TP5_Simulation_Ordre_1.ipynb

{
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   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "# INFO-F-305 - TP 5\n",
    "# Simulation à l'aide d'Octave\n",
    "# Système d'ordre 1\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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Projet\n",
    "\n",
    "- Analyse d'un ensemble des systèmes avec Octave\n",
    "- Groupes de 1-2 personnes.\n",
    "- 2 x Séances TP en salle machine + Travail personnel\n",
    "- Durée: 2 semaines\n",
    "- A remettre: Jupyter Notebook/Rapport LaTeX sur UV\n",
    "- Note: 0-2/20"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Comment installer Octave / Jupyter?\n",
    "\n",
    "- **Octave**: Plus d'informations ici: https://www.gnu.org/software/octave/\n",
    "- **Jupyter Notebook**: https://jupyter.readthedocs.io/en/latest/install.html\n",
    "- **Octave kernel for Jupyter**: https://github.com/Calysto/octave_kernel\n",
    "- **Symbolic package (Octave forge)**: https://octave.sourceforge.io/symbolic/index.html\n",
    "\n",
    "\n",
    "# Comment utiliser les notebooks?\n",
    "\n",
    "1. Installer Octave, Jupyter Notebook et le kernel Octave pour Jupyter.\n",
    "2. Se déplacer dans le dossier où les notebooks sont stockés.\n",
    "3. Démarrer ```jupyter notebook``` \n",
    "4. Selectionner le notebook dans l'interface web. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Cas d'étude 1 - Intégrateur simple\n",
    "\n",
    "$\\begin{cases}\n",
    "\\dot{x}(t)=cu(t),\\\\\n",
    "y(t)=x(t)\\end{cases}$\n",
    "\n",
    "- Conditions initiales: $x(0)=1$ \n",
    "- Valeurs paramètres: $c=1$ \n",
    "- Temps de simulation: $[0,10]$ \n",
    "- Fonction(s) d'entrée: $u(t)=\\sin(t)$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  0.50000\n",
      "ode_sys =\n",
      "\n",
      "@(t, x) [c * sin(t)]\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 1\n",
    "c = 1 # Paramètre\n",
    "ode_sys = @(t,x) [c*sin(t)] # Définition système\n",
    "[t,x] = ode23 (ode_sys, [0, 10], [1]); # Résolution système\n",
    "plot(t,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  0.50000\n",
      "ode_sys = f(t, x) = [0.500000*sin(t)]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 2\n",
    "c = 1 # Paramètre\n",
    "ode_sys = inline(sprintf('[%f*sin(t)]',c),'t','x') # Définition système\n",
    "[ts,ys] = ode45(ode_sys,[0,10],[1]); # Résolution système\n",
    "plot(ts,ys)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Bonus - Function by S.Tysebaert et H.Lloreda\n",
    "function integrateur_simple(condition_initiale, param_c, temps_simulation, fonction_entree)\n",
    "  u = fonction_entree;\n",
    "  c = param_c;\n",
    "  ode_sys = @(t, x) [c*u(t)];\n",
    "  [t, x] = ode23 (ode_sys, temps_simulation, condition_initiale);\n",
    "  plot(t, x)\n",
    "endfunction\n",
    "\n",
    "# Simulation 1\n",
    "#integrateur_simple([1], 1, [0, 10], @(t) sin(t))\n",
    "# Simulation 2\n",
    "#integrateur_simple([0], 0.1, [0, 10], @(t) t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Cas d'étude 2: croissance et décroissance exponentielle\n",
    "\n",
    "$\\begin{cases}\n",
    "\\dot{x}(t)=cx(t)\\\\\n",
    "y(t)=x(t)\\end{cases}$\n",
    "\n",
    "- Conditions initiales: $x(0)=1$ \n",
    "- Valeurs paramètres: $c=0.1$ \n",
    "- Intervalle de temps de simulation $[0,10]$. \n",
    "- Fonction(s) d'entrée: néant \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  0.10000\n",
      "ode_sys =\n",
      "\n",
      "@(t, x) [c * x(1)]\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 1\n",
    "c = 0.1 # Paramètre\n",
    "ode_sys = @(t,x) [c*x(1)] # Définition système\n",
    "[t,x] = ode23 (ode_sys, [0, 10], [1]); # Résolution système\n",
    "plot(t,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  0.10000\n",
      "ode_sys = f(t, x) = [0.100000*x(1)]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 2\n",
    "c = 0.1 # Paramètre\n",
    "ode_sys = inline(sprintf('[%f*x(1)]',c),'t','x') # Définition système\n",
    "[ts,ys] = ode45(ode_sys,[0,10],[1]); # Résolution système\n",
    "plot(ts,ys)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Bonus - Function by S.Tysebaert et H.Lloreda\n",
