{ "cells": [ { "cell_type": "markdown", "metadata": {}, "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": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from IPython.html.widgets import interact, interactive\n", "from IPython.display import clear_output, display, HTML" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "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": 4, "metadata": { "collapsed": false }, "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 Lorentz system.\"\"\"\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.jet(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. 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"AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAQ+L/AQTPKT0WqXhXAAAAAElFTkSuQmCC\n" ], "text/plain": [ "" ] }, "metadata": { "image/png": { "height": 459, "width": 734 } }, "output_type": "display_data" } ], "source": [ "w = interactive(solve_lorenz, angle=(0.,360.), 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": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "t, x_t = w.result" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "{u'N': 25,\n", " u'angle': 180.0,\n", " u'beta': 2.6666666666666665,\n", " u'max_time': 4.0,\n", " u'rho': 25.0,\n", " u'sigma': 25.0}" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "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": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "xyz_avg = x_t.mean(axis=1)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(25, 3)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "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": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABJQAAAMPCAYAAACDrI9LAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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