Linear_ODE_complex_evs.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "# Use pretty printing for SymPy `expr`essions\n",
    "init_printing()\n",
    "\n",
    "def noop(*args, **kwargs):\n",
    "    pass\n",
    "if __name__ != '__main__':\n",
    "    display = noop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear ODEs: example with complex eigenvalues\n",
    "\n",
    "In this lecture we solve the linear ODE $\\dot x=Ax$, where $A=\\begin{bmatrix}6&-5\\\\13&-10\\end{bmatrix}$.\n",
    "I will solve this ODE manually verifying each step using [Python] library [SymPy].\n",
    "\n",
    "[Python]: https://www.python.org \"Python programming language\"\n",
    "[SymPy]: https://www.sympy.org \"Symboic math in Python\"\n",
    "\n",
    "## Eigenvalues and eigenvectors\n",
    "\n",
    "The eigenvalues of this matrix are $-2\\pm i$, and the corresponding eigenvectors are $\\begin{pmatrix}8\\pm i\\\\13\\end{pmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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dzQ8tNNRpHVP6fEre0xI3fCalz6fkBTLln8wPp2TFSAKvC/4+WCChpCQB93XBs6Bk0rXOvDoCKe2fknf1iq2kALMOju1Po/OGTvxWZT/U9mgnfiUMjsSaH47gaG0npc+n5G0Nh1z6ju7zEjg6bwZ+6JX1f1Lh+IRjf3/dAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( -2 - i, \\  1, \\  \\left[ \\left[\\begin{matrix}\\frac{8}{13} - \\frac{i}{13}\\\\1\\end{matrix}\\right]\\right]\\right), \\  \\left( -2 + i, \\  1, \\  \\left[ \\left[\\begin{matrix}\\frac{8}{13} + \\frac{i}{13}\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛           ⎡⎡8    ⅈ ⎤⎤⎞  ⎛           ⎡⎡8    ⅈ ⎤⎤⎞⎤\n",
       "⎢⎜           ⎢⎢── - ──⎥⎥⎟  ⎜           ⎢⎢── + ──⎥⎥⎟⎥\n",
       "⎢⎜-2 - ⅈ, 1, ⎢⎢13   13⎥⎥⎟, ⎜-2 + ⅈ, 1, ⎢⎢13   13⎥⎥⎟⎥\n",
       "⎢⎜           ⎢⎢       ⎥⎥⎟  ⎜           ⎢⎢       ⎥⎥⎟⎥\n",
       "⎣⎝           ⎣⎣   1   ⎦⎦⎠  ⎝           ⎣⎣   1   ⎦⎦⎠⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix(2, 2, [6, -5, 13, -10])\n",
    "A.eigenvects()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normal form\n",
    "Therefore, $A = PCP^{-1}$, where $P=\\begin{bmatrix}8&-1\\\\13&0\\end{bmatrix}$, $C=\\begin{bmatrix}-2&-1\\\\1&-2\\end{bmatrix}$. In general, if $v=\\begin{bmatrix}v_1\\\\v_2\\end{bmatrix}$ is an eigenvector of $A$ with eigenvalue $λ$, then $C=\\begin{bmatrix}\\Re\\lambda&-\\Im\\lambda\\\\\\Im\\lambda&\\Re\\lambda\\end{bmatrix}$, $P=\\begin{bmatrix}\\Re v_1&-\\Im v_1\\\\\\Re v_2&-\\Im v_2\\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\left( -2 + i, \\  \\left[\\begin{matrix}8 + i\\\\13\\end{matrix}\\right], \\  \\left[\\begin{matrix}8 & -1\\\\13 & 0\\end{matrix}\\right], \\  \\left[\\begin{matrix}-2 & -1\\\\1 & -2\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛        ⎡8 + ⅈ⎤  ⎡8   -1⎤  ⎡-2  -1⎤⎞\n",
       "⎜-2 + ⅈ, ⎢     ⎥, ⎢      ⎥, ⎢      ⎥⎟\n",
       "⎝        ⎣ 13  ⎦  ⎣13  0 ⎦  ⎣1   -2⎦⎠"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(λ, m, (v,)) = A.eigenvects()[1]\n",
    "v *= 13 # get rid of the denominator; it's still an eigenvector\n",
    "assert (A * v).expand() == (λ * v).expand()\n",
    "P = re(v).row_join(-im(v))\n",
    "C = Matrix(2, 2, [re(λ), -im(λ), im(λ), re(λ)])\n",
    "assert P * C * P ** -1 == A\n",
    "λ, v, P, C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Formula for the solution\n",
    "\n",
    "### Solution of the normalized equation\n",