    "function croiss_exp(condition_initiale, param_c, temps_simulation, fonction_ode)\n",
    "  c = param_c;\n",
    "  if fonction_ode == \"ode23\"\n",
    "    ode_sys = @(t, x) [c*x(1)]; # x(1) == x(t)\n",
    "    [t, x] = ode23 (ode_sys, temps_simulation, condition_initiale);\n",
    "  elseif fonction_ode == \"ode45\"\n",
    "    ode_sys = inline(sprintf('[%f*x(1)]', c), 't', 'x')\n",
    "    [t, x] = ode45 (ode_sys, temps_simulation, condition_initiale);\n",
    "  end\n",
    "  plot(t, x)\n",
    "endfunction\n",
    "\n",
    "# Simulation 1\n",
    "#croiss_exp([1], 0.1, [0, 10], \"ode23\") # ImplÃ©mentation 1\n",
    "#croiss_exp([1], 0.1, [0, 10], \"ode45\") # ImplÃ©mentation 2\n",
    "# Simulation 2\n",
    "#croiss_exp([2], -0.1, [0, 10])\n",
    "# Simulation 3\n",
    "#croiss_exp([1], -1, [0, 10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Cas d'étude 3: système avec retard d'ordre 1\n",
    "\n",
    "$\\begin{cases}\n",
    "\\dot{x}(t)=u(t)-cx(t)\\\\\n",
    "y(t)=x(t)\\end{cases}$\n",
    " \n",
    "- Conditions initiales: $x(0)=1$ \n",
    "- Valeurs paramètres: $c=2$ \n",
    "- Intervalle de temps de simulation $[0,10]$. \n",
    "- Fonction(s) d'entrée: $u(t)=1$ pour $t>0$. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  2\n",
      "ode_sys =\n",
      "\n",
      "@(t, x) [1 - c * x(1)]\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 1\n",
    "c = 2 # Paramètre\n",
    "ode_sys = @(t,x) [1-c*x(1)] # Définition système\n",
    "[t,x] = ode23 (ode_sys, [0, 10], [1]); # Résolution système\n",
    "plot(t,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  2\n",
      "ode_sys = f(t, x) = [1-2.000000*x(1)]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 2\n",
    "c = 2 # Paramètre\n",
    "ode_sys = inline(sprintf('[1-%f*x(1)]',c),'t','x') # Définition système\n",
    "[ts,ys] = ode45(ode_sys,[0,10],[1]); # Résolution système\n",
    "plot(ts,ys)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Bonus - Function by S.Tysebaert et H.Lloreda\n",
    "function retard(condition_initiale, param_c, temps_simulation, fonction_entree)\n",
    "  u = fonction_entree;\n",
    "  c = param_c;\n",
    "  if isa(u, \"numeric\")\n",
    "    ode_sys = @(t, x) [1 - c*x(1)];\n",
    "  else\n",
    "    ode_sys = @(t, x) [u(t) - c*x(1)];\n",
    "  end\n",
    "  [t, x] = ode23(ode_sys, temps_simulation, condition_initiale);\n",
    "  plot(t, x)\n",
    "endfunction\n",
    "\n",
    "# Simulation 1\n",
    "#retard([1], 2, [0, 10], 1) # ImplÃ©mentation 1\n",
    "# Simulation 2\n",
    "#retard([1], 0.02, [0, 50], @(t) 0.1*t, \"ode23\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Cas d'étude 4: croissance logistique\n",
    "\n",
    "$\\begin{cases}\n",
    "\\dot{x}(t)=cx(t)(1-\\frac{x(t)}{k})\\\\\n",
    "y(t)=x(t)\\end{cases}$\n",
    "\n",
    "- Conditions initiales: $x(0)=0.3$ \n",
    "- Valeurs paramètres: $k=1.5$, $c=0.2$ \n",
    "- Intervalle de temps de simulation $[0,50]$. \n",
    "- Fonction(s) d'entrée: néant \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  0.20000\n",
      "k =  1.5000\n",
      "ode_sys =\n",
      "\n",
      "@(t, x) [c * x(1) * (1 - x(1) / k)]\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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1SEmS5KtoNBrlOynX6/Vms9lyuZzNZnt7exs5R4B16Fz+oUYxK3HJbjgc5g+eXq/r9XrL5fL8/DzLsvfv36/57ADWRIriV+JODVmW5Q/m83lxcDKZHB8f397efvr06e7u7rnndv6j8okCVGMYbUHnicpfZNWFNBgM8g4tFoter/f0l7rdbv6/X758ee7pj4+PlU8RoDIp2o6nL/L9fr/aF1k1SIeHh/lnGZIkOTs7CyGkaXpycvKPf/zj5ORkPp/PZrOjo6NqJwGwCWpUL51S2yVN016vt7+/v+LxXL/ff3h4qH6OACX5MaMdqvyaX+4HY5/7MSM/fgTEwzCqKXdqAJrDMKo1QQIawjCqO/+BPqAJ1KgBLCSg3lymawxBAmrMMGoSQQJqyTBqHkEC6scwaiQfagBqRo2aykICasNlumYTJKAeDKPGEyQgdoZRSwgSEDXDqD18qAGIlxq1ioUExMhluhYSJCA6hlE7CRIQEcOozQQJiIVh1HI+1ABEQY2wkIAdc5mOnCABu2QYURAkYDcMI74iSMAOGEZ8y4cagG1TI77LQgK2x2U6XiBIwJYYRrxMkICNM4xYhSABm2UYsSIfagA2SI1YnYUEbITLdJQlSMD6GUZUIEjAOhlGVCZIwNoYRryGDzUA66FGvJKFBLyWy3SshSABr2IYsS6CBFRkGLFeggRUYRixdoIElGMYsSGCBJRgGLE5PvYNrEqN2KgSC2k8HidJEkK4uroaDAb5wY8fP37+/Dl/PJ/Pb29vi18CGsNlOrZg1SClaZpl2f39fZqmk8nk5uYmP/7u3bt3796FEObz+XQ6VSNoHsOI7Vj1kl2SJHlsRqNRvpO+cn19fXZ2ts5TA3atc/mHGrE1JS7ZDYfD/MG3M+jjx4/D4bDb7T733E6nUzx+fHwsc4bAbkgRq3v6Iv/mzZtqX6TEhxqyLMsfzOfzr37pw4cPL8+jxyfKniKwfWpEKWt5kV91IQ0Gg7xDi8Wi1+s9/aX5fP7yPAJqxOcX2JVVg