scientific_computing_heath 3.ipynb

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   "id": "a51da41b-8c93-4c78-8707-6f8ab52333df",
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy.integrate import odeint\n",
    "\n",
    "# plt.style.use('seaborn-poster')\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "deaba849-38d8-444f-a25a-60e500147f58",
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    "params = (0, 1, -1, 0)  # eager beaver. he gets excited by Juliet’s love for him, and\n",
    "# is further spurred on by his own affectionate feelings for her.\n",
    "# Center, endless cycle of love and hate\n",
    "init_conditions = [[1.0, 1.0], [2.0, 2.0], [3.0, 3.0]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5e13c3f0-2f37-4247-af5a-2ad11440ec63",
   "metadata": {},
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   "source": [
    "params = (-1, 2, 2, -1) # cautious lover case 1: a**2 < b**2 \n",
    "# the relationship always fizzles out to mutual indifference. The lesson\n",
    "# seems to be that excessive caution can lead to apathy.\n",
    "# \n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "31291038-0930-45bd-a369-d6eb8aa6fec9",
   "metadata": {},
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   "source": [
    "params = (-2, 1, 1, -2) # cautious lover case 1: a**2 > b**2 \n",
    "\n",
    "# the lovers are more daring, or perhaps more sensitive to each other.\n",
    "# Now the relationship is explosive. Depending on their feelings initially, their relationship\n",
    "# either becomes a love fest or a war. In either case, all trajectories approach\n",
    "# the line R 􀀞 J , so their feelings are eventually mutual\n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0]]"
   ]
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  {
   "cell_type": "markdown",
   "id": "04bb1d79-7ea2-437e-ace8-4808b9d3fae0",
   "metadata": {},
   "source": [
    "$\\dot{R}=aR+bJ, \\dot{J}=-bR-aJ$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "09f8c818-cc41-44c3-bcab-14b5b4f2bfd3",
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   "source": [
    "params = (2, 1, -1, -2) # Fire and Ice case 1 |a|>|b|:\n",
    "\n",
    "# Do opposites attract?\n",
    "# saddle point\n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0],\n",
    "                   [2.0, -5.0], [2.0, 5.0], [-2.0, -5.0], [-2.0, 5.0]]"
   ]
  },
  {
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   "execution_count": null,
   "id": "372c2e6e-7797-4e7e-a11c-2a72f14ea0f7",
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   "source": [
    "params = (1, 2, -2, -1) # Fire and Ice case 2 |a|<|b|:\n",
    "\n",
    "# Do opposites attract?\n",
    "# center\n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0],\n",
    "                   [2.0, -5.0], [2.0, 5.0], [-2.0, -5.0], [-2.0, 5.0]]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c60d569-a189-4bcd-8d4b-a97b567cf4ac",
   "metadata": {},
   "source": [
    "$\\dot{R}=0, \\dot{J}=aR+bJ$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4b461a7d-10ca-4659-8553-ffd2313abd85",
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   "source": [
    "params = (0, 0, 2, 1) # Romeo the robot case 1 ab > 0\n",
    "\n",
    "init_conditions = [[-4.0, 4.0], [4.0, -4.0], [-3.0, 3.0], [3.0, -3.0]]"
   ]
  },
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   "cell_type": "code",
