{
 "cells": [
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   "source": [
    "# Pynamical: phase diagrams of the logistic map\n",
    "\n",
    "**Citation info**: Boeing, G. 2016. \"[Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction](http://geoffboeing.com/publications/nonlinear-chaos-fractals-prediction/).\" *Systems*, 4 (4), 37. doi:10.3390/systems4040037.\n",
    "\n",
    "Pynamical documentation: http://pynamical.readthedocs.org\n",
    "\n",
    "This notebook demonstrates several ways to create phase diagrams to visualize system attractors and differentiate random noise from chaos. Phase diagrams embed 1-dimensional data (like time series) from a dynamical system into 2- or 3-dimensional phase space by plotting the value at time t vs the value at time t+1 (vs the value at time t+2, if it's a 3-D plot)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import IPython.display as IPdisplay\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import pynamical\n",
    "from pynamical import simulate, save_fig, phase_diagram, phase_diagram_3d\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "title_font = pynamical.get_title_font()\n",
    "label_font = pynamical.get_label_font()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## First, let's a create a phase diagram to show the logistic map's fixed-point attractor at 0.655 when the growth rate parameter is set to 2.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# draw a phase diagram for 100 generations for the growth rate parameter 2.9\n",
    "# it shows points converging on 0.655 because the logistic map has a fixed-point attractor at 0.655 when r=2.9\n",
    "pops = simulate(num_gens=100, rate_min=2.9, num_rates=1, num_discard=100)\n",
    "phase_diagram(pops, title='Logistic Map Attractor, r=2.9', size=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Next, let's create phase diagrams to show the logistic map's limit cycle attractor when the growth rate parameter is set to 3.50 - 3.57"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# draw a phase diagram for 100 generations for the growth rate parameter 3.5\n",
    "# it shows 4 points because the logistic map has a period of 4 when r=3.5\n",
    "pops = simulate(num_gens=100, rate_min=3.5, num_rates=1, num_discard=100)\n",
    "phase_diagram(pops, title='Logistic Map Attractor, r=3.5', size=20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# draw a phase diagram for 100 generations for the growth rate parameter 3.56\n",
    "# it shows 8 points because the logistic map has a period of 8 when r=3.56\n",
    "pops = simulate(num_gens=100, rate_min=3.56, num_rates=1, num_discard=100)\n",
    "phase_diagram(pops, title='Logistic Map Attractor, r=3.56', size=20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# draw a phase diagram for 100 generations for the growth rate parameter 3.57\n",
    "# it shows n points because the logistic map has a period of n when r=3.57\n",
    "pops = simulate(num_gens=100, rate_min=3.57, num_rates=1, num_discard=100)\n",
    "phase_diagram(pops, title='Logistic Map Attractor, r=3.57', size=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now, let's create a phase diagram to show the logistic map's strange attractor when the growth rate parameter is set to 3.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# draw a phase diagram for 2,000 generations for the growth rate parameter 3.9\n",
    "# the plot reveals the strange attractor - the logistic map is chaotic when r=3.9\n",
    "pops = simulate(num_gens=2000, rate_min=3.9, num_rates=1)\n",
    "phase_diagram(pops, xmin=0.25, xmax=0.75, ymin=0.8, ymax=1.01, size=20, title='Logistic Map Attractor, r=3.9')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now, let's create a phase diagram to show the logistic map's strange attractors across the chaotic regime (from r=3.6 to r=4.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# draw a phase diagram for 2,000 generations across 50 growth rate steps from 3.6 to 4.0\n",
    "# each chaotic growth rate has its own parabola\n",
    "pops = simulate(num_gens=2000, rate_min=3.6, rate_max=4.0, num_rates=50)\n",
    "phase_diagram(pops, xmin=0.25, xmax=0.75, ymin=0.8, ymax=1.01, size=7, \n",
    "              title='Logistic Map Attractor, r=3.6 to r=4.0', color='viridis')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Next, let's demonstrate how to use phase diagrams to differentiate chaos from random noise in time series data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# sometimes it is hard to tell if a time series is chaotic or random\n",