3R4eDiZTEIISZLkYyhN05OTk4eHB59lgMYwjNihTql5laZpr9fb398v9Wf0+/2Hh4eSJwZslWHEulR+zS93p4bRaFThzwAiZxgRA3dqgLZTIyLhXnbQXi7TERVBgpYyjIiNIEHrGEbESZCgXQwjouVDDdAiakTMLCRoBZfpiJ8gQfMZRtSCIEGTGUbUiCBBYxlG1IsPNUAzqRG1YyFB07hMR00JEjSKYUR9CRI0hGFE3QkSNIFhRAMIEtSbYURjCBLUmGFEk/jYN9SVGtEwFhLUj8t0NJIgQc0YRjSVIEFtGEY0myBBPRhGNJ4gQewMI1pCkCBqhhHtIUgQKcOIthEkiJFhRAv5wViIjhrRThYSRMRlOtpMkCAWhhEtJ0iwe4YRBEGCnTOMICdIsDOGETwlSLAbhhF8xce+YQfUCL5lIcFWuUwHzxEk2B7DCF4gSLANhhH8kCDBxhlGsApBgg0yjGB1JYI0Ho+TJAkhXF1dDQaDb48fHx+/e/du7acINWUYQSmrfuw7TdMsy+7v7y8vLyeTSXF8Pp/PZrP7+/vb29vZbLaZk4Sa6Vz+oUZQ1qoLKUmSfBWNRqPz8/Pi+HQ6PTg4SNM0hHB1dbWJU4R6kSKopsQlu+FwmD94er0uhHB3d7dcLkMInz9/1iRaTo2gshJByrJsNBqFEObz+dPjR0dHFxcXIYS3b98+99xOp5M/eHx8rHKaED2fX6Dlitf5N2/eVPsKq76HNBgMsiwLISwWi16vVxx/+vjLly/PPf3xP6qdJUQuH0ZqRJu9/nV+1YV0eHiYf5YhSZKzs7MQQpqmJycnDw8P+TCazWbHx8eVzwNqyjCCdemUqlmapr1eb39/f8XjuX6///DwUP0cIVbeMYJvVX7NL/eDsfl7SKsfh6YyjGDt3KkBSjOMYBMECUowjGBzBAlWZRjBRgkS/JhhBFsgSPADhhFshyDBswwj2CZBgu8zjGDLBAm+ZhjBTggS/BfDCHZFkODfDCPYLUGCEAwjiIAg0XaGEURCkGg1wwjiIUi0lGEEsREk2sgwgggJEu1iGEG0BIkWMYwgZoJEKxhGED9BovkMI6gFQaLJDCOoEUGisQwjqBdBooEMI6gjQaJpDCOoKUGiOQwjqDVBoiEMI6g7QaL2DCNoBkGi3gwjaAxBoq4MI2gYQaKWDCNoHkGiZgwjaCpBok4MI2gwQaIeDCNoPEGiBgwjaANBImqGEbSHIBEvwwhaRZCIkWEELSRIRMcwgnYSJCJiGEGbCRJRkCJAkNg91+iAUCpI4/E4SZIQwtXV1WAwKI7//PPP+T8Oh8OLi4u1nyINZhgBhVWDlKZplmX39/dpmk4mk5ubm/z4YrEYDAa3t7cbO0MayzACnlo1SEmS5DNoNBqdn58Xx7Ms29vb++233/b29s7Ozrrd7kZOk2YxjIBv/bT6bx0Oh/mDp9frvnz50uv1jo6Out3u6enpc8/tPFH5XGmAzuUf+TBSI2iStbzIl3gPKcuy0WgUQpjP58XBw8PDw8PDEMJoNJpOp8899/HxsfIp0hiu0UFTPX2R7/f71b7IqkEaDAZ5hxaLRa/XK45PJpPhcJiHCp7jGh3wQ6sG6fDwcDKZhBCSJDk7OwshpGl6cnLy6dOn8/Pzo6Oj2Wx2fHy8wTOltgwjYBWdUhfT0jTt9Xr7+/srHs/1+/2Hh4fq50htGUbQQpVf88v9YOxzl+ZcsuMrUgSU5U4NrJ9rdEAFgsQ6GUZAZYLEekgR8EqCxBq4Rge8niDxKoYRsC6CRHWGEbBGgkQVhhGwdoJEOVIEbIggUYJrdMDmCBIrMYyATRMkfswwArZAkHiJYQRsjSDxfVIEbJkg8R2u0QHbJ0j8F8MI2BVB4k+GEbBDgkQIhhEQAUFqOykCIiFIreYaHRAPQWopwwiIjSC1jhQBcRKkdnGNDoiWILWFYQRETpCaT4qAWhCkJpMioEYEqbG8XQTUiyA1kGEE1JEgNYoUAfUlSA0hRUDdCVITeLsIaABBqjfDCGgMQaorKQIaRpBqyTU6oHkEqWYMI6CpBKk2pAhoNkGqASkC2kCQYuftIqAlBClehhHQKoIUIykCWqhEkMbjcZIkIYSrq6vBYPD0l5bL5S+//PL3v/99zWfXPlIEtNaqQUrTNMuy+/v7NE0nk8nNzc3TX72+vl4ulxs4vXbxdhHQZqsGKUmSfBWNRqPz8/OnvzSdTvf29r7aTJRiGAGUuGQ3HA7zB0/bs1gsptPpzc3N6enpmk+tHaQIIPfT6r81y7L8wXw+Lw5eX193u93xeJxl2Xg8XiwW331u5z9ec67Nk1+jUyOgAV7/Or/qQhoMBnmHFotFr9crjp+dneXvHiVJcnBw0O12v/v0x8fHyqfYSIYR0DDF63y/36/2FVYN0uHh4WQyCSEkSXJ2dhZCSNP05OTk4eEh/w3dbnc0GlU7iVaRIoDv6pTaLmma9nq9/f39Un9Gv98vutVmUgS0QeXX/HI/GGsDVSNFAD/kTg2bJUUAKxKkDfKDrgCrE6SNMIwAyhKkNZMigGoEaW2kCOA1BGkNpAjg9QTpVaQIYF0EqSIpAlgvQarC57kB1k6QyjGMADZEkFYlRQAbJUg/JkUAWyBIL5EigK0RpO+TIoAtE6SvSRHATgjSn6QIYIcEKQQpAohA24MkRQCRaG+QpAggKm0MkhQBRKh1QXIbOoA4tShIhhFAzFoRJCkCiF/DgyRFAHXR2CBJEUC9NDBIUgRQR40KkhQB1FdDgiRFAHVX+yBJEUAz1DhIUgTQJPULUt6hIEUAzVKnIJlEAA1WjyBJEUDjxR4kKQJoiXiDJEUArRJjkKQIoIXiCpIUAbRWLEGSIoCW232QpAiAsNsgSREAhRJBGo/HSZKEEK6urgaDQX5wuVyen5+HELIs+9vf/jYajX74ddxqAYBvrRqkNE2zLLu/v0/TdDKZ3Nzc5Mfv7u6Gw+HFxcVisfj1119fDpJJBMBzVg1SkiT5KhqNRvkkyh0dHXW73RDCbDbb29t77ulSBMDLSlyyGw6H+YPiel0IYX9/P4QwHo8/f/78/v37Z588/t8QQmccQgiPj49VzhSAWHU6neLxmzdvqn2Rn1b/rVmW5Q/m8/lXv3RxcfHp06cPHz4899zHJyqcJQAxW8uL/KpBGgwGeZAWi0Wv1yuO//bbb9PpNITQ7XaXy2Xl8wCg5Va9ZHd4eDiZTEIISZKcnZ2FENI0PTk5+fTp0/n5+Xw+n81m+XEAqKBTal6ladrr9fL3jQrL5XI+n397vNDv9x8eHl51mgDUROXX/HI/GPvdT3V3u91VfvwIAF5Q4kMNALA5ggRAFAQJgCgIEgBRECQAoiBIAERBkACIgiABEAVBAiAKggRAFAQJgCgIEgBRECQAoiBIAERBkACIgiABEAVBAiAKggRAFAQJgCgIEgBRECQAoiBIAERBkACIgiABEAVBAiAKggRAFAQJgCgIEgBRECQAoiBIAERBkACIgiABEAVBAiAKggRAFAQJgCgIEgBRECQAoiBIAERBkKLT6XR2fQo14ztWlu9YWb5j2/GX1X/reDxOkiSEcHV1NRgM8oPL5fL8/Hy5XIYQjo+P3717t4mzBKDxVl1IaZpmWXZ/f395eTmZTIrj0+l0OBze39/f3t5++PBhMycJQPOtGqQkSfJVNBqN8p2U6/V6BwcHIYRut7uJ8wOgJUpcshsOh/mD4npdCGE0GoUQFovFr7/++v79+/WeHADt0Xl8fFzl943H416vl79F9PPPP//zn/98+kuz2ez9+/d5nL7V7/fXcq4A1MLDw0OFZ626kAaDwXw+DyEsFoter1cc//jxY5Zlt7e3az8zAFpl1YUUQnj79u3BwUGSJGdnZ4eHh2manpycHB8fz+fz4g2kl8sEAM8pEaQQQpqmvV5vf39/cycEQDuVCxIAbIg7NQAQhc0GaTwev3379u3bt/kHInjZcrk8PT3Nv2MfP37c9enUxnK5/Otf/7rrs6iN4m+lf8dWcXp6mv+tTNN01+cSu+l0Oh6Pi3+s8Pq/wSA9d3MHnuO2F9VcX1/nN6/ih+bz+Ww2y/8dm81muz6d2OV/JW9vb3///Xd/JV92enp6fn5e/GO11/8NBum5mzvwHLe9qGA6ne7t7T39YW1eMJ1ODw4O0jSdz+dXV1e7Pp3Y9Xq92Wy2XC5ns9ne3t6uTydqt7e3T++NUO31v8SdGir47s0deI7bXpS1WCym0+nNzc3p6emuz6U27u7u8kH5+fNnTXpZr9fL7x+dZZm/kmVVeP3f7HtIWZblD7yHtKLxeJzXyH3TV3F9fd3tdsfjcZZl4/F4sVjs+oxq4Ojo6OLi4uLiwt/KH5pMJsfHx7e3t58+fbq7u9v16dRMhdf/DS6k527uwHNWue0FT52dneX/Zz9JkoODA5c6f6jX6xUvE1++fNntydRC/i9Vt9v17Sql2uv/BoN0eHiYv5eV39xhc39QY8xmsyzLiqtPyvRDxaWAbrf73K0Ueerdu3dv374NIcxms+Pj412fTuyOj49PTk7yT4IcHR3t+nTqpNrr/8Z/MNbNHSA2/laW4ttVWdlvnTs1ABAFd2oAIAqCBEAUBAmAKAgSAFEQJACi8P8ctOSvPCzu1wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 1\n",