    "Recall that the solution of $\\dot y=\\begin{pmatrix}a&-b\\\\b&a\\end{pmatrix}y$, $y(0)=c$, is given by $y(t)=e^{at}\\begin{pmatrix}\\cos(bt) & -\\sin(bt)\\\\\\sin(bt) & \\cos(bt)\\end{pmatrix}c$, hence solutions of $\\dot y=Cy$ are given by $y(t)=e^{-2t}\\begin{pmatrix}\\cos t&-\\sin t\\\\\\sin t&\\cos t\\end{pmatrix}y(0)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}e^{- 2 t} \\cos{\\left(t \\right)} & - e^{- 2 t} \\sin{\\left(t \\right)}\\\\e^{- 2 t} \\sin{\\left(t \\right)} & e^{- 2 t} \\cos{\\left(t \\right)}\\end{matrix}\\right], \\  \\left[\\begin{matrix}c_{0} e^{- 2 t} \\cos{\\left(t \\right)} - c_{1} e^{- 2 t} \\sin{\\left(t \\right)}\\\\c_{0} e^{- 2 t} \\sin{\\left(t \\right)} + c_{1} e^{- 2 t} \\cos{\\left(t \\right)}\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡ -2⋅t           -2⋅t       ⎤  ⎡    -2⋅t              -2⋅t       ⎤⎞\n",
       "⎜⎢ℯ    ⋅cos(t)  -ℯ    ⋅sin(t)⎥  ⎢c₀⋅ℯ    ⋅cos(t) - c₁⋅ℯ    ⋅sin(t)⎥⎟\n",
       "⎜⎢                           ⎥, ⎢                                 ⎥⎟\n",
       "⎜⎢ -2⋅t          -2⋅t        ⎥  ⎢    -2⋅t              -2⋅t       ⎥⎟\n",
       "⎝⎣ℯ    ⋅sin(t)  ℯ    ⋅cos(t) ⎦  ⎣c₀⋅ℯ    ⋅sin(t) + c₁⋅ℯ    ⋅cos(t)⎦⎠"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('c0 c1') # Coordinates of $x(0)$\n",
    "var('t') # Time\n",
    "c = Matrix([c0, c1])\n",
    "M = exp(re(λ) * t) * Matrix(2, 2, [cos(im(λ) * t), -sin(im(λ) * t), sin(im(λ) * t), cos(im(λ) * t)])\n",
    "y = M * c\n",
    "assert y.diff(t) == (C * y).expand()\n",
    "M, y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solution of the original equation\n",
    "Solutions of $\\dot x=Ax$, $x(0)=c$, are given by \n",
    "$$\n",
    "x(t)=PM(t)P^{-1}c=e^{-2t}\\begin{bmatrix}8\\sin t+\\cos t&-5\\sin t\\\\13\\sin t&\\cos t-8\\sin t\\end{bmatrix}x(0).\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}8 e^{- 2 t} \\sin{\\left(t \\right)} + e^{- 2 t} \\cos{\\left(t \\right)} & - 5 e^{- 2 t} \\sin{\\left(t \\right)}\\\\13 e^{- 2 t} \\sin{\\left(t \\right)} & - 8 e^{- 2 t} \\sin{\\left(t \\right)} + e^{- 2 t} \\cos{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡   -2⋅t           -2⋅t                     -2⋅t               ⎤\n",
       "⎢8⋅ℯ    ⋅sin(t) + ℯ    ⋅cos(t)          -5⋅ℯ    ⋅sin(t)        ⎥\n",
       "⎢                                                              ⎥\n",
       "⎢           -2⋅t                     -2⋅t           -2⋅t       ⎥\n",
       "⎣       13⋅ℯ    ⋅sin(t)         - 8⋅ℯ    ⋅sin(t) + ℯ    ⋅cos(t)⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0} \\left(8 e^{- 2 t} \\sin{\\left(t \\right)} + e^{- 2 t} \\cos{\\left(t \\right)}\\right) - 5 c_{1} e^{- 2 t} \\sin{\\left(t \\right)}\\\\13 c_{0} e^{- 2 t} \\sin{\\left(t \\right)} + c_{1} \\left(- 8 e^{- 2 t} \\sin{\\left(t \\right)} + e^{- 2 t} \\cos{\\left(t \\right)}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    ⎛   -2⋅t           -2⋅t       ⎞         -2⋅t         ⎤\n",
       "⎢ c₀⋅⎝8⋅ℯ    ⋅sin(t) + ℯ    ⋅cos(t)⎠ - 5⋅c₁⋅ℯ    ⋅sin(t)  ⎥\n",
       "⎢                                                         ⎥\n",
       "⎢       -2⋅t             ⎛     -2⋅t           -2⋅t       ⎞⎥\n",
       "⎣13⋅c₀⋅ℯ    ⋅sin(t) + c₁⋅⎝- 8⋅ℯ    ⋅sin(t) + ℯ    ⋅cos(t)⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = P * M * P ** -1 * c\n",
    "assert x.diff(t).expand() == (A * x).expand()\n",
    "assert x.subs(t, 0) == c\n",
    "display(P * M * P ** -1, x)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "myenv",
   "language": "python",
   "name": "myenv"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}