    "# generate two time series of 1,000 steps, one chaotic and one random\n",
    "# generate 30,000 time steps for the chaotic series but only keep the final 1,000 (when system is fully evolved)\n",
    "total_gens = 30000\n",
    "gens = 1000\n",
    "np.random.seed(1)\n",
    "\n",
    "chaos_pops = simulate(num_gens=total_gens, rate_min=3.99, num_rates=1)\n",
    "chaos_pops = chaos_pops.iloc[total_gens-gens:].reset_index().drop(labels='index', axis=1)\n",
    "\n",
    "random_pops = pd.DataFrame(np.random.random(gens), columns=['value'])\n",
    "time_series = pd.concat([chaos_pops, random_pops], axis=1)\n",
    "time_series.columns = ['chaos', 'random']\n",
    "time_series.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot the chaotic and random time series to show how they are sometimes tough to differentiate\n",
    "ax = time_series.plot(kind='line', figsize=[10, 6], linewidth=3, alpha=0.6, style=['#003399','#cc0000'])\n",
    "ax.grid(True)\n",
    "ax.set_xlim(40, 90)\n",
    "ax.set_ylim(0, 1)\n",
    "ax.set_title('Time Series, Deterministic Chaos vs Random Data', fontproperties=title_font)\n",
    "ax.set_xlabel('Generation', fontproperties=label_font)\n",
    "ax.set_ylabel('Population', fontproperties=label_font)\n",
    "ax.legend(loc=3)\n",
    "\n",
    "save_fig('chaos-vs-random-line')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot same data as 2D phase diagram instead\n",
    "pops = pd.concat([chaos_pops, random_pops], axis=1)\n",
    "pops.columns = ['chaos', 'random']\n",
    "phase_diagram(pops, size=20, color=['#003399','#cc0000'], ymax=1.005, legend=True, \n",
    "              filename='logistic-attractor-chaos-random')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now, let's create 3-D phase diagrams to show the same chaos vs random noise, in three dimensions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot same data as 3D phase diagram instead\n",
    "phase_diagram_3d(pops, color=['#003399','#cc0000'], filename='logistic-attractor-chaos-random-3d',\n",
    "                 legend=True, legend_bbox_to_anchor=(0.94, 0.9))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create a 3-D phase diagram to show the logistic map's strange attractors across the chaotic regime (from r=3.6 to r=4.0), twisting and curling around their state space in three dimensions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# run logistic model for 4,000 generations across 50 growth rate steps from 3.6 to 4.0\n",
    "pops = simulate(num_gens=4000, rate_min=3.6, rate_max=4.0, num_rates=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# phase diagram: each chaotic growth rate has its own strange attractor curling through state space\n",
    "phase_diagram_3d(pops, title='Logistic Map Attractor, r=3.6 to r=4.0', alpha=0.1, color='viridis', color_reverse=False, \n",
    "                 azim=230, filename='3d-logistic-map-attractor-1')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# phase diagram: each chaotic growth rate has its own strange attractor curling through state space\n",
    "phase_diagram_3d(pops, title='Logistic Map Attractor, r=3.6 to r=4.0', alpha=0.1, color='viridis', color_reverse=False, \n",
    "                 elev=7, azim=320, filename='3d-logistic-map-attractor-2')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## In another notebook, I create animated gifs by panning and rotating around my 3-D phase diagrams to reveal more of their structure:\n",
    "[pynamical-demo-3d-animation.ipynb](pynamical-demo-3d-animation.ipynb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# here's an example of the animated phase diagrams that I create in pynamical-demo-3d-animation.ipynb\n",
    "IPdisplay.Image(url='images/phase-animate/05-logistic-3d-phase-diagram-chaotic-regime.gif')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## For more info:\n",
    "  - [Read the journal article](http://geoffboeing.com/publications/nonlinear-chaos-fractals-prediction/)\n",
    "  - [Pynamical documentation](http://pynamical.readthedocs.org)\n",
    "  - [Chaos Theory and the Logistic Map](http://geoffboeing.com/2015/03/chaos-theory-logistic-map/)\n",
    "  - [Visualizing Chaos and Randomness with Phase Diagrams](http://geoffboeing.com/2015/04/visualizing-chaos-and-randomness/)\n",
    "  - [Animated 3D Plots in Python](http://geoffboeing.com/2015/04/animated-3d-plots-python/)"
   ]
  }
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