TP3_Portrait_de_phase_II.ipynb

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    "# INFO-F-305 - TP 3\n",
    "# Dessin qualitiatif - Portrait de phase II\n",
    "### Jacopo De Stefani - __[jdestefa@ulb.ac.be](mailto:jdestefa@ulb.ac.be)__"
   ]
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    "# Système linéaire, regulier, à dimensions finies - Th 5.1\n",
    "\n",
    "$\\begin{cases}\n",
    "\\dot{\\mathbf{X}}(t) & = \\mathbf{A}(t)\\mathbf{X}(t) + \\mathbf{B}(t)\\mathbf{U}(t) \\\\\n",
    "\\mathbf{Y}(t) & = \\mathbf{C}(t)\\mathbf{X}(t) \\\\\n",
    "\\end{cases}$\n",
    "\n",
    "où $\\mathbf{A}(\\cdot),\\mathbf{B}(\\cdot),\\mathbf{C}(\\cdot)$ sont des matrices continues en $T$.\n",
    "\n",
    "Si on considère le mouvement libre (c-à-d $\\mathbf{U}(t) = 0$ $\\forall t$), et on se concentre sur l'état, avec $\\mathbf{A}(t)$ constante:\n",
    "\n",
    "$\\dot{\\mathbf{X}}(t) = \\mathbf{A}\\mathbf{X}(t)$"
   ]
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   "source": [
    "# Rappels Théoriques - Système du second ordre\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",
    "### Equation caractéristique\n",
    "\n",
    "$\\det(A-\\lambda I) = \\lambda^{2}-\\tau\\lambda+\\Delta=0$\n",
    "\n",
    "$\\lambda_{1,2} = \\frac{\\tau \\pm \\sqrt{\\tau^2 - 4 \\Delta}}{2}$\n",
    "\n",
    "où:\n",
    "- $\\tau=a_{11}+a_{22}=tr(A)$\n",
    "- $\\Delta=a_{11}a_{22}-a_{12}a_{21}=\\det(A)$"
   ]
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    "# Mouvement et trajectoire\n",
    "\n",
    "### Mouvement\n",
    "Ensemble des paires $\\{(t,x(t))\\}$ pour un système $\\mathbf{X}(t)=\\varphi(t,t_0,\\mathbf{X_{t_0}},u(\\cdot))$ \n",
    "\n",
    "### Trajectoire\n",
    "Ensemble des valeurs $\\{x(t)\\}$ pour $t\\geq t_0$ pour un système $\\mathbf{X}(t)=\\varphi(t,t_0,\\mathbf{X_{t_0}},u(\\cdot))$\n",
    "\n",
    "### Calcul d'une trajectoire\n",
    "1. Calcul équation solution générale de $\\dot{\\mathbf{X}}(t) = \\mathbf{A}\\mathbf{X}(t)$\n",
    "2. Détermination $c_1$ et $c_2$ dans la solution générale en fonction de $\\mathbf{X}(0)$\n",
    "3. Calcul de la trajectoire pour différent valeurs de $t$"
   ]
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   "source": [
    "# Stabilité\n",
    "\n",
    "### Stabilité asymptotique\n",
    "Si le mouvement libre du système (c-à-d en absence d'entrée $\\mathbf{U}(t)$) tends vers l'origine pour chaque valeur de la condition initiale.\n",
    "\n",
    "### Stabilité simple\n",
    "Si le mouvement libre du système (c-à-d en absence d'entrée $\\mathbf{U}(t)$) est limité pour chaque valeur de la condition initiale.\n",
    "\n",
    "### Instabilité\n",
    "S'il existe au moins une condition initiale telle que le mouvement libre du système (c-à-d en absence d'entrée $\\mathbf{U}(t)$) ne soit pas limité."
