importer les bons packages de Julia"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# import Pkg; Pkg.add(\"DifferentialEquations\")\n",
+ "# Pkg.add(\"Plots\")\n",
+ "using DifferentialEquations\n",
+ "using Plots"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Exemple 1\n",
+ "\n",
+ "Nous allons ici résoudre le problème à valeur initiale\n",
+ "$$(IVP)\\left\\{\\begin{array}{l}\n",
+ "\\dot{x}_1(t)=x_1(t)+x_2(t)+\\sin t\\\\\n",
+ "\\dot{x}_2(t)=-x_1(t)+3x_2(t)\\\\\n",
+ "x_1(0)=-9/25\\\\\n",
+ "x_2(0)=-4/25,\n",
+ "\\end{array}\\right.\n",
+ "$$\n",
+ "\n",
+ "- Ecrire la fonction $f$ qui permet d'écrire l'équation différentielle sous la forme $\\dot{x}(t)=f(t,x(t))$\n",
+ "- Vérifier que la solution de $(IVP$ est\n",
+ "$$\\begin{align*}\n",
+ "x_1(t)&= (-1/25)(13\\sin t+9\\cos t)\\\\\n",
+ "x_2(t)&= (-1/25)(3\\sin t+4\\cos t).\n",
+ "\\end{align*}$$\n",
+ "- Coder la fonction $f$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
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+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function exemple1(x,p,t)\n",
+ "# second member of the IVP\n",
+ "# x : state\n",
+ "# p : paramter vector\n",
+ "# t : time, variable not use here\n",
+ " xpoint = zeros(size(x))\n",
+ "\n",
+ "# TO COMPLETE\n",
+ " return xpoint\n",
+ "end\n",
+ "p = []\n",
+ "t0 = 0.; tf = 10\n",
+ "tspan = (t0,tf)\n",
+ "x0 = [-9/25 -4/25]\n",
+ "prob = ODEProblem(exemple1,x0,tspan,p) # défini le problème en Julia\n",
+ "sol = solve(prob)\n",
+ "p1 = plot(sol, label = [\"x_1(t)\" \"x_2(t)\"], title = \"default options\", lw = 0.5) # lw = linewidth"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Le contrôle du pas\n",
+ "Les bons intégrateurs numériques ont une mise à jour automatique du pas. Pour les méthodes de Runge-Kutta par exemple, sur un pas $[t_0, t_{1}]$ on calcule par 2 schémas différents deux solutions approchées à l'instant $t_{1}~: x_{1}$ et $\\hat{x}_{1}$ dont l'erreur local est respectivement en $O(h^p)$ et $O(h^{p+1})$ (les err\n",
+ "eurs globales sont elles en $O(h^{p-1})$ et $O(h^p)$). Ainsi on peut estimer l'erreur locale du schéma le moins précis à l'aide de la différence $x_{1}-\\hat{x}_{1}$. On peut alors estimer le pas de façon à avoir la norme de cette différence inférieur à une tolérance donnée $Tol$ ou encore à avoir \n",
+ "$$\\frac{\\|x_{1}-\\hat{x}_{1}\\|}{Tol}<1.$$\n",
+ "\n",
+ "En pratique on considère des tolérences absolue et relative composante par composante et on définit : $sc_i = Atol_i + \\max(|x_{0i}|,|x_{1i}|)Rtol_i$. On calcule alors le pas de façon à avoir \n",
+ "\n",
+ "$$err = \\sqrt{\\frac{1}{n}\\sum_{i=1}^n\\left(\\frac{x_{1i}-\\hat{x}_{1i}}{sc_i}\\right)^2}<1.$$\n",
+ "\n",
+ "\n",
+ "Référence : [Hairer, Nørsett, Wanner (2008) Solving Ordinary Differential Equations I Nonstiff Problems DOI: 10.1007/978-3-540-78862-1](https://link.springer.com/book/10.1007/978-3-540-78862-1)\n",
+ "\n",
+ "Dans `Julia`, on peut modifier les valeurs par défaut : reltol = 1.e-3 et abstol = 1.e-6\n",
+ "(lorsque ces valeurs sont des scalaires, on considère les mêmes quantités pour toutes les composantes).\n",
+ "\n",
+ "Réaliser l'intégration numérique pour \n",
+ "- reltol = 1.e-3 et abstol = 1.e-6\n",
+ "- reltol = 1.e-6 et abstol = 1.e-9\n",
+ "- reltol = 1.e-10 et abstol = 1.e-15\n",
+ "\n",
+ "et afficher les résultats en ajoutant la solution."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Commentaires?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