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   "id": "0af9aef1-6f0b-4f23-b969-4de30182118c",
   "metadata": {},
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   "source": [
    "params = (0, 0, 2, -1) # Romeo the robot case 2 ab < 0\n",
    "\n",
    "init_conditions = [[-4.0, 4.0], [4.0, -4.0], [-3.0, 3.0], [3.0, -3.0]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c0b544e9-5898-4417-a313-667fcb40bee2",
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   "source": [
    "def romeo_juliet(z, t, a, b, c, d):\n",
    "    \"\"\"love model\n",
    "\n",
    "    Args:\n",
    "        z (list): list of parameters\n",
    "        t (float): time\n",
    "        a (float): parameter\n",
    "        b (float): parameter\n",
    "        c (float): parameter\n",
    "        d (float): parameter\n",
    "\n",
    "    Returns:\n",
    "        list: list of parameters\n",
    "    \"\"\"\n",
    "    R, J = z\n",
    "    return [a*R + b*J, c*R + d*J]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1ad42605-7832-4d8c-8dea-2f3848c20321",
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   "source": [
    "def plot_phase_portrait(func, params, init_conditions, xlabel, ylabel, xlim, ylim):\n",
    "    \"\"\"plots phase portrait of a 2D system of ODEs\n",
    "\n",
    "    Args:\n",
    "        func (fucntion): fuction that returns the derivatives of the system\n",
    "        params (list): list of parameters for the system\n",
    "        init_conditions (list): list of initial conditions\n",
    "        xlabel (string): label for x-axis\n",
    "        ylabel (string): label for y-axis\n",
    "        xlim (int): limits for x-axis\n",
    "        ylim (int): limits for y-axis\n",
    "    \"\"\"\n",
    "    x = np.linspace(-10, 10, 20)\n",
    "    y = np.linspace(-10, 10, 20)\n",
    "    xv, yv = np.meshgrid(x, y)\n",
    "    \n",
    "    t = 0\n",
    "    \n",
    "    U, V = np.zeros(xv.shape), np.zeros(yv.shape)\n",
    "    \n",
    "    nx, ny = xv.shape\n",
    "    \n",
    "    for i in range(nx):\n",
    "        for j in range(ny):\n",
    "            x = xv[i, j]\n",
    "            y = yv[i, j]\n",
    "            # yprime = f([x, y], t)\n",
    "            U[i,j] = func([x, y], t, *params)[0]\n",
    "            V[i,j] = func([x, y], t, *params)[1]    \n",
    "    # plot\n",
    "    fig, ax = plt.subplots(figsize=(6.0, 6.0))\n",
    "    \n",
    "    ax.quiver(xv, yv, U, V, color=\"C0\", \n",
    "              angles='xy',\n",
    "              scale_units='xy', scale=5, width=.015)\n",
    "    \n",
    "    ax.set(xlabel=xlabel, ylabel=ylabel, xlim=xlim, ylim=ylim)\n",
    "    \n",
    "    # for y20 in [0, 0.5, 1, 1.5, 2, 2.5]:\n",
    "    #     tspan = np.linspace(0, 50, 200)\n",
    "    #     y0 = [0.0, y20]\n",
    "    #     ys = odeint(f, y0, tspan)\n",
    "    #     plt.plot(ys[:,0], ys[:,1], 'b-') # path\n",
    "    #     plt.plot([ys[0,0]], [ys[0,1]], 'o') # start\n",
    "    #     plt.plot([ys[-1,0]], [ys[-1,1]], 's') # end\n",
    "    \n",
    "    for y0 in init_conditions:\n",
    "        t_span = np.linspace(0, 15, 200)\n",
    "        # sol = solve_ivp(romeo_juliet, [0, 200], [y0, y0], args=params, \n",
    "        #                 dense_output=True)\n",
    "        sol = odeint(func, y0, t_span, args=params)\n",
    "        plt.plot(sol[:,0], sol[:,1], 'r-') # path\n",