   ]
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   "source": [
    "# Stabilité et valeurs propres\n",
    "\n",
    "### Stabilité asymptotique - Th 5.3\n",
    "Un système $\\dot{\\mathbf{X}}(t) = \\mathbf{A}\\mathbf{X}(t)$ est asymptotiquement stable si la partie réelle de tous les valeurs propres de A est negative.\n",
    "\n",
    "### Stabilité simple - Th 5.4\n",
    "Un système $\\dot{\\mathbf{X}}(t) = \\mathbf{A}\\mathbf{X}(t)$ est (simplement) stable si la partie réelle de tous les valeurs propres de A est negative ou nulle et celles ayant partie réelle nulle ont une multiplicité égale à 1. \n",
    "\n",
    "\n",
    "### Instabilité - Th 5.5\n",
    "Un système $\\dot{\\mathbf{X}}(t) = \\mathbf{A}\\mathbf{X}(t)$ est instable s'il existe soit une valeur propre de A avec partie réelle positive soit une valeur propre ayant partie réelle nulle et une multiplicité supérieure à 1. "
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   "source": [
    "# Points d'équilibre\n",
    "\n",
    "### Definition\n",
    "Un point d’équilibre est un point où la trajectoire coincide avec la condition initiale. \n",
    "\n",
    "- Si $\\det(\\mathbf{A}) \\neq 0$ alors le seul état d’équilibre est (0, 0) et le système corréspondant est dit *simple*\n",
    "- Si $\\det(\\mathbf{A}) = 0$ alors il existe plusieurs état d'équilibre et le système corréspondant est dit *non simple*\n"
   ]
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   "source": [
    "# Cas 1 - Racines réelles et distinctes - $\\lambda_1,\\lambda_2$\n",
    "\n",
    "### Solution générale - Théoreme 5.11\n",
    "$\\mathbf{X}(t) = C_1 \\mathbf{X}^{(1)}(t) + C_2 \\mathbf{X}^{(2)}(t) $\n",
    "\n",
    "### Solution particulières\n",
    "$\\mathbf{X}^{(1)}(t) = e^{\\lambda_1 t} \\mathbf{v_1} $\n",
    "\n",
    "$\\mathbf{X}^{(2)}(t) = e^{\\lambda_2 t} \\mathbf{v_2} $\n",
    "\n",
    "### Exemple\n",
    "$\\mathbf{\\dot{x}}=\\begin{bmatrix}1 & 3\\\\\n",
    "3 & 1\\end{bmatrix}\\mathbf{x}$"
   ]
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     "text": [
      "A =\n",
      "\n",
      "   1   3\n",
      "   3   1\n",
      "\n",
      "ode_sys = (sym 2x1 matrix)\n",
      "\n",
      "  [d                          ]\n",
      "  [--(x1(t)) = x1(t) + 3*x2(t)]\n",
      "  [dt                         ]\n",
      "  [                           ]\n",
      "  [d                          ]\n",
      "  [--(x2(t)) = 3*x1(t) + x2(t)]\n",
      "  [dt                         ]\n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "                4*t         -2*t\n",
      "  x1(t) = 3*C1*e    + 3*C2*e    \n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "                4*t         -2*t\n",
      "  x2(t) = 3*C1*e    - 3*C2*e    \n",
      "\n"
     ]
    }
   ],
   "source": [
    "pkg load symbolic\n",
    "\n",
    "A = [1 3;3 1]\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}"
   ]
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   "source": [
    "# Stabilité -  - Racines réelles et distinctes - $\\lambda_1,\\lambda_2$\n",
    "\n",
    "### Noeud stable\n",
    "Si les valeurs propres de $\\mathbf{A}$ sont réelles et négatives.\n",
    "\n",
    "### Noeud instable\n",
    "Si les valeurs propres de $\\mathbf{A}$ sont réelles et positive.\n",
    "\n",
    "### Selle\n",
    "Si les valeurs propres de $\\mathbf{A}$ sont réelles et ayant signe discorde."