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+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "sol = solve(prob, reltol = 1.e-3, abstol = 1.e-6)\n",
+ "p1 = plot(sol, label = [\"x1(t)\" \"x2(t)\"], title = \"reltol=1e-3, abstol=1e-6\", lw = 0.5) # lw = linewidth\n",
+ "#\n",
+ "# TO COMPLETE\n",
+ "p2 = plot()\n",
+ "p3 = plot()\n",
+ "\n",
+ "#\n",
+ "T = t0:(tf-t0)/100:tf\n",
+ "sinT = sin.(T) # opération vectorielle\n",
+ "cosT = cos.(T)\n",
+ "p4 = plot(T,[(-1/25)*(13*sinT+9*cosT) (-1/25)*(3*sinT+4*cosT)], label = [\"x_1(t)\" \"x_2(t)\"], title = \"Solution exacte\")\n",
+ "plot(p1,p2,p3,p4, size=(1200, 600))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Pendule simple\n",
+ "### Introduction\n",
+ "On s'intéresse ici au pendule simple. Les principes physiques de la mécanique classique donnent comme équation qui régit l'évolution du mouvement\n",
+ "\n",
+ "$$ ml^2\\ddot{\\alpha}(t)+mlg\\sin(\\alpha(t)) +k\\dot{\\alpha}(t)=0,$$\n",
+ "où $\\ddot{\\alpha}(t)$ désigne la dérivée seconde de l'angle $\\alpha$ par rapport au temps $t$. \n",
+ "\n",
+ "- En prenant comme variable d'état qui décrit le système $x(t)=(x_1(t),x_2(t))=(\\alpha(t), \\dot{\\alpha}(t))$, écrire la fonction $f$ qui permet d'écrire l'équation différentielle sous la forme $\\dot{x}(t) = f(t,x(t))$\n",
+ "- Coder cette fonction"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function pendule(x,p,t)\n",
+ "# second member of the IVP\n",
+ "# x : state\n",
+ "# p : paramter vector\n",
+ "# t : time, variable not use here\n",
+ " xpoint = zeros(size(x))\n",
+ "\n",
+ "# TO COMPLETE\n",
+ " \n",
+ " return xpoint \n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {},
+ "outputs": [
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+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "# Main\n",
+ "#\n",
+ "g = 9.81; l = 10; k = 0.;\n",
+ "p = [g,l,k] # constantes\n",
+ "theta0 = pi/3\n",
+ "t0 = 0.\n",
+ "tf = 3*pi*sqrt(l/g)*(1 + theta0^2/16 + theta0^4/3072) # 2*approximation of the period\n",
+ "tspan = (t0,tf) # instant initiel et terminal\n",
+ "x0 = [theta0,0] # état initial\n",
+ "prob = ODEProblem(pendule,x0,tspan,p) # défini le problème en Julia\n",
+ "sol = solve(prob) # réalise l'intégration numérique\n",
+ "plot(sol, label = [\"x1(t)\" \"x2(t)\"], lw = 0.5) "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Diagrammes de phases"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
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+ ],
+ "source": [
+ "# Main\n",
+ "#\n",
+ "g = 9.81; l = 10; k = 0.;\n",
+ "p = [g l k]\n",
+ "plot()\n",
+ "for theta0 in 0:(2*pi)/10:2*pi\n",
+ " theta0_princ = theta0\n",
+ " tf = 3*pi*sqrt(l/g)*(1 + theta0_princ^2/16 + theta0_princ^4/3072) # 2*approximation of the period\n",
+ " tspan = (0.0,tf)\n",
+ " x0 = [theta0 0]\n",
+ " prob = ODEProblem(pendule,x0,tspan,p)\n",
+ " sol = solve(prob)\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ "end\n",
+ "theta0 = pi-10*eps()\n",
+ "x0 = [theta0 0]\n",
+ "tf = 70 # problem for tf=50 1/4 of the period!\n",
+ "tspan = (0.0,tf)\n",
+ "prob = ODEProblem(pendule,x0,tspan,p)\n",
+ "sol = solve(prob)\n",
+ "plot!(sol,vars=(1,2), xlims = (-2*pi,4*pi), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ "\n",
+ "theta0 = pi+10*eps()\n",
+ "x0 = [theta0 0]\n",
+ "tf = 70\n",
+ "tspan = (0.0,tf)\n",
+ "prob = ODEProblem(pendule,x0,tspan,p)\n",
+ "sol = solve(prob)\n",
+ "plot!(sol,vars=(1,2), xlims = (-2*pi,4*pi), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ "\n",
