TP9_Equations_aux_differences_II.ipynb

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    "# INFO-F-305 - TP 9\n",
    "# Equations aux différences II\n",
    "\n",
    "### Jacopo De Stefani - __[jdestefa@ulb.ac.be](mailto:jdestefa@ulb.ac.be)__\n",
    "### Github: https://github.com/jdestefani/ulb-infof305\n",
    "### Slides: https://jdestefani.github.io/ulb-infof305"
   ]
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   "source": [
    "# Rappels Théoriques - Equation aux différences affines\n",
    "\n",
    "### Equation affine aux différences, linéaire, d’ordre 1 et à coefficients constants\n",
    "\n",
    "<center> <h3>$x(k+1) = a x(k) + b$</h3> </center>\n",
    "\n",
    "- $ a \\in \\mathbb{R}, a \\neq 0$\n",
    "- $ b \\in \\mathbb{R}$\n",
    "\n"
   ]
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   "source": [
    "# Rappels Théoriques - Equation aux différences affines\n",
    "\n",
    "### Théorème 7.9\n",
    "\n",
    "Pour chaque valeur $x^0 = x(0) $ l'equation a une solution unique\n",
    "\n",
    "- $a=1$ $\\Rightarrow$ $x(k) = x(0) + kb$\n",
    "    - $b=0$ $\\Rightarrow$ Aucun équilibre, trajectoires divergentes\n",
    "    - $b \\neq 0$ $\\Rightarrow$ Toutes les valeurs de $x$ sont des valeurs d'équilibres, trajectoires constantes\n",
    "\n",
    "- $a \\neq 1$ $\\Rightarrow$ $x(k) = (x(0) - \\alpha)a^k + \\alpha$\n",
    "    - $\\bar{x} = \\alpha = \\frac{b}{1-a}$ seul équilibre\n",
    "    - $\\vert a \\vert < 1$ $\\Rightarrow$ Toutes trajectoires convergent vers $\\alpha$\n",
    "    - $\\vert a \\vert > 1$ $\\Rightarrow$ Toutes trajectoires divergentes sauf celle pour $x(0) = \\alpha$\n",
    "    - $a = -1$ $\\Rightarrow$ Equilibre $\\alpha = \\frac{b}{2}$ et toutes solutions oscillantes, à part celle constante pour $x(0) = \\alpha = \\frac{b}{2}$\n"
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   "source": [
    "# Rappels Théoriques - Equation aux différences non linéaires\n",
    "\n",
    "### Systèmes discrets non-linéraires d'ordre 1\n",
    "\n",
    "$x(k+1) = f(x(k))$\n",
    "\n",
    "- $ f \\in C^{\\infty}$\n",
    "- $ x(k) \\in \\mathbb{I} \\subset \\mathbb{R}$\n",
    "- $x(1) = f(x(0))$\n",
    "- $x(2) = f(x(1)) = f(f(x(0))) = f^2(x(0))$\n",
    "- $\\cdots$\n",
    "- $x(k) = f(x(k-1)) = \\cdots = f^k(x(0))$\n",
    "\n",
    "**N.B**\n",
    "- $x(k) = f^k(x(0)) \\neq (f(x(0)))^k \\neq f^{(k)}(x(0))$ \n",
    "\n",
    "\n",
    "    "
   ]
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   "source": [
    "# Rappels Théoriques - Equation aux différences non linéaires\n",
    "\n",
    "$x(k+1) = f(x(k))$\n",
    "\n",
    "### Etat d'équilibre\n",
    "$\\bar{x}$ est un état d'équilibre du système ${f,I}$ si $\\bar{x} = f(\\bar{x})$\n",
    "\n",
    "- **En pratique - Graphique:** Intersection entre $y=f(x)$ et $y=x$ sur le plan $x,y$\n",
    "\n",
    "### Théorème 7.10\n",
    "Soit $I=[a,b]$ un intervalle clos et borné et $f: I \\mapsto I$.\n",
    "Alors il existe toujours $\\bar{x}$ et $\\bar{x} = f(\\bar{x})$"
   ]
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   "source": [
    "# Rappels Théoriques - Cycles\n",
    "\n",
    "\n",
    "Un cycle d'ordre $s$ est un ensemble de $s$ valeurs différents $\\{\\bar{x}_0,\\cdots,\\bar{x}_{s-1}\\}$ un système discret $\\{f,I\\}$ telles que:\n",
    "\n",
    "<center> $\\bar{x}_1 = f(\\bar{x}_0)$   $\\bar{x}_2 = f(\\bar{x}_1)$ $\\cdots$ $\\bar{x}_0 = f(\\bar{x}_{s-1})$</center>\n",
    "\n",
