{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Lorenz Differential Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we start, we import some preliminary libraries. We will also import (below) the accompanying `lorenz.py` file, which contains the actual solver and plotting routine."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from ipywidgets import interactive, fixed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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",
    "Let's change (\\\\(\\sigma\\\\), \\\\(\\beta\\\\), \\\\(\\rho\\\\)) with ipywidgets and examine the trajectories."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "acd69dff8aca4639a53fc6a8fdd60c28",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=10.0, description='sigma', max=50.0), FloatSlider(value=2.666666666666…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from lorenz import solve_lorenz\n",
    "w=interactive(solve_lorenz,sigma=(0.0,50.0),rho=(0.0,50.0))\n",
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the default set of parameters, we see the trajectories swirling around two points, called attractors. "
   ]
  },
  {
   "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": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t, x_t = w.result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'sigma': 10.0, 'beta': 2.6666666666666665, 'rho': 28.0}"
      ]
     },
     "execution_count": 4,
     "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": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xyz_avg = x_t.mean(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(30, 3)"
      ]
     },
     "execution_count": 6,
     "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": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(xyz_avg[:,0])\n",
    "plt.title('Average $x(t)$');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(xyz_avg[:,1])\n",
    "plt.title('Average $y(t)$');"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}