    "        plt.plot([sol[0, 0]], [sol[0, 1]], 'o') # start\n",
    "        plt.plot([sol[-1,0]], [sol[-1,1]], 's') # end\n",
    "    \n",
    "    plt.show()\n"
   ]
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   "id": "1e49cb13-065c-493d-9951-3127ca9f3366",
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   "source": [
    "plot_phase_portrait(romeo_juliet, params, init_conditions, \n",
    "                    xlabel='R', ylabel='J', xlim=(-5, 5), ylim=(-5, 5))"
   ]
  },
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  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a0ddfdc3-152d-4f76-a16a-33cc35e75758",
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   "source": [
    "x = np.linspace(-10, 10, 20)\n",
    "y = np.linspace(-10, 10, 20)\n",
    "xv, yv = np.meshgrid(x, y)\n",
    "\n",
    "t = 0\n",
    "\n",
    "U, V = np.zeros(xv.shape), np.zeros(yv.shape)\n",
    "\n",
    "nx, ny = xv.shape\n",
    "\n",
    "# plot\n",
    "fig, axs = plt.subplots(3, 2, figsize=(12.0, 12.0))\n",
    "\n",
    "params = (-1, 2, 2, -1) # cautious lover case 1: a**2 < b**2 \n",
    "# the relationship always fizzles out to mutual indifference. The lesson\n",
    "# seems to be that excessive caution can lead to apathy.\n",
    "# \n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0]]\n",
    "\n",
    "for i in range(nx):\n",
    "    for j in range(ny):\n",
    "        x = xv[i, j]\n",
    "        y = yv[i, j]\n",
    "        # yprime = f([x, y], t)\n",
    "        U[i,j] = romeo_juliet([x, y], t, *params)[0]\n",
    "        V[i,j] = romeo_juliet([x, y], t, *params)[1]    \n",
    "# first\n",
    "axs[0, 0].quiver(xv, yv, U, V, color=\"C0\",\n",
    "                angles='xy',\n",
    "                scale_units='xy', scale=5, width=.015)\n",
    "\n",
    "axs[0, 0].set(xlabel='R', ylabel='J', xlim=(-5, 5), ylim=(-5, 5), title='cautious lover case 1: a**2 < b**2 ')\n",
    "\n",
    "for y0 in init_conditions:\n",
    "    t_span = np.linspace(0, 15, 200)\n",
    "    sol = odeint(romeo_juliet, y0, t_span, args=params)\n",
    "    axs[0, 0].plot(sol[:,0], sol[:,1], 'r-') # path\n",
    "    axs[0, 0].plot([sol[0, 0]], [sol[0, 1]], 'o') # start\n",
    "    axs[0, 0].plot([sol[-1,0]], [sol[-1,1]], 's') # end\n",
    "\n",
    "# second\n",
    "\n",
    "params = (-2, 1, 1, -2) # cautious lover case 2: a**2 > b**2 \n",
    "\n",
    "# the lovers are more daring, or perhaps more sensitive to each other.\n",
    "# Now the relationship is explosive. Depending on their feelings initially, their relationship\n",
    "# either becomes a love fest or a war. In either case, all trajectories approach\n",
    "# the line R 􀀞 J , so their feelings are eventually mutual\n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0]]\n",
    "\n",
    "for i in range(nx):\n",
    "    for j in range(ny):\n",
    "        x = xv[i, j]\n",
    "        y = yv[i, j]\n",
    "        # yprime = f([x, y], t)\n",
    "        U[i,j] = romeo_juliet([x, y], t, *params)[0]\n",
    "        V[i,j] = romeo_juliet([x, y], t, *params)[1]    \n",
    "\n",
    "axs[0, 1].quiver(xv, yv, U, V, color=\"C0\",\n",
    "                angles='xy',\n",
    "                scale_units='xy', scale=5, width=.015)\n",
    "\n",
    "axs[0, 1].set(xlabel='R', ylabel='J', xlim=(-5, 5), ylim=(-5, 5), title='cautious lover case 1: a**2 > b**2 ')\n",
    "\n",