+ "# circulation case\n",
+ "for thetapoint0 in 0:0.2:2 \n",
+ " tf = 10\n",
+ " tspan = (0.,tf)\n",
+ " x0 = [-pi thetapoint0] # thetapoint0 > 0 so theta increases from -pi to ...\n",
+ " prob = ODEProblem(pendule,x0,tspan,p)\n",
+ " sol = solve(prob)\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ "end\n",
+ "for thetapoint0 in -2:0.2:0\n",
+ " tf = 10\n",
+ " tspan = (0.,tf)\n",
+ " x0 = [3*pi thetapoint0] # thetapoint0 < 0 so theta decreases from 3pi to ...\n",
+ " prob = ODEProblem(pendule,x0,tspan,p)\n",
+ " sol = solve(prob)\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ "end\n",
+ "plot!([-pi 0 pi 2*pi 3*pi], [0 0 0 0 0], seriestype=:scatter)\n",
+ "plot!(xlims = (-pi,3*pi))\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "- Interprétez ces graphiques\n",
+ "- On considère le cas où $k=0.15$ (on introduit un frotement). Que se passe-t-til?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Modèle de Van der Pol\n",
+ "### Exemple de cycle limite\n",
+ "L'équation différentielle considérée est l'équation de Van der Pol\n",
+ "$$(IVP)\\left\\{\\begin{array}{l}\n",
+ "\\dot{y}_1(t)=y_2(t)\\\\\n",
+ "\\dot{y}_2(t)=(1-y_1^2(t))y_2(t)-y_1(t)\\\\\n",
+ "y_1(0)=2.00861986087484313650940188\\\\\n",
+ "y_2(0)=0\n",
+ "\\end{array}\\right.\n",
+ "$$\n",
+ "$t_f=T=6.6632868593231301896996820305$\n",
+ "\n",
+ "\n",
+ "La solution de ce problème de Cauchy est périodique de période $T$.\n",
+ "\n",
+ "Résoudre et de visualiser la solution pour les points de départ :\n",
+ "- x0 = [2.00861986087484313650940188,0]\n",
+ "- [x01 , 0] pour x01 dans -2:0.4:0\n",
+ "- [0 , x02] pour x02 dans 2:1:4\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function vdp(x,p,t)\n",
+ "# Van der Pol model\n",
+ "# second member of the IVP\n",
+ "# x : state\n",
+ "# real(2)\n",
+ "# p : parameter vector\n",
+ "# t : time, variable not use here\n",
+ "# real\n",
+ "# Output\n",
+ "# xpoint : vector of velocity\n",
+ "# same as x\n",
+ "\n",
+ " xpoint = zeros(size(x))\n",
+ "\n",
+ "# TO COMPLETE\n",
+ " \n",
+ " return xpoint\n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "x0 = [2.0086198608748433, 0.0], p = [1], tspan = (0.0, 6.66328685932313)\n"
+ ]
+ },
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+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "# Main\n",
+ "#\n",
+ "tf = 6.6632868593231301896996820305\n",
+ "tspan = (0.0, tf)\n",
+ "x0 = [2.00861986087484313650940188,0]\n",
+ "mu = 1\n",
+ "p = [mu]\n",
+ "t0 = 0.;\n",
+ "tspan = (t0,tf)\n",
+ "println(\"x0 = $x0, p = $p, tspan = $tspan\")\n",
+ "prob = ODEProblem(vdp,x0,tspan,p)\n",
+ "sol = solve(prob)\n",
+ "plot(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false)\n",
+ "for x01 in -2:0.4:0 #4.5:4.5\n",
+ " x0 = [x01,0]\n",
+ " prob = ODEProblem(vdp,x0,tspan,p)\n",
+ " sol = solve(prob)#,force_dtmin=true)# \n",
+ " plot!(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false) \n",
+ "end\n",
+ "for x02 in 2:1:4\n",
+ " x0 = [0.,x02]\n",
+ " prob = ODEProblem(vdp,x0,tspan,p)\n",
+ " sol = solve(prob)#solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8)\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false)\n",
+ "end\n",
+ "plot!([0], [0], seriestype=:scatter) # point visualisation\n",
+ "plot!(xlims = (-2.5,5))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Flots en dimension 2\n",
+ "\n",
+ "### Flot des extrémales pour le problème de conrôle optimal \n",
+ "\n",
