docs/source/examples/Lorenz Differential Equations.ipynb

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   "source": [
    "# Exploring the Lorenz System of Differential Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this Notebook we explore the Lorenz system of differential equations:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\dot{x} & = \\sigma(y-x) \\\\\n",
    "\\dot{y} & = \\rho x - y - xz \\\\\n",
    "\\dot{z} & = -\\beta z + xy\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "This is one of the classic systems in non-linear differential equations. It exhibits a range of different behaviors as the parameters (\\\\(\\sigma\\\\), \\\\(\\beta\\\\), \\\\(\\rho\\\\)) are varied."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Imports"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we import the needed things from IPython, NumPy, Matplotlib and SciPy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Imports for JupyterLite\n",
    "%pip install -q ipywidgets matplotlib numpy scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from ipywidgets import interact, interactive\n",
    "from IPython.display import clear_output, display, HTML"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import integrate\n",
    "\n",
    "from matplotlib import pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.colors import cnames\n",
    "from matplotlib import animation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing the trajectories and plotting the result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define a function that can integrate the differential equations numerically and then plot the solutions. This function has arguments that control the parameters of the differential equation (\\\\(\\sigma\\\\), \\\\(\\beta\\\\), \\\\(\\rho\\\\)), the numerical integration (`N`, `max_time`) and the visualization (`angle`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):\n",
    "\n",
    "    fig = plt.figure()\n",
    "    ax = fig.add_axes([0, 0, 1, 1], projection='3d')\n",
    "    ax.axis('off')\n",
    "\n",
    "    # prepare the axes limits\n",
    "    ax.set_xlim((-25, 25))\n",
    "    ax.set_ylim((-35, 35))\n",
    "    ax.set_zlim((5, 55))\n",
    "    \n",
    "    def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho):\n",
    "        \"\"\"Compute the time-derivative of a Lorenz system.\"\"\"\n",
    "        x, y, z = x_y_z\n",
    "        return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]\n",
    "\n",
    "    # Choose random starting points, uniformly distributed from -15 to 15\n",
    "    np.random.seed(1)\n",
    "    x0 = -15 + 30 * np.random.random((N, 3))\n",
    "\n",
    "    # Solve for the trajectories\n",
    "    t = np.linspace(0, max_time, int(250*max_time))\n",
    "    x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)\n",
    "                      for x0i in x0])\n",
    "    \n",
    "    # choose a different color for each trajectory\n",
    "    colors = plt.cm.viridis(np.linspace(0, 1, N))\n",
    "\n",
    "    for i in range(N):\n",
    "        x, y, z = x_t[i,:,:].T\n",
    "        lines = ax.plot(x, y, z, '-', c=colors[i])\n",
    "        plt.setp(lines, linewidth=2)\n",
    "\n",
    "    ax.view_init(30, angle)\n",
    "    plt.show()\n",
    "\n",
    "    return t, x_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's call the function once to view the solutions. For this set of parameters, we see the trajectories swirling around two points, called attractors. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t, x_t = solve_lorenz(angle=0, N=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using IPython's `interactive` function, we can explore how the trajectories behave as we change the various parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "w = interactive(solve_lorenz, angle=(0.,360.), max_time=(0.1, 4.0), \n",
    "                N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The object returned by `interactive` is a `Widget` object and it has attributes that contain the current result and arguments:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t, x_t = w.result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "w.kwargs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After interacting with the system, we can take the result and perform further computations. In this case, we compute the average positions in \\\\(x\\\\), \\\\(y\\\\) and \\\\(z\\\\)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xyz_avg = x_t.mean(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xyz_avg.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Creating histograms of the average positions (across different trajectories) show that on average the trajectories swirl about the attractors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(xyz_avg[:,0])\n",
    "plt.title('Average $x(t)$');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(xyz_avg[:,1])\n",
    "plt.title('Average $y(t)$');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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