examples/pynamical-demo-cobweb-plots.ipynb

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    "# Pynamical: cobweb plots 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"
   ]
  },
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   "source": [
    "### To draw these cobweb plots, pynamical does the following:\n",
    "\n",
    "1. Let x = 0.5\n",
    "1. Start on the x-axis at the point (x, 0)\n",
    "1. Draw a vertical line to the red function curve: this point has the coordinates (x, f(x))\n",
    "1. Draw a horizontal line from this point to the gray diagonal line: this point has the coordinates (f(x), f(x))\n",
    "1. Draw a vertical line from this point to the red function curve: this point has the coordinates (f(x), f(f(x)))\n",
    "1. Repeat steps 4 and 5 recursively one hundred times\n",
    "\n",
    "### To draw the red function curve and gray diagonal line, pynamical does the following:\n",
    "\n",
    "Run the logistic map once each for 1000 population values evenly spaced between 0 and 1. This gives us the results of the logistic equation (y values) across the entire range of possible population values (x values). The gray diagonal line is just a plot of y=x."
   ]
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  {
   "cell_type": "code",
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   "source": [
    "import glob\n",
    "import IPython.display as IPdisplay\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from pynamical import cobweb_plot\n",
    "from PIL import Image\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Draw cobweb plots for several values of r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cobweb_plot(r=1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cobweb_plot(r=2.7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cobweb_plot(r=3.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cobweb_plot(r=3.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Demonstrate a basin of attraction with three different initial values and same r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cobweb_plot(r=2.7, cobweb_x=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cobweb_plot(r=2.7, cobweb_x=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cobweb_plot(r=2.7, cobweb_x=0.9)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now animate it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# configure the animation: 70 rates between 0 and 3.999 plus 15 paused frames each at start and end\n",
    "save_folder = 'images/cobweb-animate'\n",
    "gif_filepath = 'images/animated-logistic-cobweb.gif'\n",
    "start_r = 0\n",
    "end_r = 3.999\n",
    "steps = 70\n",
    "rates = np.linspace(start_r, end_r, steps)"
   ]
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   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "# create a plot for each growth rate\n",
    "for n, r in enumerate(rates):\n",
    "    filename = '{:03d}'.format(n)\n",
    "    title = 'Logistic Map Cobweb Plot, r={:.2f}'.format(r)\n",
    "    cobweb_plot(r=r, filename=filename, folder=save_folder, dpi=90, title=title, show=False)\n",
    "    plt.close()"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a tuple of display durations, one for each frame\n",
    "first_last = 100 #show the first and last frames for 100 ms\n",
    "standard_duration = 10 #show all other frames for 10 ms\n",
    "durations = tuple([first_last] + [standard_duration] * (len(rates) - 2) + [first_last])"
   ]
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   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "# load all the static images into a list then save as an animated gif\n",
    "images = [Image.open(image) for image in sorted(glob.glob('{}/*.png'.format(save_folder)))]\n",
    "gif = images[0]\n",
    "gif.info['duration'] = durations #ms per frame\n",
    "gif.info['loop'] = 0 #how many times to loop (0=infinite)\n",
    "gif.save(fp=gif_filepath, format='gif', save_all=True, append_images=images[1:])"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# verify that the number of frames in the gif equals the number of image files and durations\n",
    "Image.open(gif_filepath).n_frames == len(images) == len(durations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "IPdisplay.Image(url=gif_filepath)"
   ]
  },
  {
   "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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