    "La quantité $s$ est le période de l'orbite.\n",
    "\n",
    "### Théorème 7.11\n",
    "La paire $\\{\\bar{x}_0,\\bar{x}_{1}\\}$ du système $\\{f,I\\}$ est un cycle d'ordre $2$ si et seulement si $\\bar{x}_0$ ,$\\bar{x}_{1}$ sont des points d'équilibre du système $\\{f^2,I\\}$ mais pas de $\\{f,I\\}$.\n"
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   "source": [
    "# Rappels Théoriques - Stabilité\n",
    "\n",
    "### Stabilité de l'équilibre\n",
    "\n",
    "- **Stable:** Si pour chaque $\\varepsilon$, il existe $\\delta > 0$ telle quelle\n",
    "$\\vert x(0) - \\bar{x} \\vert < \\delta \\Rightarrow \\vert x(k) - \\bar{x} \\vert < \\varepsilon$\n",
    "- **Globalement asymptotiquement stable/attractif:** Si pour chaque $x(0) \\in I$\n",
    "$\\lim\\limits_{k \\rightarrow \\infty} x(k) = \\bar{x}$ \n",
    "- **Localement asymptotiquement stable/attractif:** Si il existe $\\eta$ telle quelle pour chaque $x(0) \\in I \\cap \\{x-\\eta,x+\\eta\\}$\n",
    "$\\lim\\limits_{k \\rightarrow \\infty} x(k) = \\bar{x}$\n",
    "\n",
    "### Théorème 7.12\n",
    "\n",
    "Soit $\\bar{x}$ un équilibre du système $\\{f,I\\}$ où $f \\in C^1$:\n",
    "\n",
    "- **Instable:** $\\vert f'(\\bar{x}) \\vert > 1$.\n",
    "- **Aucune information:** $\\vert f'(\\bar{x}) \\vert = 1$. \n",
    "- **Localement asymptotiquement stable/attractif:** $\\vert f'(\\bar{x}) \\vert < 1$.\n"
   ]
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   "source": [
    "# Rappels Théoriques - Stabilité ordre 1\n",
    "\n",
    "### Théorème 7.13\n",
    "\n",
    "Soit $\\bar{x}$ un équilibre du système $\\{f,I\\}$ où $f \\in C^2$ et $f'(\\bar{x})=1$:\n",
    "\n",
    "- $f''(\\bar{x}) > 0$ $\\Rightarrow$ $\\bar{x}$ inférieurement asymptotiquement stable et supérieurement instable (où répulsif)\n",
    "- $f''(\\bar{x}) < 0$ $\\Rightarrow$ $\\bar{x}$  supérieurement asymptotiquement stable et inférieurement instable (où répulsif)\n",
    "\n",
    "### Théorème 7.14\n",
    "\n",
    "Soit $\\bar{x}$ un équilibre du système $\\{f,I\\}$ où $f \\in C^3$ et $f'(\\bar{x})=1$ et $f''(\\bar{x})=0$:\n",
    "\n",
    "- $f'''(\\bar{x}) > 0$ $\\Rightarrow$ $\\bar{x}$ instable\n",
    "- $f'''(\\bar{x}) < 0$ $\\Rightarrow$ $\\bar{x}$ localement asymptotiquement stable\n"
   ]
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   "source": [
    "# Rappels Théoriques - Stabilité ordre 1\n",
    "\n",
    "### Théorème 7.15\n",
    "\n",
    "Soit $\\bar{x}$ un équilibre du système $\\{f,I\\}$ où $f \\in C^3$ et $f'(\\bar{x})=-1$:\n",
    "\n",
    "- $2 f'''(\\bar{x}) + 3(f''(\\bar{x}))^2 > 0$ $\\Rightarrow$ $\\bar{x}$ localement asymptotiquement stable\n",
    "- $2 f'''(\\bar{x}) + 3(f''(\\bar{x}))^2 < 0$ $\\Rightarrow$ $\\bar{x}$ instable\n"
   ]
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   "source": [
    "# Exercice 1\n",
    "\n",
    "Trouver les cycles du système suivant :\n",
    "<h2><center> $x(k + 1) = x(k)^2 − 1$ </center></h2>"
   ]
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   "source": [
    "# Solution - Exercice 1\n",
    "\n",
    "$f(x) = x^2 - 1$\n",
    "\n",
    "$f^2(x)=f(f(x))= f(x^2 - 1) = (x^2 - 1)^2 - 1 = x^4 - 2x^2$"
   ]
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = -2 : 0.01 : 2;\n",
    "f = @(x) x.^2 - 1;\n",
    "\n",
    "hold on;\n",
    "plot(x,f(x))\n",
    "plot(x,f(f(x)))\n",
    "plot(x,x)\n",
    "set(gca, \"xaxislocation\", \"origin\");\n",
    "set(gca, \"yaxislocation\", \"origin\");\n",