    "for y0 in init_conditions:\n",
    "    t_span = np.linspace(0, 15, 200)\n",
    "    sol = odeint(romeo_juliet, y0, t_span, args=params)\n",
    "    axs[0, 1].plot(sol[:,0], sol[:,1], 'r-') # path\n",
    "    axs[0, 1].plot([sol[0, 0]], [sol[0, 1]], 'o') # start\n",
    "    axs[0, 1].plot([sol[-1,0]], [sol[-1,1]], 's') # end\n",
    "# third\n",
    "params = (2, 1, -1, -2) # Fire and Ice case 1 |a|>|b|:\n",
    "\n",
    "# Do opposites attract?\n",
    "# saddle point\n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0],\n",
    "                   [2.0, -5.0], [2.0, 5.0], [-2.0, -5.0], [-2.0, 5.0]]\n",
    "\n",
    "for i in range(nx):\n",
    "    for j in range(ny):\n",
    "        x = xv[i, j]\n",
    "        y = yv[i, j]\n",
    "        # yprime = f([x, y], t)\n",
    "        U[i,j] = romeo_juliet([x, y], t, *params)[0]\n",
    "        V[i,j] = romeo_juliet([x, y], t, *params)[1]    \n",
    "\n",
    "axs[1, 0].quiver(xv, yv, U, V, color=\"C0\",\n",
    "                angles='xy',\n",
    "                scale_units='xy', scale=5, width=.015)\n",
    "\n",
    "axs[1, 0].set(xlabel='R', ylabel='J', xlim=(-5, 5), ylim=(-5, 5), title='Fire and Ice case 1 |a|>|b|')\n",
    "    \n",
    "for y0 in init_conditions:\n",
    "    t_span = np.linspace(0, 15, 200)\n",
    "    sol = odeint(romeo_juliet, y0, t_span, args=params)\n",
    "    axs[1, 0].plot(sol[:,0], sol[:,1], 'r-') # path\n",
    "    axs[1, 0].plot([sol[0, 0]], [sol[0, 1]], 'o') # start\n",
    "    axs[1, 0].plot([sol[-1,0]], [sol[-1,1]], 's') # end\n",
    "# fourth\n",
    "\n",
    "params = (1, 2, -2, -1) # Fire and Ice case 2 |a|<|b|:\n",
    "\n",
    "# Do opposites attract?\n",
    "# center\n",
    "init_conditions = [[-1.0, 4.0], [-4.0, 4.0], [-4.0, 1.0], [4.0, -1.0], [4.0, -4.0], [1.0, -4.0], [4.0, 4.0], [-4.0, -4.0],\n",
    "                   [2.0, -5.0], [2.0, 5.0], [-2.0, -5.0], [-2.0, 5.0]]\n",
    "\n",
    "for i in range(nx):\n",
    "    for j in range(ny):\n",
    "        x = xv[i, j]\n",
    "        y = yv[i, j]\n",
    "        # yprime = f([x, y], t)\n",
    "        U[i,j] = romeo_juliet([x, y], t, *params)[0]\n",
    "        V[i,j] = romeo_juliet([x, y], t, *params)[1]    \n",
    "\n",
    "axs[1, 1].quiver(xv, yv, U, V, color=\"C0\",\n",
    "                angles='xy',\n",
    "                scale_units='xy', scale=5, width=.015)\n",
    "\n",
    "axs[1, 1].set(xlabel='R', ylabel='J', xlim=(-5, 5), ylim=(-5, 5), title='Fire and Ice case 2 |a|<|b|')\n",
    "\n",
    "\n",
    "for y0 in init_conditions:\n",
    "    t_span = np.linspace(0, 15, 200)\n",
    "    sol = odeint(romeo_juliet, y0, t_span, args=params)\n",
    "    axs[1, 1].plot(sol[:,0], sol[:,1], 'r-') # path\n",
    "    axs[1, 1].plot([sol[0, 0]], [sol[0, 1]], 'o') # start\n",
    "    axs[1, 1].plot([sol[-1,0]], [sol[-1,1]], 's') # end\n",
    "# fifth\n",
    "params = (0, 0, 2, 1) # Romeo the robot case 1 ab > 0\n",
    "\n",
    "init_conditions = [[-4.0, 4.0], [4.0, -4.0], [-3.0, 3.0], [3.0, -3.0]]\n",
    "for i in range(nx):\n",
    "    for j in range(ny):\n",
    "        x = xv[i, j]\n",
    "        y = yv[i, j]\n",
    "        # yprime = f([x, y], t)\n",
    "        U[i,j] = romeo_juliet([x, y], t, *params)[0]\n",
    "        V[i,j] = romeo_juliet([x, y], t, *params)[1]    \n",
    "\n",
    "axs[2, 0].quiver(xv, yv, U, V, color=\"C0\",\n",
    "                angles='xy',\n",