   ]
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   "source": [
    "# Système non simple\n",
    "\n",
    "Considérons le cas où $\\det(A) = \\lambda_1\\lambda_2 = 0$ et la valeur propre $\\lambda_2 \\neq 0$.\n",
    "\n",
    "Tous les états qui appartiennent à la droite $a_{11}x_1 + a_{12}x_2 = 0$ sont des états d’équilibre. \n",
    "Aussi, toutes les trajectoires sont des droites parallèles à la droite $v_2$. \n",
    "\n",
    "Deux configurations sont possibles:\n",
    "- $0 = \\lambda_1 > \\lambda_2$ : ceci implique que tous les états d’équilibre sont stables.\n",
    "- $ 0 = \\lambda_1 < \\lambda_2$: ceci implique que tous les états d’équilibre sont instables."
   ]
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   "source": [
    "# Cas 2 - Racines réelles et multiples - $\\lambda=\\lambda_1=\\lambda_2$\n",
    "\n",
    "### Solution générale - Théoreme 5.11\n",
    "$\\mathbf{X}(t) = C_1 \\mathbf{X}^{(1)}(t) + C_2 \\mathbf{X}^{(2)}(t) $\n",
    "\n",
    "### Solution particulières\n",
    "$\\mathbf{X}^{(1)}(t) = e^{\\lambda t} \\mathbf{v_1} $\n",
    "\n",
    "$\\mathbf{X}^{(2)}(t) = e^{\\lambda t} (\\mathbf{v_2} + t\\mathbf{v_1}) $\n",
    "\n",
    "### Exemple\n",
    "$\\mathbf{\\dot{x}}=\\begin{bmatrix}3 & -4\\\\\n",
    "1 & -1\\end{bmatrix}\\mathbf{x}$"
   ]
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     "text": [
      "A =\n",
      "\n",
      "   3  -4\n",
      "   1  -1\n",
      "\n",
      "ode_sys = (sym 2x1 matrix)\n",
      "\n",
      "  [d                            ]\n",
      "  [--(x1(t)) = 3*x1(t) - 4*x2(t)]\n",
      "  [dt                           ]\n",
      "  [                             ]\n",
      "  [  d                          ]\n",
      "  [  --(x2(t)) = x1(t) - x2(t)  ]\n",
      "  [  dt                         ]\n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "                                   t\n",
      "  x1(t) = (-4*C1 + C2*(-4*t - 2))*e \n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "                            t\n",
      "  x2(t) = (-2*C1 - 2*C2*t)*e \n",
      "\n"
     ]
    }
   ],
   "source": [
    "pkg load symbolic\n",
    "\n",
    "A = [3 -4;1 -1]\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}"
   ]
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   "source": [
    "# Stabilité - Racines réelles et multiples - $\\lambda=\\lambda_1=\\lambda_2$\n",
    "\n",
    "### Systéme simple - $\\lambda_1 = \\lambda_2 = \\lambda \\neq 0$\n",
    "    \n",
    "- A diagonalisable - : Tous les vecteurs sont vecteurs propres. Chaque droite qui passe par l’origine est une\n",
    "trajectoire. Si $\\lambda < 0(> 0)$ nous avons la (in)stabilité asymptotique. L’origine est dénommée **noeud singulier**.\n",
    "- A non diagonalisable: il existe un seul vecteur propre et donc seul une droite qui contient une trajectoire. L’origine est dénommée **noeud dégénéré**.\n",
    "    \n",
    "### Systéme non simple - $\\lambda_1 = \\lambda_2 = \\lambda = 0$\n",
    "\n",
    "- Il y a une infinité de points d’équilibres instables (multiplicité plus grande que 1) et toutes les trajectoires se trouvent sur des droites parallèles."