    "xlabel(\"x(k)\");\n",
    "ylabel(\"x(k+1)\");\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$\\bar{x} = f(\\bar{x})$ $\\Rightarrow$ $\\bar{x} = \\bar{x}^2 - 1$ $\\Rightarrow$ $\\bar{x}^2 - \\bar{x} - 1$\n",
    "\n",
    "$\\bar{x} = f^2(\\bar{x})$ $\\Rightarrow$ $\\bar{x} = \\bar{x}^4 - 2\\bar{x}^2$ $\\Rightarrow$ $\\bar{x}^4 - 2\\bar{x}^2 - \\bar{x}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:06.591407Z",
     "start_time": "2020-12-01T20:52:06.393770Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "  -0.61803\n",
      "   1.61803\n",
      "\n",
      "ans =\n",
      "\n",
      "   1.61803\n",
      "  -1.00000\n",
      "  -0.61803\n",
      "   0.00000\n",
      "\n",
      "ans =\n",
      "\n",
      "  -1\n",
      "   0\n",
      "\n"
     ]
    }
   ],
   "source": [
    "c_f = [1 -1 -1]; # Coeffiecients de x=f(x)\n",
    "c_f_2 = [1 0 -2 -1 0]; # Coeffiecients de x=f^2(x)\n",
    "roots(c_f) # Racines de x=f(x)\n",
    "roots(c_f_2) # Racines de x=f^2(x)\n",
    "setdiff(single(roots(c_f_2)),single(roots(c_f))) # Racines de f^2 mais pas de f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:07.440321Z",
     "start_time": "2020-12-01T20:52:06.594399Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "function plot_system_simulation(f,x,x_0,max_steps=5)\n",
    "    hold on;\n",
    "    plot(x,x);\n",
    "    plot(x,f(f(x)));\n",
    "    plot(x,f(x));\n",
    "    set(gca, \"xaxislocation\", \"origin\");\n",
    "    set(gca, \"yaxislocation\", \"origin\");\n",
    "    xlabel(\"x(k)\");\n",
    "    ylabel(\"x(k+1)\");\n",
    "\n",
    "    x_k = [];\n",
    "    # Initial drawing - x_0\n",
    "    x_i = f(x_0);\n",
    "    line([x_0,x_0],[0,x_i]); # (x(0),0) -> (x(0),f(x(0))) = (x(0),x(1)) \n",
    "    line([x_0,x_i],[x_i,x_i]); # (x(0),x(1)) -> (x(1),x(1))\n",
    "\n",
    "    for (i = 2:max_steps)\n",
    "         x_k = [x_k;x_i]; # Concatenate x_i to x_k\n",
    "         x_i_1 = f(x_i); # x(k+1) = f(x(k))\n",
    "         line([x_i,x_i],[x_i,x_i_1]); # (x(k),x(k)) -> (x(k),f(x(k))) = (x(k),x(k+1))\n",
    "         line([x_i,x_i_1],[x_i_1,x_i_1]); # (x(k),x(k+1)) -> (x(k+1),x(k+1))\n",
    "         x_i = x_i_1; # x(k+1) = x(k) for next iteration\n",
    "    end\n",
    "\n",
    "    x_k = [x_k;x_i];\n",
    "endfunction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:08.601494Z",
     "start_time": "2020-12-01T20:52:07.444309Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_system_simulation(f,0:0.01:1,0.5,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:09.494906Z",
     "start_time": "2020-12-01T20:52:08.603492Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "error: 'max_steps' undefined near line 1 column 8\n"
     ]
    },
    {
     "data": {
      "image/png": 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yPDk5WSwWs9msqqr7NShlSs/P1f1KS6fTaZY6/3C/vuju5+r4+Pjo6CilNJvN+v3+hgcawIq56vf7JycnzWUgh4eHmx7pu7LhD8a6icNq5uf3mavftGKizOEj5uqVuVMDACG4UwMAIQgSACEIEgAhCBIAIQgSACEIEgAhCBKsXVmWL7sbP/xRBAmAEAQJXs/t7a1TJXjOJn/9BPxRbm9vb25u3H4QniNI8BqKopjNZn///bebjsNzLNnBK/n27Zvfpw4rCBK8hv39/V6v1+/3m19qAPxKkOD1DIfDHz9+lGW56YFARH79BAAhOEMCIARBAiCE/we9x6K0jt2uCgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(1:max_steps,x_k);\n",
    "xlabel(\"k\");\n",
    "ylabel(\"x(k)\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Exercice 2\n",
    "\n",
    "<h2><center> $x(k + 1) = 6x(k)^2(1 − x(k))$ </center></h2>\n",
    "\n",
    "où $x(k) \\in [0, 1]$.\n",