+ "On considère le problème de contrôle optimal suivant:\n",
+ "\n",
+ "$$ \\frac{1}{2}\\int_0^{1} u^2(t)\\,\\mathrm{d}t \\to \\min, $$\n",
+ "\n",
+ "où\n",
+ "\n",
+ "$$ \\dot{x}(t) = -x(t) + u(t)$$\n",
+ "\n",
+ "$$ x(0)=x_0=-1,\\quad x(1)=0. $$\n",
+ "\n",
+ "Le but est ici de tracer les extrémales et le flot associé pour des points initiaux $z(0) = (x_0, p(0))$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "plot_traj (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 18,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function plot_traj(plt,f,Z0,tf; par=[])\n",
+ " \"\"\" \n",
+ " Plot the trajectories in the phase plan of the ode dot{x}(t)=f(t,z(t))\n",
+ " The initial points are in the matrix `Z0`\n",
+ " -------\n",
+ " \n",
+ " parameters (input)\n",
+ " ------------------\n",
+ " plt : plot object\n",
+ " f : function f(x,p,t)\n",
+ " Z0 : array of float(n,nbpoints)\n",
+ " Z0[:,i] = initial point number i\n",
+ " returns\n",
+ " -------\n",
+ " nothing\n",
+ " \"\"\"\n",
+ " nbpoints = size(Z0,2)\n",
+ " tspan = (0. , tf)\n",
+ " for i in 1:nbpoints \n",
+ " prob = ODEProblem(f,Z0[:,i],tspan,par) # défini le problème en Julia\n",
+ " sol = solve(prob)\n",
+ " plot!(plt, sol, vars = (1,2),legend = false, color = :blue) # lw = linewidth\n",
+ " end\n",
+ " return nothing\n",
+ "end "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function plot_flow(plt,f,Z0,tf; par=[])\n",
+ " \"\"\" \n",
+ " plot the flows of the ode dot{x}(t)=f(t,z(t))\n",
+ " The initial points are in the matrix `Z0`\n",
+ " -------\n",
+ " \n",
+ " parameters (input)\n",
+ " ------------------\n",
+ " plt : plot object\n",
+ " f : function f(x,p,t)\n",
+ " Z0 : array of float(n,nbpoints)\n",
+ " Z0[:,i] = initial point number i\n",
+ " returns\n",
+ " -------\n",
+ " nothing\n",
+ " \"\"\"\n",
+ " nbpoints = size(Z0,2)\n",
+ " tspan = (0. , tf)\n",
+ " T = range(0.,tf,length=4)\n",
+ " for i in 1:nbpoints \n",
+ " # TO COMPLETE\n",
+ " \n",
+ " end\n",
+ " return nothing\n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "- Compléter la fonction suivante pour tracer le flot pour 4 valeurs de $t$\n",
+ "\n",
+ "Remarque : `sol(t)` donne la valeur de l'approximation de la solution à l'instant $t$"
+ ]
+ },
+ {
+ "cell_type": "code",
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+ "execution_count": 23,
+ "metadata": {},
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+ }
+ ],
+ "source": [
+ "function fun1(z,p,t)\n",
+ " zpoint = similar(z)\n",
+ " zpoint[1] = -z[1] + z[2]\n",
+ " zpoint[2] = z[2]\n",
+ " return zpoint\n",
+ "end\n",
+ "\n",
+ "t0 = 0; tf = 1.;\n",
+ "\n",
+ "seg = range(-1.,1.,length=10)\n",
+ "Z0 = transpose([-ones(size(seg)) seg])\n",
+ "plt1 = plot()\n",
+ "plot_traj(plt1,fun1,Z0,tf)\n",
+ "\n",
+ "seg = range(-1.,1.,length=200)\n",
+ "Z0 = transpose([-ones(size(seg)) seg])\n",
+ "plot_flow(plt1,fun1,Z0,tf)\n",
+ "plot!(plt1, xlims = (-2.,1))\n",
+ "plot!(plt1,[0.,0.],[-3.,3], linestyle = :dot, color = :black)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "- Retrouvez graphiquement la solution du problème de contrôle optimal\n",
+ "- La droite de départ se transforme par le plot en une droite, pourquoi?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Flot des extrémales pour le problème de conrôle optimal \n",
+ "\n",
+ "On considère le problème de contrôle optimal suivant:\n",
+ "\n",
+ "$$ \\frac{1}{2}\\int_0^{1} u^2(t)\\,\\mathrm{d}t \\to \\min, $$\n",
+ "\n",
+ "où\n",
+ "\n",