    "c = 0.2 # Paramètre\n",
    "k = 1.5 # Paramètre\n",
    "ode_sys = @(t,x) [c*x(1)*(1-x(1)/k)] # Définition système\n",
    "[t,x] = ode23 (ode_sys, [0, 10], [0.3]); # Résolution système\n",
    "plot(t,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  0.20000\n",
      "k =  1.5000\n",
      "ode_sys = f(t, x) = [0.200000*x(1)*(1-x(1)/1.500000)]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Implémentation 2\n",
    "c = 0.2 # Paramètre\n",
    "k = 1.5 # Paramètre\n",
    "ode_sys = inline(sprintf('[%f*x(1)*(1-x(1)/%f)]',c,k),'t','x') # Définition système\n",
    "[ts,ys] = ode45(ode_sys,[0,10],[0.3]); # Résolution système\n",
    "plot(ts,ys)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Bonus - Function by S.Tysebaert et H.Lloreda\n",
    "function croissance_logistique(condition_initiale, param_k, param_c, temps_simulation)\n",
    "  title_plot = \"Croissance logistique\"\n",
    "  c = param_c;\n",
    "  k = param_k;\n",
    "  ode_sys = @(t, x) [c*x(1)*(1-(x(1)/k))];\n",
    "  [t, x] = ode23(ode_sys, temps_simulation, condition_initiale);\n",
    "  plot(t, x)\n",
    "  endfunction\n",
    "\n",
    "# Simulation 1\n",
    "#croissance_logistique([0.3], 1.5, 0.2, [0, 50])\n",
    "# Simulation 2\n",
    "#croissance_logistique([3], 1.5, 0.2, [0, 50])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Système du second ordre - Bases\n",
    "\n",
    "$\\dot{\\mathbf{X}}(t) = \\mathbf{A}\\mathbf{X}(t)$\n",
    "\n",
    "$\\begin{bmatrix}\\dot{x_{1}}\\\\\n",
    "\\dot{x_{2}}\\end{bmatrix}=\\begin{bmatrix}a_{11} & a_{12}\\\\\n",
    "a_{21} & a_{22}\\end{bmatrix}\\begin{bmatrix}x_{1}\\\\\n",
    "x_{2}\\end{bmatrix}$\n",
    "\n",
    "1. Dessin d'une trajectoire\n",
    "2. Dessin des droites invariantes\n",
    "3. Dessin des isoclines\n",
    "4. Dessin du portrait de phase\n",
    "5. Dessin du diagramme tr(A) - det(A)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "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": [
    "# Implémentation 1\n",
    "A = [-5 -3; 1 -1];\n",
    "ode_sys = @(t,x) [A(1,1)*x(1)+A(1,2)*x(2);A(2,1)*x(1)+A(2,2)*x(2)]; # Définition système\n",
    "[t,x] = ode23 (ode_sys, [0, 10], [2, 0]); # Résolution système\n",
    "plot(t,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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XhmexWPz222/Hx8fdbvfm5qaODQagnSpN2WVZVo570jQtx0mlJEmOj49DCJ1OJ4RQFMX+/v719fV4PC6HTQDwTNXPkHq9XvlgOV8XQkjTNE3T+Xw+GAwuLi4eHh6SJDk5Oel0OoPBoOIaAWilqp8hFUWRpmkIYTabPV0+Ho+n0+nFxUX5036/H0JI0/Tu7u61t9rb2ysfPD4+VtwqPsje1b9Xf/Lj118/bkuAqCxP4CGET58+rfcmlYLU7XbLDs3n8yRJlsv//PPPoiiWHxpNJpNer1eW6Q06FJsf8/OuxrxRL62Clnl6Aj88PFzvTSoFqd/vTyaTEEKWZcPhMM/z8/Pz+/v76XRaFMVydu7q6mo0Gp2cnEyn07Ozsypr5ONUzM+P3nj5a60SKthle9XHJXmeJ0lycHBQ5WmHh4f39/cVt4Q1LNsQQwyi2hhgPWufz2sIUi0EafPKs3+cp35lguZa+3xe9aIGGmrv6t8xn+uX26ZMsDsEaefEPDD6kTLB7hCkHdKsFD2jTNB6grQTGp2iZ5QJ2kqQWq5NKXpGmaBlBKnNIr9yoS7KBO0gSK21IzV6Spmg0QSphVo8TbciZYImEqS22cGB0RuelcmegZgJUquo0WvK3SJLEDNBag81+ilZgpgJUkuo0epkCeIkSG2gRmuQJYiNIDWeGlUhSxAPQWo2NaqFLEEMBAn+lyzBdglSgxkefQRZgm0RpKZSow8lS7B5gtRIarQZsgSbJEjwE7IEmyFIzWN4tBWyBB9NkBpGjbZLluDjCBK8myzBRxCkJjE8ioosQb0EqTHUKE6yBHURJKiBLEF1/7PtDWAlhkeN8Pj118evvy7/bjrwLoIENSubJEvwXqbsGsDwqHHM4MEaBAk+iizBuwhS7AyPmk6WYEU+Q4JNcL0D/JQgRc3wqGVc7wBvECTYKEMleI0gwRYYKsGPBCle5uvazVAJnhEk2CZDJVgSpEgZHu0OQyUoCRJEwVAJBAliYajEjqv5Tg3j8TjLshDCly9fut1uuXCxWIxGo8ViEUI4Ozv7/PlzvSttH/N1u2zZJMcAu6bOEVKe50VR3N7eXl1dTSaT5fK7u7ter3d7e3tzc/PHH3/UuEZoJUMldlOdQcqyrBwVpWlajpNKSZIcHx+HEDqdTo2rg3bzqRK7puYpu16vVz5YzteFENI0DSHM5/Pff//94uLitdfu7e0tHz8+Pta7YQ1ivo4lN2alKZ6ewD99+rTem9QcpKIoyvzMZrOny8fj8XQ6vbi4KH/6ol2OELxhmSVNIlpPT+CHh4frvUmdQep2u2WH5vN5kiTL5X/++WdRFDc3NzWuC3aNix1ovTqD1O/3y2sZsiwbDochhDzPz8/Pz87OiqIYDAbl05QJ1mOoRLvt1T5Rlud5kiQHBwfvetXh4eH9/X29W9JEzjWswlCJmK19Pq//L8a+8SkRUAtDJVrJnRqgqVwXTssIEjSYr9DSJoIUETMwrMdQiXYQJGgDQyVaQJCgPQyVaDRBglYxVKK5BAlayFCJJqr/e0isxxUN1MuNWWkcIyRoMzN4NIggQftpEo0gSLATfKpE/AQJdoXpOyInSFFwRQMbo0lES5Bg55i+I06CBLvI9B0REiTYXZpEVAQJdprpO+IhSLDrTN8RCUHaPpfYEQNNYusECfhfpu/YLkEC/o/pO7ZIkIDnNImtECTgBabv2DxBAl5m+o4NEyTgLZrExggS8BOm79gMQQJ+zvQdGyBIW+ZbsTSIJvGhBAl4B9N3fBxBAt7H9B0fRJCAdWgStRMkYE2m76iXIAHrM31HjQQJqEqTqIUgATUwfUd1ggTUw/QdFQkSUCdNYm2CBNTM9B3rESSgfqbvWEPVII3H49PT09PT09ls9uxHd3d34/G4fHx0dDQYDAaDwXIJ0HqaxLv8UuXFeZ4XRXF7e5vn+WQy+fbt2/JHg8Egy7KLi4sQwnw+73a7Nzc3VTcWaJplk9xEmJ+qNELKsqzb7YYQ0jTNsuzpj25ubsoahRCKotjf37++vh6Px4vFosoagcYxfceKqk7Z9Xq98kFZphc9PDwkSXJyctLpdAaDQcU1tom/PcHu0CR+qmqQiqIoH/z4GdJSv9+/vLxM03Q4HL7xVnv/VXGTgDi5+q7dqp/DKwWp2+2WQZrP50mSvPa0yWSS5/lP3+3xv6psEhAz03ctVv0cXumihn6/P5lMQghZlg2HwzzPz8/P7+/vnz3t+Ph4NBqdnJxMp9Ozs7MqawRaoGyS+Wqe2as+IsnzPEmSg4ODKk87PDz8sWSt53eSXeb4b6u1z+c1fDE2TdOf1mj1pwE7wtwdz7hTA7A1LnPgKUECtsllDiwJErB9mkQQJCASmoQgAbHQpB0nSEBEXOawywQJiIvLHHaWIAEx0qQdJEhApDRp1wgSEC8fKe0UQQKi5iOl3SFIQANo0i4QJKAZNKn1BAloDE1qN0ECmsRlDi0mSEDDuMyhrQRpm/xSwdr8+rSPIAFNpUktI0hAg2lSmwgS0Gwuc2gNQQIaz2UO7SBIQEtoUtMJEtAemtRoggS0iiY1lyABbeMyh4YSJKCFXObQRIIEtJYmNYsgAW2mSQ0iSEDLaVJTCBLQfi5zaARBAnaCyxziJ0jADtGkmAkSsFs0KVqCBOwcHynFSZCAXeQjpQgJErC7NCkqgrRlfh9gu/wOxkOQgF2nSZEQJABNioIgAYSgSRGoOUjj8fj09PT09HQ2mz370d3d3Xg8rnd1ADXSpO2qM0h5nhdFcXt7e3V1NZlMnv5oMBiMRqMa1wXwEXxFaYvqDFKWZd1uN4SQpmmWZU9/dHNzc3FxUeO6AD6IryhtS81Tdr1er3xQluld9p6od6sA3kuT3qWWE3jNQSqKonzw42dIP/X4RL1bBbAGTVpdLSfwOoPU7XbLIM3n8yRJanxngK3QpE2qM0j9fj/LsvF4PBqNhsNhCCHP88PDwxpXAbBhmrQxe7XPj+V5niTJwcHBu151eHh4f39f75Y0xd7Vvx+//rrtrQDe4vd0dWufz+v/Ymyapu+t0Y7z/y+In9/TDXCnBoCV+IrSRxMkgFX5itKHEiSA99GkDyJIAO+mSR9BkADWoUm1EySANWlSvQQJYH2aVCNBAqhEk+oiSABV+YpSLQQpCv6HBU3nK0rVCRJAbTSpCkECqJMmrU2QAGqmSesRJID6adIaBAngQ2jSewkSwEfRpHcRJIAP5CtKqxMkgI/lK0orEqRYOF6h3fyO/5QgAWyIJr1NkAA2R5PeIEgAG6VJrxEkgE3TpBcJEsAWuBz8R4IEsB0uB39GkCLi0IQd5Bd/SZAAtkyTSoIEsH2aFAQJIBKaJEgAsdjxJgkSQER2uUmCBBCXnW2SIMVlZw9E4KndPBUIEkCMdrBJggQQqV1rkiABxGunmiRIAFHbnSYJEkDsduTW4IIE0AC7cGvwqkEaj8enp6enp6ez2eyN5UdHR4PBYDAYjMfjimtsvdYfc8Da2n1++KXKi/M8L4ri9vY2z/PJZPLt27cXl19dXXW73Zubmzo2GGCnlU16/PrrtjekfpVGSFmWdbvdEEKaplmWvba8KIr9/f3r6+vxeLxYLCpuMcCOa+s4qeqUXa/XKx+UBXpx+cPDQ5IkJycnnU5nMBhUXCMArWxS1SAVRVE+ePYZ0tPl/X7/8vIyTdPhcPjGW+39V8VNAtgFsTWp+jm8UpC63W4Znvl8niTJa8snk0me5z99t8f/qrJJALsjqiZVP4dXuqih3+9PJpMQQpZlw+Ewz/Pz8/P7+/tny5MkGY1GJycn0+n07OysyhoBeKpN1zjsVR+R5HmeJMnBwcHby197Wunw8PD+/r7ilrRJa44wYAOiOmOsfT6vNEIqpWm6yvLXngZARe0YJ7lTA0AbRPV50noECaAlmt4kQQJoj0bfhlWQAFqlubdhFaRINfR4AiLRxHOIIAG0U+OaJEgArdWsJgkSQJs1qEmCBNByTWmSIAG0XyOaJEjxasQBBDRF/KcUQQLYFZE3SZAAdkjMTRIkgN0SbZMECWDnxNkkQYpanAcN0AIRnl4ECWBHxdYkQQLYXVE1SZAAdlo8TRIkgF0XyZ/1EyQAovizfoIUu60fIsDu2O4JR5AA+D9bbJIgAfD/2VaTBAmA57bSJEEC4AWbb5IgNYDrGoCt2PDJR5AAeNUmmyRIALxlY00SJAB+YjNNEiQAfm4DTRIkAFby0U0SpGZwoR0Qgw89FwkSAO/wcU0SJADe54OaJEgAvNtHNEmQAFhH7U0SpMZwXQMQm3rPS4IEwPpqbJIgAVBJXU3aUJDG4/Hp6enp6elsNtvMGgHYmFqatIkg5XleFMXt7e3V1dVkMtnAGtvKx0hAtKqfoDYRpCzLut1uCCFN0yzLNrBGADavYpN+qXFT3tDr9coHZZletLe3t3z8+Pj44dsEQE2ensDDp0/rvcmGPkMqiqJ88MZnSI9PbGarmujx66/b3gSA52o5gW8iSN1utwzSfD5PkmQDawSgcTYxZdfv98trGbIsGw6HG1gjAI2zoc+Qbm9v8zw/Ozs7ODjYzBoBaJYNBSmEkKbpxtYFQOO4UwMAURAkAKIgSABEQZAAiIIgARAFQQIgCoIEQBQECYAoCBIAURAkAKIgSABEQZAAiIIgARAFQQIgCoIEQBQECYAoCBIAURAkAKIgSABEQZAAiIIgARAFQQIgCoIEQBQECYAoCBIAURAkAKIgSABEQZAAiIIgARAFQQIgCoIEQBQECYAoCBIAURAkAKIgSABEQZAAiIIgARAFQQIgCoLUMHt7e9vehGawo1ZnX63Ijvpov1R8/Xg8zrIshPDly5dut/va8qOjo/KnvV7v8vKy4koBaJ9KQcrzvCiK29vbPM8nk8m3b99eXH51ddXtdm9uburYYADaqdKUXZZl5bgnTdNyPPTi8qIo9vf3r6+vx+PxYrGouMUAtFLVz5B6vV754Ol83bPlDw8PSZKcnJx0Op3BYFBxjQC00run7PI8z7IsSZLPnz+HEIqiSNM0hDCbzZ4+7enyfr/f7/dDCGma3t3dvfbOh4eH792YHfTp0yc7ahV21OrsqxXZUR/t3UFK07QsTQih2+2WHZrP50mSLJ/zbPlkMun1estXvej+/v69WwJAm+w9Pj5Wef3p6enx8XGWZcPhcH9///z8vEzL0+VJkoxGo5OTk+l0enJyUg6tAOCpqkEKIeR5niTJwcHB28tfexoAhFqCBADVuVMDAFGoeqeGKl67y4PbOrzmtT1GcDit6+7ubjab2TOveXH/OKhes1gsRqNR+X3Ts7Ozd18x8LglWZaNRqOnD0pFUZyfn29rq2L22h7j0eG0rvPz80+fPn39+nXbGxKpF/ePg+oN379/L3fXw8PD3/72t/e+fGtTdq/d5cFtHV7z2h4jOJzWdXNzc3Fxse2tiNeL+8dB9YYkSY6Pj0MInU5njZdv8zOkF+/y4LYOb3jtvhgEhxOb4qB6Q/lF1fl8PhgM1vi/zkY/Q1rlLg8r3tZhN712XwyCw4lNcVC9bTweT6fTi4uLt2+G8KKNBmmVuzyscluH3fTaHiM4nNggB9Ub/vzzz6Io1v7bDlu7yq7f708mkxBCeTeHPM/LuzwcHx8vb+twdna2rc2L0LM9tu3NiYvDiY/moFrFdDotimI5k/neMm35i7Er3uWBJXvmDQ4nNsZB9RHcqQGAKLhTAwBRECQAoiBIAERBkACIgiABEIX/BzWWqZweBItmAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(x(:,1),x(:,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "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": [