    "\n",
    "1. Calculer analytiquement $f^2(x)$ et tracer les graphiques respectifs de $f(x)$ et de $f^2(x)$\n",
    "2. Trouver les points d’équilibre\n",
    "3. Etudier la stabilité des points d’équilibre\n",
    "4. Pour chaque point d’équilibre, simuler graphiquement et numériquement une trajectoire de 5 étapes qui soit compatible avec l’analyse de stabilité\n",
    "5. Trouver graphiquement s’il existe un cycle d’ordre 2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "1. Calculer analytiquement f^2(x) et tracer les graphiques respectifs de f(x) et de f^2(x)\n",
    "\n",
    "$f(x) = 6x^2(1 − x)$\n",
    "\n",
    "$f^2(x)=f(f(x))= f(6x^2(1 − x)) = 6(6x^2(1 − x))^2(1- (6x^2(1 − x)) = 216x^4(1-2x+x^2)(1 - 6x^2 + 6x^3)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:10.674693Z",
     "start_time": "2020-12-01T20:52:09.498151Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = 0 : 0.01 : 1;\n",
    "f = @(x) ((6*x.^2).*(1-x));\n",
    "\n",
    "hold on;\n",
    "plot(x,f(x))\n",
    "plot(x,f(f(x)))\n",
    "plot(x,x)\n",
    "set(gca, \"xaxislocation\", \"origin\");\n",
    "set(gca, \"yaxislocation\", \"origin\");\n",
    "xlabel(\"x(k)\");\n",
    "ylabel(\"x(k+1)\");\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "2. Trouver les points d’équilibre\n",
    "\n",
    "$\\bar{x} = f(\\bar{x})$ $\\Rightarrow$ $\\bar{x} = 6\\bar{x}^2(1-\\bar{x})$ $\\Rightarrow$ $-6\\bar{x}^3 + 6\\bar{x}^2 - x$\n",
    "\n",
    "- $x^{(1)} = \\frac{-6-\\sqrt{12}}{12}$\n",
    "- $x^{(2)} = \\frac{-6+\\sqrt{12}}{12}$\n",
    "- $x^{(3)} = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:10.799891Z",
     "start_time": "2020-12-01T20:52:10.676858Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_eq =\n",
      "\n",
      "   0.78868\n",
      "   0.21132\n",
      "   0.00000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "c_f = [-6 6 -1 0]; # Coeffiecients de x=f(x)\n",
    "x_eq = roots(c_f) # Racines de x=f(x)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "3. Etudier la stabilité des points d’équilibre\n",
    "\n",
    "$f(x) = 6x^2(1 − x)$ $\\Rightarrow$ $f'(x) = 12x(1 − x) + -6x^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:11.294048Z",
     "start_time": "2020-12-01T20:52:10.804855Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f_sym = (sym)\n",
      "\n",
      "     2        \n",
      "  6*x *(1 - x)\n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "       2               \n",
      "  - 6*x  + 12*x*(1 - x)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "pkg load symbolic\n",
    "syms x;\n",
    "f_sym = 6*x^2*(1-x)\n",
    "diff(f_sym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:11.449545Z",
     "start_time": "2020-12-01T20:52:11.299253Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "diff_f =\n",
      "\n",
      "@(x) -6 * x .^ 2 + 12 * x .* (1 - x)\n",
      "\n",
      "x_eq =\n",
      "\n",
      "   0.78868\n",
      "   0.21132\n",
      "   0.00000\n",
      "\n",
      "ans =\n",
      "\n",
      "  -1.73205\n",
      "   1.73205\n",
      "   0.00000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "diff_f = @(x) -6*x.^2+12*x.*(1-x)\n",
    "x_eq\n",
    "diff_f(x_eq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "### Théorème 7.12\n",
    "- $\\vert f'x^{(1)} \\vert $ > 1 $\\Rightarrow$ $x^{(1)}$ instable\n",
    "- $\\vert f'x^{(2)} \\vert $ > 1 $\\Rightarrow$ $x^{(2)}$ instable\n",