   ]
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   "source": [
    "# Cas 3 - Racines complexes et conjuguées - $\\lambda_{1,2} = \\alpha \\pm i\\beta$\n",
    "\n",
    "### Solution générale - Théoreme 5.11\n",
    "$\\mathbf{X}(t) = C_1 \\mathbf{X}^{(1)}(t) + C_2 \\mathbf{X}^{(2)}(t) $\n",
    "\n",
    "### Solution particulières - $\\mathbf{v_{1,2}} = \\mathbf{u} + i \\mathbf{v}$\n",
    "$\\mathbf{X}^{(1)}(t) = e^{\\alpha t} (\\cos(\\beta t)\\mathbf{u} - \\sin(\\beta t)\\mathbf{v} ) $\n",
    "\n",
    "$\\mathbf{X}^{(2)}(t) = e^{\\alpha t} (\\sin(\\beta t)\\mathbf{u} + \\cos(\\beta t)\\mathbf{v}) $\n",
    "\n",
    "### Exemple\n",
    "$\\mathbf{\\dot{x}}=\\begin{bmatrix}0 & 4\\\\\n",
    "-1 & 0\\end{bmatrix}\\mathbf{x}$"
   ]
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     "text": [
      "A =\n",
      "\n",
      "   0   4\n",
      "  -1   0\n",
      "\n",
      "ode_sys = (sym 2x1 matrix)\n",
      "\n",
      "  [d                  ]\n",
      "  [--(x1(t)) = 4*x2(t)]\n",
      "  [dt                 ]\n",
      "  [                   ]\n",
      "  [d                  ]\n",
      "  [--(x2(t)) = -x1(t) ]\n",
      "  [dt                 ]\n",
      "\n",
      "ans = (sym) x1(t) = 4*C1*cos(2*t) + 4*C2*sin(2*t)\n",
      "ans = (sym) x2(t) = -2*C1*sin(2*t) + 2*C2*cos(2*t)\n"
     ]
    }
   ],
   "source": [
    "pkg load symbolic\n",
    "\n",
    "A = [0 4;-1 0]\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}"
   ]
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   "source": [
    "# Stabilité - Racines complexes et conjuguées - $\\lambda_{1,2} = \\alpha \\pm i\\beta$\n",
    "\n",
    "### Centre\n",
    "- $\\alpha = 0$: \n",
    "Les trajectoires sont des ellipses fermées avec période $T = \\frac{2\\pi}{b}$. L’origine est dit un **centre**.\n",
    "\n",
    "### Foyer stable\n",
    "- $\\alpha < 0$: Le système est asymptotiquement stable et les trajectoires convergent vers l’origine en suivant des spirales.\n",
    "\n",
    "### Foyer instable\n",
    "- $\\alpha > 0$: Le système est instable et les trajectoires s’éloignent de l’origine en suivant des spirales. "
   ]
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   "source": [
    "# Exercice 1 - Système non simple\n",
    "\n",
    "$ \\mathbf{A} = \\begin{bmatrix} 0 & 1 \\\\ 0 & 1 \\end{bmatrix} $ \n",
    "\n",
    "1. Calculer les valeurs propres et vecteurs propres et en déduire le type du système \n",
    "2. Vérifier le lien entre la trace et le déterminant de la matrice\n",
    "3. Dessiner les vecteurs vitesse pour les points suivants (en normalisant leurs tailles):\n",
    "    - $(0,1),(0,-1),(1,0),(-1,0)$"
   ]
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     "text": [
      "A =\n",
      "\n",
      "   0   1\n",
      "   0   1\n",
      "\n",
      "V =\n",
      "\n",
      "   1.00000   0.70711\n",
      "   0.00000   0.70711\n",
      "\n",
      "L =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      "   0   0\n",
      "   0   1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 1. Calcul valeur propres et vecteurs propres\n",
    "A = [0 1; 0 1]\n",
    "[V,L] = eig(A)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tr(A) = 1\n",
      "det(A) = 0\n",
      "Tr(A) > 4*det(A)? = 1\n"
     ]
    }
   ],
   "source": [
    "printf(\"Tr(A) = %d\\n\",trace(A));\n",
    "printf(\"det(A) = %d\\n\",det(A));\n",
    "printf(\"Tr(A) > 4*det(A)? = %d\\n\",(trace(A) > 4*det(A)));"
   ]
  },
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   "execution_count": 7,