    "                scale_units='xy', scale=5, width=.015)\n",
    "\n",
    "axs[2, 0].set(xlabel='R', ylabel='J', xlim=(-5, 5), ylim=(-5, 5), title='Romeo the robot case 1 ab > 0')\n",
    "\n",
    "for y0 in init_conditions:\n",
    "    t_span = np.linspace(0, 15, 200)\n",
    "    sol = odeint(romeo_juliet, y0, t_span, args=params)\n",
    "    axs[2, 0].plot(sol[:,0], sol[:,1], 'r-') # path\n",
    "    axs[2, 0].plot([sol[0, 0]], [sol[0, 1]], 'o') # start\n",
    "    axs[2, 0].plot([sol[-1,0]], [sol[-1,1]], 's') # end\n",
    "# sixth\n",
    "params = (0, 0, -2, 1) # Romeo the robot case 2 ab < 0\n",
    "\n",
    "init_conditions = [[-4.0, 4.0], [4.0, -4.0], [-3.0, 3.0], [3.0, -3.0]]\n",
    "\n",
    "for i in range(nx):\n",
    "    for j in range(ny):\n",
    "        x = xv[i, j]\n",
    "        y = yv[i, j]\n",
    "        # yprime = f([x, y], t)\n",
    "        U[i,j] = romeo_juliet([x, y], t, *params)[0]\n",
    "        V[i,j] = romeo_juliet([x, y], t, *params)[1]    \n",
    "\n",
    "axs[2, 1].quiver(xv, yv, U, V, color=\"C0\",\n",
    "                angles='xy',\n",
    "                scale_units='xy', scale=5, width=.015)\n",
    "\n",
    "axs[2, 1].set(xlabel='R', ylabel='J', xlim=(-5, 5), ylim=(-5, 5), title='Romeo the robot case 1 ab < 0')\n",
    "\n",
    "for y0 in init_conditions:\n",
    "    t_span = np.linspace(0, 15, 200)\n",
    "    sol = odeint(romeo_juliet, y0, t_span, args=params)\n",
    "    axs[2, 1].plot(sol[:,0], sol[:,1], 'r-') # path\n",
    "    axs[2, 1].plot([sol[0, 0]], [sol[0, 1]], 'o') # start\n",
    "    axs[0, 1].plot([sol[-1,0]], [sol[-1,1]], 's') # end\n",
    "\n",
    "plt.show()\n",
    "# plt.savefig('love_dynamics_6.jpg')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "54739dfc-22c0-4f86-b499-41c0deffdabd",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c536b1be-e889-4be3-8df3-7028bc373f42",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "32a292ea-051e-4a06-88cc-035900b7067d",
   "metadata": {},
   "outputs": [],
   "source": [
    "def rhett_scarlett(z, t, A_s, A_r, k):\n",
    "    \"\"\"dynamics of love between Rhett and Scarlett\n",
    "\n",
    "    Args:\n",
    "        z (float): current state of the system\n",
    "        t (float): current time\n",
    "        A_s (float): Attraction of Scarlett\n",
    "        A_r (float): Attraction of Rhett\n",
    "        k (float): constant\n",
    "\n",
    "    Returns:\n",
    "        list: derivatives of the system\n",
    "    \"\"\"\n",
    "    R, S = z\n",
    "    return [-R + A_s + k*S*math.exp(-S), -S + A_r + k*R*math.exp(-R)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "91dc4bb7-dccd-4320-a00d-d7a847ab8d7f",
   "metadata": {},
   "outputs": [],
   "source": [
    "params = (1.2, 1, 15)\n",
    "init_conditions = [[0.0, 0.0], [1.0, 10.0], [10.0, 1.0], [10.0, 4.0], [4.0, 10.0], [6.0, 10.0], [10.0, 6.0], [0.0, 2.0], [2.0, 0.0], \n",
    "                   [10.0, 10.0]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f5388d69-9c28-49fd-84b0-aa69a40d9501",
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_phase_portrait(rhett_scarlett, params, init_conditions, xlabel='R', ylabel='S', xlim=(0, 10), ylim=(0, 10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3ac52af5-ad3b-4bb4-b8d1-d7b7fc4593f5",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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