    "- $\\vert f'x^{(3)} \\vert $ < 1 $\\Rightarrow$ $x^{(3)}$ asymptotiquement stable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:12.743124Z",
     "start_time": "2020-12-01T20:52:11.452511Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 4. Pour chaque point d’équilibre, simuler graphiquement et numériquement une trajectoire de 5 étapes qui soit compatible avec l’analyse de stabilité.\n",
    "# Pour x(0) = 0.05\n",
    "plot_system_simulation(f,0:.001:.05,0.05,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:13.992292Z",
     "start_time": "2020-12-01T20:52:12.746117Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 4. Pour chaque point d’équilibre, simuler graphiquement et numériquement une trajectoire de 5 étapes qui soit compatible avec l’analyse de stabilité.\n",
    "# Pour x(0) = 0.1\n",
    "plot_system_simulation(f,0:.01:1,0.25,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:15.280170Z",
     "start_time": "2020-12-01T20:52:13.995284Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    "# 4. Pour chaque point d’équilibre, simuler graphiquement et numériquement une trajectoire de 5 étapes qui soit compatible avec l’analyse de stabilité.\n",
    "# Pour x(0) = 0.7\n",
    "plot_system_simulation(f,0:.01:1,0.7,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:16.406599Z",
     "start_time": "2020-12-01T20:52:15.282178Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f_sym = (sym)\n",
      "\n",
      "     2        \n",
      "  6*x *(1 - x)\n",
      "\n",
      "f2_sym = (sym)\n",
      "\n",
      "       4 / 2          \\ /   3      2    \\\n",
      "  216*x *\\x  - 2*x + 1/*\\6*x  - 6*x  + 1/\n",
      "\n",
      "c1 = (sym) [-6  6]  (1x2 matrix)\n",
      "t1 = (sym 1x2 matrix)\n",
      "\n",
      "  [ 3   2]\n",
      "  [x   x ]\n",
      "\n",
      "c2 = (sym) [1296  -3888  3888  -1080  -432  216]  (1x6 matrix)\n",
      "t2 = (sym 1x6 matrix)\n",
      "\n",
      "  [ 9   8   7   6   5   4]\n",
      "  [x   x   x   x   x   x ]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 5. Trouver analytiquement s’il existe un cycle d’ordre 2.\n",
    "syms x;\n",
    "f_sym = 6*x^2*(1-x)\n",
    "f2_sym = 216*x.^4.*(1-2*x+x.^2).*(1 - 6*x.^2 + 6*x.^3)\n",
    "[c1,t1] = coeffs(f_sym)\n",
    "[c2,t2] = coeffs(f2_sym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-12-01T20:52:17.095595Z",
     "start_time": "2020-12-01T20:52:16.411826Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "   0.78868\n",
      "   0.21132\n",
      "   0.00000\n",
      "\n",
      "ans =\n",
      "\n",
      "   1.05393 + 0.00000i\n",
      "   0.86603 + 0.00000i\n",
      "   0.78868 + 0.00000i\n",
      "   0.60287 + 0.00000i\n",
      "  -0.35946 + 0.00000i\n",
      "   0.21132 + 0.00000i\n",
      "  -0.08169 + 0.12935i\n",
      "  -0.08169 - 0.12935i\n",
      "   0.00000 + 0.00000i\n",
      "\n",
      "ans =\n",
      "\n",
      "  -0.08169 - 0.12935i\n",
      "  -0.08169 + 0.12935i\n",
      "  -0.35946 + 0.00000i\n",
      "   0.60287 + 0.00000i\n",
      "   0.86603 + 0.00000i\n",
      "   1.05393 + 0.00000i\n",
      "\n"
     ]
    }
   ],
   "source": [
    "c_f = [double(c1) -1 0]; # Coeffiecients de x=f(x) => f(x) - x = 0\n",
    "c_f_2 = [double(c2) 0 0 -1 0]; # Coeffiecients de x=f^2(x) => f^2(x) - x = 0\n",
    "roots(c_f) # Racines de x=f(x)\n",
    "roots(c_f_2) # Racines de x=f^2(x)\n",
    "setdiff(single(roots(c_f_2)),single(roots(c_f))) # Racines de f^2 mais pas de f"
   ]
  }
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