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     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#3. Dessiner les droites correspondants aux vecteurs propres et le sense des trajectoires associés\n",
    "line_range = -1.5:.1:1.5;\n",
    "line_1 = (V(2,1)/V(1,1)) * line_range;\n",
    "line_2 = (V(2,2)/V(1,2)) * line_range;\n",
    "hold on;\n",
    "plot(line_range,line_1,\"linewidth\",10);\n",
    "plot(line_range,line_2,\"linewidth\",10);\n",
    "quiver([0;0],[0;0],V(1,:),V(2,:),\"linewidth\",10,\"color\",\"k\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "velocity_vec =\n",
      "\n",
      "   1\n",
      "   1\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "  -1\n",
      "  -1\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "   0\n",
      "   0\n",
      "\n",
      "warning: division by zero\n",
      "velocity_vec =\n",
      "\n",
      "   0\n",
      "   0\n",
      "\n",
      "warning: division by zero\n"
     ]
    }
   ],
   "source": [
    "# 4.b Dessiner les vecteurs vitesse pour les points suivants (en normalisant leurs tailles) $(0,1),(1,0),(0,-1),(-1,0),(-0.5,-1)$\n",
    "point_sequence = [[0,1];[0,-1];[1,0];[-1,0]]'; #cbind like\n",
    "velocity_sequence = [];\n",
    "for point_vec = point_sequence # Column-wise iteration\n",
    "    velocity_vec = A*point_vec\n",
    "    velocity_norm = velocity_vec/norm(velocity_vec);\n",
    "    velocity_sequence = [velocity_sequence,velocity_norm];\n",
    "endfor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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": [
    "quiver(point_sequence(1,:),point_sequence(2,:),velocity_sequence(1,:),velocity_sequence(2,:),0.2,\"linewidth\",5,\"color\",\"k\");\n",
    "hold on;\n",
    "plot(point_sequence(1,:),point_sequence(2,:),\"ok\")\n",
    "plot(line_range,line_1,\"linestyle\",\":\",\"color\",\"k\");\n",
    "plot(line_range,line_2,\"linestyle\",\":\",\"color\",\"k\");\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# 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",
    "arrow=sqrt(x1p.^2+x2p.^2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "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": [
    "# Vector field plot\n",
    "hold on;\n",
    "quiver(x1,x2,x1p./arrow,x2p./arrow,0.5);\n",
    "# Trajectory plot\n",
    "#sys_str = sprintf('[%d*x(1)+%d*x(2);%d*x(1)+%d*x(2)]',A(1,1),A(1,2),A(2,1),A(2,2))\n",
    "#sys_ode = inline(sys_str,'t','x')\n",
    "#for x_0=0:0.1:1\n",
    "# [ts,ys] = ode45(sys_ode,[0,10],[0;x_0]);\n",
    "# plot(ys(:,1),ys(:,2))\n",
    "#end\n",
    "grid on;\n",
    "axis tight;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ode_sys = (sym 2x1 matrix)\n",
      "\n",
      "  [d                ]\n",
      "  [--(x1(t)) = x2(t)]\n",
      "  [dt               ]\n",
      "  [                 ]\n",
      "  [d                ]\n",
      "  [--(x2(t)) = x2(t)]\n",
      "  [dt               ]\n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "              t     \n",
      "  x1(t) = C1*e  + C2\n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "              t\n",
      "  x2(t) = C1*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": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Exercice 2 - Valeurs propres complexes\n",
    "\n",
    "$ \\mathbf{A} = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix} $ \n",
    "\n",
    "1. Calculer les valeurs propres et vecteurs propres et en déduire le type du système \n",
    "2. Vérifier le lien entre la trace et le déterminant de la matrice\n",