    "# Implémentation 2\n",
    "A = [-5 -3; 1 -1];\n",
    "sys_str = sprintf('[%f*x(1)+%f*x(2);%f*x(1)+%f*x(2)]',A(1,1),A(1,2),A(2,1),A(2,2)); # Définition système (1)\n",
    "sys_ode = inline(sys_str,'t','x'); # Définition système (2)\n",
    "[ts,ys] = ode45(sys_ode,[0,20],[2,0]); # Résolution système\n",
    "plot(ys(:,1),ys(:,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "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": [
    "# 2. Dessiner les droites correspondants aux vecteurs propres et le sense des trajectoires associés\n",
    "[V,L] = eig(A);\n",
    "line_range = -1.5:.1:1.5;\n",
    "eigenvector_1 = (V(2,1)/V(1,1)) * line_range;\n",
    "eigenvector_2 = (V(2,2)/V(1,2)) * line_range;\n",
    "hold on;\n",
    "plot(line_range,eigenvector_1,\"linewidth\",10);\n",
    "plot(line_range,eigenvector_2,\"linewidth\",10);\n",
    "quiver([0;0],[0;0],V(1,:),V(2,:),\"linewidth\",10,\"color\",\"k\");\n",
    "legend(\"v_1\",\"v_2\",\"location\",\"south\");\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "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": [
    "# 3. Calculer les isoclines\n",
    "line_range = -1.5:.1:1.5;\n",
    "isocline_1 = -(A(1,1)/A(1,2)) * line_range;\n",
    "isocline_2 = -(A(2,1)/A(2,2)) * line_range;\n",
    "hold on;\n",
    "plot(line_range,isocline_1,\"linewidth\",5);\n",
    "plot(line_range,isocline_2,\"linewidth\",5);\n",
    "legend(\"isocline_1\",\"isocline_2\",\"location\",\"south\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "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. Portrait de phase\n",
    "#Define grid for plotting\n",
    "x1range=-1.5:.1:1.5;\n",
    "x2range=-1.5:.1:1.5;\n",
    "[x1,x2] = meshgrid(x1range, x2range);\n",
    "\n",
    "# Define the system to plot (based on matrix A)\n",
    "x1p = A(1,1)*x1+A(1,2)*x2;\n",
    "x2p = A(2,1)*x1+A(2,2)*x2;\n",
    "\n",
    "#Normalize values for plotting\n",
    "norms=sqrt(x1p.^2+x2p.^2);\n",
    "# Vector field plot\n",
    "hold on;\n",
    "quiver(x1,x2,x1p./norms,x2p./norms,0.5);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ode_sys = (sym 2×1 matrix)\n",
      "\n",
      "  ⎡d                             ⎤\n",
      "  ⎢──(x₁(t)) = -5⋅x₁(t) - 3⋅x₂(t)⎥\n",
      "  ⎢dt                            ⎥\n",
      "  ⎢                              ⎥\n",
      "  ⎢  d                           ⎥\n",
      "  ⎢  ──(x₂(t)) = x₁(t) - x₂(t)   ⎥\n",
      "  ⎣  dt                          ⎦\n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "                  -2⋅t         -4⋅t\n",
      "  x₁(t) = - 3⋅C₁⋅ℯ     - 3⋅C₂⋅ℯ    \n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "                -2⋅t       -4⋅t\n",
      "  x₂(t) = 3⋅C₁⋅ℯ     + C₂⋅ℯ    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "pkg load symbolic\n",
    "syms x1(t) x2(t);\n",
    "ode_sys = [diff(x1(t),t) == A(1,1)*x1(t) + A(1,2)*x2(t);  diff(x2(t),t) == A(2,1)*x1(t) + A(2,2)*x2(t)]\n",
    "solutions = dsolve(ode_sys);\n",
    "solutions{1}\n",
    "solutions{2}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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63a5hGOvr67NPMe0byaL5MVavQBJ6pzcwhUBCUiTi0KpFyS6IUSUAUCspR1Tteki+ROQ5AOgsWQdSfQNJSkqwYzVRsoPOEnf81K5kN4UJeADwVAk9bOreQ/Il9P8vAEQp0YfKxASSlLgeKNKNkh20kvQjpO4luylU8ABgVjoOjAnrIfmS/ocAAKiSmuNhUgNJpOUvAiQaJTvEK2WHwYSV7IL8U2hFirYHADxGKg99CQ4kiSs7AFg1aT3iJT6QJCY7AFgF6T7QpSSQBBU8AKm2Cge39ASSRCwBSJnVOaClLZAkBpYApMNKHccSPO37MVbnLwsAKbOCh6+UB5K0gtsV0eA8JCzDyh6y0lmym8LAEoBEWPHD1Er0kIJWqiALIEE4Oq1cIImV/xsEClGygxIclKSVKNlNoYIHQBMciIJWsYcUxK8BQCw4+Mxa9UCS+GUgHEp2CIEDzl1WsWQ3iyIegAhwkLkfgfQLru8AYHk4tjyIQJrGhcMBqMUh5ZEIpDmo4AFQgsPIkxBIdyKWAITGoSMEAukBxBKAJ+FwERrTvp+A3xmAe3CIWBCB9GT85uDjPCRIHBaUoGT3ZBTxAPg4FChED2kh/BaBlcXurxyBpID8XQp+mquHkt0KYn9fHkp2Cvi/S/5iAlKMHXzZ6CGpx68WSBl26mgQSMvCL3gVULJLPXbkKFGyWxYm4wGJxs4bPQJpuYKxJPhxA9pjb40RgRQFZj0A+mP3jB2BFCnqeICG2CU1QSDFgFgCNMFuqBUCKTYMLwFxYb/TE9O+dcFfakAE2NF0RiDphT/ckoXzkJKCPSsRKNnphfl4gFrsSglCD0lr/FkHhMO+k0QEUjLwV96yffGbZ0KIT/79/ZPeRclOQ+wsyUXJLhmYkrc8wSj64jfPnppJ0AR7RwrQQ0ok9j2FpkKITEoW9oU0oYeUSFNzHwR7Y1gLxg8lu7jwy08lAinZmJWHVcNPPcUIpJRgkCmcT/79PTW6ROC3vQoIpFShlBeOn0mEk274Ja8UJjWkHPvzIzHtWytP/d2++1teLnzwh+Gy2oTlI5BWhb+HC8JJHQJJodA/0Xd/y/s5JJOJWEooSnarIriH022CPhb8NQbTSAjxwR+Gfm8JiUMgrSKGmlShexSa8t/eVDIhiQiklfb2/34iF9YOv5ALJNOTULJ7kqXWjUmjFCCQVtcP2x9uXP1LLr/d/lAIsXH1L4aaoNxSO+KyRkcapQOTGlZUMI3uWkM4IbSIfzzBcSPCKbnoIa262RzyzZ0HIQinAEp2QTH+SAihdCCQVt1daTSFcMJc/BigECW71TXVN7qnq3QPjkcriI2OJSGQVtoP2x/6yyHSaMoKHqdWp2S3ghsX0SOQsBTB45dI7yEsxYG0IlsQWiGQEIVUHt0+/uadXPj77z6ItyVKpHIbIVkIJMQgBce+j7959/fffSB7SHI57hY9zdQmEMncCkgZAgnxmz04Cr2Pj34C+SU7zTOJ+EEiMO0b8Zt7cEzEMXTjrBt3E+ZIxP86YBaBBE3NHkPndqTmvlKhZrPpuq4Q4tWrV6ZpTj37w15JKJrU8OmXH/nLf/njPx/zllj+hwDLQ8kOiXfXcVla5Ojc7XYvLy9fv37tL/hPBWt0i9frPv3yo2AI+Q+X958GaIhAQsrdf0wPmj2+N5vNTCZTr9eFEM+fP//222+Dz4aeZTfVpMb/PDj+P825a4gcrJTHBlKhUBA/3/0luMxreE1qXvP46FrQVMx8+uVHX/2v/yeE+O///T/+8sd/yvbI5QfbPPXJvIbXaPKacOfnPTaQ1tboS+mCbRGZZrNpGEatVhPzekhC0baYqtfNXYMHsV/oI/S2+C/lTQFSwzRNz/OEEKPRyDCM2RdsbGws/i1/+eM/gzMagJXFLDvgTpVK5fz8XAjhuq4cSVqSqUyie4TVRMkuedgWEet2u4ZhrK+vzz7FttAH20IfobcFmxAIL8UXVwWixxgSAEALBBIAQAsEEgBACwQSAEALBBIAQAsPB1Kn02k2f7nQVrPZrFar1Wp1MBgss2G40/Pnz3d3d3d3d4PbBdGT///ZF3TATqGDxcPigRNjd3d3Xde1bVs+7Ha7nuddX193u93z8/PgxY8RjdFoZJrmxcVF3A1ZdXJfePv2LftC7NgpdKAkLB7oIV1cXPhfIIRwXVfeEqZUKsmbxCBinudls9nPP/+82WxOJpO4m7O65L5QKBTYF2LHTqEDJWHx5DGkzc1NuTB7szJE4Pb21jCMra2tTCazu7sbd3NWGvuCJtgp9BRiB5lTsut2u67r+hc5nuJ5XqlUEkJQN49ScKNUKhUhRKlU6nQ6cbdrpcmSnWBfiFulUmGn0FCIsJgTSKVSSX7KLNM05UffdfFjLIm/Uc7Pzzc3N+/aQIiM3BcKhcLXX3/NvhAvdgoNhQuLp13tO7KLH+Mu5XJ5f39/a2ur3+/v7OzE3ZzV5e8L+/v77AvxYqfQULiwCHNx1XsufoxosAk0wYbQB9tCQ0/dKFztGwiPq30DCnGlBgCAFggkAIAWCCQAgBYIJACAFggkAIAWCCQAgBbmTPtut9v9ft/zPCFENps1TXNra4vZ/QCApXovkDqdTqfTMQyjXC4bhpHJZAaDged58ipq9Xo9k8nE2FZAN5yHBCj0XiB1u93Z60FNJpNMJjMajYQQ9JOAIAIJUOi+KzV0Oh3XdS8vL9nlAADLNufiqjKHOp1OJpPZ2dn5+uuvo28WkAj0kACF3ushNZvNN2/eZLPZSqWSzWY9zzs4OIixcYDmCCRAofemfbdarc3NzVevXtXrdW7xAjyINAIUei+Q5K3G9vf3q9Xqmzdv4moTkBSFQiHuJgDpMX9Sw2Aw6HQ6l5eXhmFUKhXuPwbMRckOUGj+HWNN0zRN8+DgQE5wiLhNAIAV9F4PSZ5yFGNrgGShhwQo9N4Y0unp6fn5uTwHNqjdbr98+XJ2PQAAqkyPIcmrB/X7fTnLbjKZCCEqlcrOzg6dJ2AKPSRAoTuv1DAYDCaTiWma5BBwFwIJUGjO7SeazaYQwjTNUqmUyWTa7fZgMIi8YUACkEaAQu/NshsMBqenp4PBoN/vB1d+9dVXkTcMSAB6SIBCc0p2zWaTKwYBj0EgAQrNH0Nqt9ue58lbIpXLZYaRAADLNn8MSV43yPM8z/N2d3cjbxWQDFw6CFBoTiBdXl7+9a9/LZfLQoh6vW6aZrfbjbxhAIDVMieQpgp0nudRsgPmYgAJUGjOGFK73b68vDRN0/O8bDZ7e3t7cXERS+MAzTGpAVDovqt9CyEMw6jVapG3CkgGAglQaPqOsXNfxCxwYC4CCVBozhiS53n+3fn6/f7t7W20TQIArKL3rtQge0LVavWrr77yJzJUq9XRaLS+vh5D6wAAK2N+Dyn40DCMqTUAACg3546xOzs7+/v7Ozs72WzWdV3P80qlUvQtAwCslDsvHeS67u3tbblc5k5IAIAI3Hk/JAAPYpYdoNCcMSQAAKJHDwkAoAV6SEB4XO0bUIhAAgBogZIdAEAL9JCA8CjZAQoRSAAALRBIAAAtEEgAAC0QSAAALRBIAAAtEEgAAC1wHhIAQAv0kIDwOA8JUIhAAgBogZIdAEAL9JCA8CjZAQoRSAAALVCyAwBogR4SEB4lO0ChX8XdAEAXk8lkf39/MpkIIXZ2dmq1mhCi2Wy6riuEePXqlWmaMTcRSDV6SMBPOp3O5ubm9fX1xcVFq9USQnS7Xc/zrq+vDw8Pz8/P424gkHIEEvATwzDK5bIQIpPJyDWu68peUalUkv0kAMtDIAE/KZVKpVJpNBrt7u7ati1Xbm5uygW/XrcWEE9DgZRiDAmrrtvtuq5rGEatVms2m/1+37btUqkkn/U8Ty4PBgO5JjgxlUkNgEIEElad7BgJIdrttud5FxcX/lOmacocGo1GhmHE1kRgNRBIwE/6/b7nebu7u/LhxcVFpVKRcxlc163X67Nvefv2baRNBFKNE2OBB3S7XcMw1tfXZ58qFApkEqAKPSTgAf54EoCloocEANAC076B8JhlByhEIAEAtEDJDgCgBXpIQHiU7ACFCCQAgBYIJACAFggkAIAWCCQAgBYIJACAFggkAIAWOA8JAKAFekhAeJyHBChEIAEAtBC+ZLe2tqa2KYgLZVsAOljofkgcyNJhbY2hxJC4QR+gECU7AIAWFirZ8Wd1OrApAeiAHhIQHrPsAIUIJACAFggkAIAWCCQAgBZiCKQftj8M/V7HcRqNxnA4nH3q6uoq+LDX67VaLfnKVqvVaDRCfykAIAKRBtIP2x/+sP3hxtW/5EKIT3Acx7KsfD4/tb7RaLRareDLTk5O8vl8rVYbDoe2bfd6vYWaDgBYsoVOjH0SGUVyWS4E1yxiNmyGw+HR0VGxWLQsazgczgYYAEA3iR9DGo/HJycnx8fHwZW2bReLxb29PcdxisViXG1D6nGZBkChhI0h+eRgkvx3sVh0HGc8Hk91lc7OzizLCpbyALU4DwlQKIZAUlKmsyzr+PjYsqxisTgej6cCyZ/4IJ9d/OsAAMsW3RiSnMvgp5GqASTbtuVCr9fzl4vFYq1WsyzLcZx2u734twBzUbIDFIoukMTPmeQvq/3wm5sbf3l7e9uyrF6vNzW2BKjF1b4BhaIu2W1c/Uv+E/oTGo3GY+Zw53I5y7Ke9BYAQIy42jfYlAC0kPhp30CMmGUHKEQgAQC0QCABALRAIAEAtEAgAQC0QCABALRAIAEAtMB5SGBTAtACPSQgPM5DAhQikAAAWqBkBzYlAC3QQwLCo2QHKEQgAQC0EHXJ7t3f8nLhgz8MQ3yp4ziO49i2nc/nG42GXCkfBl92dXW1vb3darWGwyG3RHoQJTsAOoi0h/Tub/kP/jCU//jJ9CSO41iWlc/nh8PhXbc4ajQarVZLCGHbNrdBwlJRsgMUiu6OsTKN/Icyk8L1k4QQw+GwWCxallUsFnO5nL+eBAKAhIphDClc32hKr9eT5bsXL174ITQej09OTqjRAUASxRBIoXtFQUdHR999993x8fHR0dHV1ZVc2Wg0isWi4zjj8ZiuEgAkS3SBFHrcaK6rq6vhcDrYisXieDwmkAAgiaIbQxIzmbRIVymXy+3t7RWLxV6vd3Z2Jlfati0Xer2evwwASIRIA0koqtcJIeR0hl6vN3fE6ObmRsm3AAAik7wTYxuNhizH5XI5y7Ie80pgSd6+fRt3E4D04Fp2YFOGVygUyCRAleT1kAAAqUQPCWxKAFqghwSEx6WDAIUIJACAFijZgU0JQAv0kIDwKNkBChFIAAAtEEgAAC0QSAAALRBIAAAtEEgAAC0QSAAALcRwHtLnz16/+v5luC+Vty23bXs4HDqOI1fm8/m77n50dXW1vb099bDVag2HQ+507uM8JAA6iLSH9Pmz1zKN5EKIT3Acx7KsfP6Xu/wNh8