+ "$$ \\dot{x}(t) = -x(t) + \\alpha x^2(t) + u(t)$$\n",
+ "\n",
+ "$$ x(0)=x_0=-1,\\quad x(1)=0. $$\n",
+ "\n",
+ "Le but est ici de tracer les extrémales et le flot associé pour des points initiaux $z(0) = (x_0, p(0))$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
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+ " 1247.78,1098.08 1250.17,1100.25 1252.57,1102.42 1254.99,1104.58 1257.43,1106.73 1259.87,1108.88 1262.33,1111.03 1264.81,1113.16 1267.29,1115.3 1269.8,1117.42 \n",
+ " 1272.31,1119.54 1274.85,1121.66 1277.39,1123.76 1279.95,1125.87 1282.53,1127.96 1285.12,1130.05 1287.72,1132.14 1290.34,1134.22 1292.97,1136.29 1295.62,1138.35 \n",
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+ " 1494.33,1251.82 1498.39,1253.46 1502.49,1255.1 1506.62,1256.73 1510.79,1258.36 1514.99,1259.97 1519.23,1261.59 1523.5,1263.19 1527.81,1264.79 1532.16,1266.37 \n",
+ " 1536.55,1267.96 1540.98,1269.53 1545.45,1271.1 1549.95,1272.66 1554.5,1274.22 1559.09,1275.77 1563.71,1277.31 1568.39,1278.84 1573.1,1280.37 1577.86,1281.89 \n",
+ " 1582.66,1283.4 1587.5,1284.91 1592.39,1286.41 1597.33,1287.91 1602.31,1289.39 1607.33,1290.87 1612.41,1292.35 1617.53,1293.81 1622.7,1295.27 1627.92,1296.73 \n",
+ " 1633.2,1298.17 1638.52,1299.61 1643.89,1301.05 1649.31,1302.48 1654.79,1303.9 1660.32,1305.31 1665.9,1306.72 1671.54,1308.12 1677.23,1309.52 1682.98,1310.9 \n",
+ " 1688.78,1312.29 1694.64,1313.66 1700.57,1315.03 1706.55,1316.4 1712.6,1317.75 1718.71,1319.1 1724.89,1320.44 1731.13,1321.78 1737.44,1323.1 1743.81,1324.43 \n",
+ " 1750.25,1325.74 1756.77,1327.05 1763.35,1328.35 1770,1329.64 1776.73,1330.92 1783.53,1332.2 1790.4,1333.47 1797.35,1334.74 1804.38,1335.99 1811.48,1337.24 \n",
+ " 1818.67,1338.49 1825.93,1339.72 1833.28,1340.95 1840.72,1342.17 1848.23,1343.39 1855.84,1344.59 1863.54,1345.79 1871.32,1346.99 1879.2,1348.17 1887.18,1349.35 \n",
+ " 1895.25,1350.52 1903.41,1351.68 1911.68,1352.84 1920.05,1353.99 1928.53,1355.13 1937.11,1356.27 1945.8,1357.39 1954.6,1358.51 1963.52,1359.63 1972.55,1360.73 \n",
+ " 1981.69,1361.83 1990.96,1362.92 2000.35,1364.01 2009.87,1365.09 2019.51,1366.16 2029.29,1367.22 2039.19,1368.27 2049.24,1369.32 2059.42,1370.36 2069.74,1371.4 \n",
+ " 2080.21,1372.43 2090.82,1373.45 2101.59,1374.46 2112.5,1375.46 2123.58,1376.46 2134.81,1377.46 2146.2,1378.44 2157.76,1379.42 2169.49,1380.39 2181.39,1381.35 \n",
+ " \n",
+ " \"/>\n",
+ "<polyline clip-path=\"url(#clip142)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" stroke-dasharray=\"2, 4\" points=\"\n",
+ " 467.732,1445.72 467.732,87.9763 \n",
+ " \"/>\n",
+ "</svg>\n"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function fun2(z,p,t)\n",
+ " α = p\n",
+ " zpoint = similar(z)\n",
+ " zpoint[1] = -z[1] + α*z[1]^2 + z[2]\n",
+ " zpoint[2] = (1. - 2*α*z[1])*z[2]\n",
+ " return zpoint\n",
+ "end\n",
+ "\n",
+ "t0 = 0; tf = 1.;\n",
+ "α = 1.5\n",
+ "seg = range(-0.,2,length=10)\n",
+ "Z0 = transpose([-ones(size(seg)) seg])\n",
+ "plt1 = plot()\n",
+ "plot_traj(plt1,fun2,Z0,tf,par=α)\n",
+ "\n",
+ "seg = range(-0.,2,length=200)\n",
+ "Z0 = transpose([-ones(size(seg)) seg])\n",
+ "plot_flow(plt1,fun2,Z0,tf,par=α)\n",
+ "#plot!(plt1, xlims = (-2.,1))\n",
+ "plot!(plt1,[0.,0.],[0.,4], linestyle = :dot, color = :black)\n",
+ "plot(plt1)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.4.1",
+ "language": "julia",
+ "name": "julia-1.4"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.4.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}