    "3. Dessiner les vecteurs vitesse pour les points suivants (en normalisant leurs tailles):\n",
    "    - $(0,1),(0,-1),(1,0),(-1,0)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A =\n",
      "\n",
      "   0  -1\n",
      "   1   0\n",
      "\n",
      "V =\n",
      "\n",
      "   0.70711 + 0.00000i   0.70711 - 0.00000i\n",
      "   0.00000 - 0.70711i   0.00000 + 0.70711i\n",
      "\n",
      "L =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      "   0 + 1i        0\n",
      "        0   0 - 1i\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 1. - Calcul valeur propres et vecteurs propres\n",
    "A = [0 -1; 1 0]\n",
    "[V,L] = eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tr(A) = 0\n",
      "det(A) = 1\n",
      "Tr(A) > 4*det(A)? = 0\n"
     ]
    }
   ],
   "source": [
    "printf(\"Tr(A) = %d\\n\",trace(A));\n",
    "printf(\"det(A) = %d\\n\",det(A));\n",
    "printf(\"Tr(A) > 4*det(A)? = %d\\n\",(trace(A) > 4*det(A)));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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hZ/xBURQbX8ekX154pEN4onmwwdO9+1fZrWqu7PD333+fnZ3NZrOfP39+/vx51wdFax5dy+PhQh5//fVX8s/40+uYpOqFK7YM4Ykm2+I7edxfjHXFh1Ste2aH8IwP4TE2Xnikw/kikL3x6Y4bJAAGJeKr7AAYIEECIARBAiAEQQIgBEECIIT/B0S7tOlNLy7PAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#3. Dessiner les droites correspondants aux vecteurs propres et le sense des trajectoires associés\n",
    "line_range = -1.5:.1:1.5;\n",
    "line_1 = (V(2,1)/V(1,1)) * line_range;\n",
    "line_2 = (V(2,2)/V(1,2)) * line_range;\n",
    "hold on;\n",
    "plot(line_range,line_1,\"linewidth\",10);\n",
    "plot(line_range,line_2,\"linewidth\",10);\n",
    "quiver([0;0],[0;0],V(1,:),V(2,:),\"linewidth\",10,\"color\",\"k\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "velocity_vec =\n",
      "\n",
      "  -1\n",
      "   0\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "   1\n",
      "   0\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "   0\n",
      "   1\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "   0\n",
      "  -1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 4.b Dessiner les vecteurs vitesse pour les points suivants (en normalisant leurs tailles) $(0,1),(1,0),(0,-1),(-1,0),(-0.5,-1)$\n",
    "point_sequence = [[0,1];[0,-1];[1,0];[-1,0]]'; #cbind like\n",
    "velocity_sequence = [];\n",
    "for point_vec = point_sequence # Column-wise iteration\n",
    "    velocity_vec = A*point_vec\n",
    "    velocity_norm = velocity_vec/norm(velocity_vec);\n",
    "    velocity_sequence = [velocity_sequence,velocity_norm];\n",
    "endfor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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eT3bYXV//PeyHt8dbvdWZUprNZo1GY758v8/7XKvVevfw99InB3sgp5tFS5z0Tf+V3SKf7HAIut3ucDhMKeV5fnl5WZ30yWTyzz//HMJ5f3X42x5Ovd4e7KGdbipLP7dH+cdYn+yw3z46vwdy3g/kMCufHOxB/R6ofOukRwkSAAdu+39lBwBJkAAIQpAACEGQAAhBkAAI4f9nDKaGEpxAuAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hold on;\n",
    "quiver(point_sequence(1,:),point_sequence(2,:),velocity_sequence(1,:),velocity_sequence(2,:),0.2,\"linewidth\",5,\"color\",\"k\");\n",
    "plot(point_sequence(1,:),point_sequence(2,:),\"ok\");\n",
    "plot(line_range,line_1,\"linestyle\",\":\",\"color\",\"k\");\n",
    "plot(line_range,line_2,\"linestyle\",\":\",\"color\",\"k\");\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Define grid for plotting\n",
    "x1range=-3:.1:3;\n",
    "x2range=-3:.1:3;\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",
    "arrow=sqrt(x1p.^2+x2p.^2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "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": [