PZW8dKjUaj1WrNPrRtm9tSQAnOQwIUiu4GfcGOkVxYpKtkWZa8GdKLFy/m9nWmIocEAgDNJXsMqdFoHB0dza4fj8cnJyd+UE09BO4xmUyCN35sNpvVarVarQ4Gg9kXczMkQKEYAilcsW4uWcELPmw0GvLfxWLRcZzxeNzr9aYeqvp2pNLp6elkMpHL3W7X87zr6+vDw8Pz8/PZF1OyAxSKIZBCl+mmtFqt4IQFIYRlWcfHx5ZlFYvF8XjsJ9DUQyXfjlTqdDrZbNY0TfnQdV25XCqVXNeNtWlA+kU3hiTnMvhptMgAktTr9e6aXOevn3rNPW8BRqNRp9N5/fr17u6uv3Jzc1Mu+Cm1trbmP8vsRECh6AJJ/JxJ/vKCn3Z2dvbga25ubu55CAghut2u67qGYbium8lkms2m53nNZnNnZ0cI4XleqVQSQvhjSMEQKhQKDCMBqkQaSEJFDjUajbOzs2KxuMgnULiDr1QqycgxTVOOHrmuWy6XM5mMaZoyh0ajkWEYMTcUSDtu0Ac25bTd3d2Liwu5XK1Wy+Wy67r1er1SqUy9kh4SoBCBBDblA7rdrmEY6+vrs08RSIBCUZfsgMSRBT0Ay5bsE2MBAKlBIAEAtMAYEtiUALRADwkIj0sHAQoRSAAALVCyA5sSgBboIQHhUbIDFOI8JABYLoXX8Ew3SnZgUwJLpPYuB+lGyQ4Ij5Id7jeVQME7HmAWgQQAS0cOPUbUgfTplx/Jf8K9Xd6efDgc+g/9ZWk4HLZaLbmy1Wo1Go0FGwwAi6NS9xiRBtKnX370lz/+U/4TLpMcx7EsK5/Pi5lwEkIMh8O9vT0hxN7e3nA4tG2b+x4BiBE1uieJbpadTCP/ocyk4JqnchxnPB4H17RaLdu2t7e3hRBXV1dHR0ehPxwAlJjKJLpK94hh2veCOeQ7Pj6e7QDlcjkhRD6fdxxn8a8AgMURQo8Uw6QGJWkEAEiZ6AIp9LjR4+VyOTmk1Ov15DgTsFTcLhZQKNKS3VQmKe8q2bZdq9WGw2Gv12u322o/HJjFLcwBhaIeQ1IbQjc3N8GHuVzu5ubGcZzj42OF3wIAiEDyToxtNBr3T+a2LOuRrwQWRPcIUIhr2YFNGR4lO0Ch5PWQAACpRA8JbEoAWqCHBITH1b4BhQgkAIAWCCQAgBYIJACAFggkAIAWCCQAgBYIJACAFjgPCWxKAFqghwSEx3lIgEIEEgBACzEE0sffvAv9XsdxGo2GvAvfcDhstVpyeeo1cmWr1Wo0Gos0FbgfV1YFFIo0kD7+5t3H37z7++8+kAshPsFxHMuy8vn8cDjc29sTQuzt7QUzKZhYtm1z+wksFSU7QKHobtAno0guy4XgmqdqtVq2bW9vbwshrq6ujo6O5HrHccbjsYr2AgAileAxpFwuJ4TI5/PBBDo+Ps7n8/E1CquFkh2gUMLGkACtULIDFIohkEKX6YJyuZwcKOr1enSJACAFohtDknMZ/DRaZABJCGHbdq1WGw6HvV6v3W4raiMAIDbRBZL4OZP85UU+KpfL3dzcOI5zfHw89dTNzc0inwwAiEWkgSRU1OsajcbZ2VmxWBRCWJZ1/yuZ9g0AScG17MCmDK9QKDDRDlAlwdO+AQBpQg8JbEoAWqCHBITHeUiAQgQSAEALlOzApgSgBXpIQHiU7ACFCCQAgBYo2YFNCUAL9JCA8CjZAQoRSAAALRBIAAAtEEgAAC0QSAAALUR9+4kvfvNMLnzy7+9DvN1xHMdxbNsO3iV2OBw6jmNZVnDleDy+uroqFovyRhWhH8pvlJ+Zz+dt2/a/otFoyIVge1R9kfzvmr3bEwCkVaQ9pC9+8+yTf38v//GT6Ulmg2c4HO7t7Qkh9vb25E3NhRDj8bhWq+VyuUaj4TjOIg+DX+R/vnw4e7MlhV9k2zY3cwKwUqI7D0mm0f1rHtRoNCzLCt6Xr9FoFIvF7e3tVqs1Ho+Pjo6EECcnJ7lczrbt8XgseyGhH/rh9+LFi+C9aGWHxrKsYrGYy+XkykW+d/aLpr5xeTgPCYAOYhhDCtc3uofMg3w+Px6P5Rp5uH/x4sXe3l4ul1vkofzARqMho87X6/VkJr148cLvyizji6AzzkMCFIohkMKNHk1xHGeq0jVle3v75uZme3v75ORkwYfi51Jh8POPjo6+++674+Pjo6Ojq6srJd8794sAYEVEF0ihx43msizr+PjYsqxcLieHdnq9nl/1yuVyfp9jwYdCiFartb29PdWAq6ur4JDS8r4IOuP+5YBCkc6ym8okJV0l27ZrtZqcYtButx3HOTk5abfbspLmOE673c7lcqEfCiF6vV5wcp2Uy+X29vaKxWKv1zs7O1v8e+/6IuisUCiQSYAyP4a1yHtDOzo6urm5mV3/mJWLPJzr3bt3yr83yLKsB9ugRCybMh02NjbibgKQHgm72rccNzo7O5Pn8aRYo9FotVrv3r2L4LuYZQdABwkLJCwDmzI0SnaAQlw6CACgBQIJAKAFAgkAoAUCCQCghYXOQ1pbW1PVDkAHzWbTdV0hxM7OTq1WC6559eqVaZoxtw9ItfCBxLwspMxgMOj3+9fX15PJ5PT0tFardbtdz/Our6+73e75+fnr16/jbiOQZsz3BX7SbDYzmczm5qYQolQq+Wvq9boQ4vnz599++23MTQRSjTEk4BeXl5eu67qu+/nnn8s1Mp+EEHPrdVztG1Ao6jvGArrpdruu6xqGIYTY2to6ODgQQlSrVfms53mytzQYDOSa4NDpxsZG1M0F0oseElZdqVQ6ODio1Woyk6Tb21shhGmanucNBoPRaOQ/Ky+69dlnn93e3srLNIxGo0Kh0Gw25Qu63W6n04n8vwNIPMaQgF9Uq9Vyudzv98vlshw6qlar//nPf37961/X6/VKpSJf1m63b29v5QuEEM1m0/O8fr/v36Dr5cuXzIAAnooeEvCL6+vrcrn85z//2Q+bcrk8Ho9/+9vfGobRbrd3d3flRPCtrS3/XW/evKnX69ls1u8YyRfH8B8AJBmBBLynVCqtr6/7Dw8ODp49e/anP/1JzgW3bfvg4KDf7/uvGQwG2WzWNM1KpSLPWBJCyG5WDK0HkoxAAh7LNM1SqTQYDIKjTZ1OxzTNbrc71UPyPC+mZgJJRSABTzOZTIIPLy8vs9ms67qe52UyGZlJmUwmptYBCca0b+BpgickdTodwzDkTHEhRCaTcV23UqnIOl5MDQSSih4S8IDBYBCcoZDJZPxOkowf/6mtra3Ly8vJZOJ5Hhe+A56Kad/AA0aj0e3tbTBgzs/Ps9msvPrqXC9fvjw8PAxOjgDwIHpIwAPW19enujs7Ozv+hLpZ8ixa0gh4KnpIQBij0eiuyBmNRplMhnkNwFP9fxgytD8O2FLAAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plan Tr(A)-det(A)\n",
    "\n",
    "x = -10:0.1:10;\n",
    "plot (x, 1/4*x.^2);\n",
    "hold on;\n",
    "A_vec = [[-2 1; 1 -2];[2 1; 2 3];[5 9; 6 2];[0 1; 0 1];[0 -1; 1 0];[1/3 -2; 3 -1]]; #cbind like\n",
    "i = 0;\n",
    "while i<(rows(A_vec)/2)# Column-wise iteration\n",
    "    A = A_vec(2*i+1:2*(i+1),1:2); # Get the proper submatrix\n",
    "    plot(trace(A),det(A),sprintf(\";[%d %d];o\",trace(A),det(A)))\n",
    "    i++;\n",
    "endwhile\n",
    "set(gca, \"xaxislocation\", \"origin\");\n",
    "set(gca, \"yaxislocation\", \"origin\");\n",
    "h = legend; legend(h,\"location\",\"southwest\");\n",
    "xlabel(\"tr(A)\");\n",
    "ylabel(\"det(A)\");\n",
    "box off;\n",
    "hold off;"
   ]
  }
 ],
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     "text": "Octave Kernel",
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