    "# Vector field plot\n",
    "hold on;\n",
    "quiver(x1,x2,x1p./arrow,x2p./arrow,0.5);\n",
    "# Trajectory plot\n",
    "sys_str = sprintf('[%d*x(1)+%d*x(2);%d*x(1)+%d*x(2)]',A(1,1),A(1,2),A(2,1),A(2,2));\n",
    "sys_ode = inline(sys_str,'t','x');\n",
    "for x_0=0:0.1:1\n",
    " [ts,ys] = ode45(sys_ode,[0,10],[0;x_0]);\n",
    " plot(ys(:,1),ys(:,2))\n",
    "end\n",
    "grid on;\n",
    "axis tight;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ode_sys = (sym 2x1 matrix)\n",
      "\n",
      "  [d                 ]\n",
      "  [--(x1(t)) = -x2(t)]\n",
      "  [dt                ]\n",
      "  [                  ]\n",
      "  [d                 ]\n",
      "  [--(x2(t)) = x1(t) ]\n",
      "  [dt                ]\n",
      "\n",
      "ans = (sym) x1(t) = -C1*cos(t) - C2*sin(t)\n",
      "ans = (sym) x2(t) = -C1*sin(t) + C2*cos(t)\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": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Exercice 3 - Valeurs propres complexes\n",
    "\n",
    "$ \\mathbf{A} = \\begin{bmatrix} \\frac{1}{3} & -2 \\\\ 3 & -1 \\end{bmatrix} $ \n",
    "\n",
    "1. Calculer les valeurs propres et vecteurs propres et en déduire le type du système \n",
    "2. Vérifier le lien entre la trace et le déterminant de la matrice\n",
    "3. Dessiner les vecteurs vitesse pour les points suivants (en normalisant leurs tailles):\n",
    "    - $(0,1),(0,-1),(1,0),(-1,0)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A =\n",
      "\n",
      "   0.33333  -2.00000\n",
      "   3.00000  -1.00000\n",
      "\n",
      "V =\n",
      "\n",
      "   0.17213 + 0.60858i   0.17213 - 0.60858i\n",
      "   0.77460 + 0.00000i   0.77460 - 0.00000i\n",
      "\n",
      "L =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      "  -0.3333 + 2.3570i                  0\n",
      "                  0  -0.3333 - 2.3570i\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 1. - Calcul valeur propres et vecteurs propres\n",
    "A = [1/3 -2; 3 -1]\n",
    "[V,L] = eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tr(A) = -0.666667\n",
      "det(A) = 5.66667\n",
      "Tr(A) > 4*det(A)? = 0\n"
     ]
    }
   ],
   "source": [
    "printf(\"Tr(A) = %d\\n\",trace(A));\n",
    "printf(\"det(A) = %d\\n\",det(A));\n",
    "printf(\"Tr(A) > 4*det(A)? = %d\\n\",(trace(A) > 4*det(A)));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "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. Dessiner les droites correspondants aux vecteurs propres et le sense des trajectoires associés\n",
    "line_range = -1.5:.1:1.5;\n",
    "line_1 = (V(2,1)/V(1,1)) * line_range;\n",
    "line_2 = (V(2,2)/V(1,2)) * line_range;\n",
    "hold on;\n",
    "plot(line_range,line_1,\"linewidth\",10);\n",
    "plot(line_range,line_2,\"linewidth\",10);\n",
    "quiver([0;0],[0;0],V(1,:),V(2,:),\"linewidth\",10,\"color\",\"k\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "velocity_vec =\n",
      "\n",
      "  -2\n",
      "  -1\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "   2\n",
      "   1\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "   0.33333\n",
      "   3.00000\n",
      "\n",
      "velocity_vec =\n",
      "\n",
      "  -0.33333\n",
      "  -3.00000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# 4.b Dessiner les vecteurs vitesses pour les points suivants (en normalisant leurs tailles) $(0,1),(1,0),(0,-1),(-1,0),(-0.5,-1)$\n",
    "point_sequence = [[0,1];[0,-1];[1,0];[-1,0]]'; #cbind like\n",
    "velocity_sequence = [];\n",
    "for point_vec = point_sequence # Column-wise iteration\n",
    "    velocity_vec = A*point_vec\n",
    "    velocity_norm = velocity_vec/norm(velocity_vec);\n",
    "    velocity_sequence = [velocity_sequence,velocity_norm];\n",
    "endfor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "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": [
    "quiver(point_sequence(1,:),point_sequence(2,:),velocity_sequence(1,:),velocity_sequence(2,:),0.2,\"linewidth\",5,\"color\",\"k\");\n",
    "hold on;\n",
    "plot(point_sequence(1,:),point_sequence(2,:),\"ok\")\n",
    "plot(line_range,line_1,\"linestyle\",\":\",\"color\",\"k\");\n",
    "plot(line_range,line_2,\"linestyle\",\":\",\"color\",\"k\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# 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",
    "arrow=sqrt(x1p.^2+x2p.^2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sys_str = [0.333333*x(1)+-2*x(2);3*x(1)+-1*x(2)]\n",
      "sys_ode = f(t, x) = [0.333333*x(1)+-2*x(2);3*x(1)+-1*x(2)]\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Vector field plot\n",
    "hold on;\n",
    "quiver(x1,x2,x1p./arrow,x2p./arrow,0.5);\n",
    "# Trajectory plot\n",
    "sys_str = sprintf('[%d*x(1)+%d*x(2);%d*x(1)+%d*x(2)]',A(1,1),A(1,2),A(2,1),A(2,2))\n",
    "sys_ode = inline(sys_str,'t','x')\n",
    "for x_0=0:0.1:1\n",
    " [ts,ys] = ode45(sys_ode,[0,10],[0;x_0]);\n",
    " plot(ys(:,1),ys(:,2))\n",
    "end\n",
    "grid on;\n",
    "axis tight;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "warning: passing floating-point values to sym is dangerous, see \"help sym\"\n",
      "warning: called from\n",
      "    double_to_sym_heuristic at line 50 column 7\n",
      "    sym at line 379 column 13\n",
      "    mtimes at line 65 column 5\n",
      "ode_sys = (sym 2x1 matrix)\n",
      "\n",
      "  [d           x1(t)          ]\n",
      "  [--(x1(t)) = ----- - 2*x2(t)]\n",
      "  [dt            3            ]\n",
      "  [                           ]\n",
      "  [d                          ]\n",
      "  [--(x2(t)) = 3*x1(t) - x2(t)]\n",
      "  [dt                         ]\n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "                                                         -t \n",
      "          /          /    ___  \\           /    ___  \\\\  ---\n",
      "          |          |5*\\/ 2 *t|           |5*\\/ 2 *t||   3 \n",
      "  x1(t) = |- 2*C1*cos|---------| - 2*C2*sin|---------||*e   \n",
      "          \\          \\    3    /           \\    3    //     \n",
      "\n",
      "ans = (sym)\n",
      "\n",
      "          /   /             /    ___  \\        /    ___  \\\\      /       /    __\n",
      "          |   |      ___    |5*\\/ 2 *t|        |5*\\/ 2 *t||      |       |5*\\/ 2\n",
      "          |   |  5*\\/ 2 *sin|---------|   2*cos|---------||      |  2*sin|------\n",
      "          |   |             \\    3    /        \\    3    /|      |       \\    3 \n",
      "  x2(t) = |C1*|- ---------------------- - ----------------| + C2*|- ------------\n",
      "          \\   \\            3                     3        /      \\         3    \n",
      "  \n",
      "  _  \\              /    ___  \\\\\\     \n",
      "   *t|       ___    |5*\\/ 2 *t|||  -t \n",
      "  ---|   5*\\/ 2 *cos|---------|||  ---\n",
      "     /              \\    3    /||   3 \n",
      "  ---- + ----------------------||*e   \n",
      "                   3           //     \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}\n"
   ]
  }
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