examples/pynamical-demo-other-models.ipynb

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   "source": [
    "# Pynamical: demo of the Cubic Map, Singer Map, and Mandelbrot 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",
    "Pynamical can simulate, visualize, and explore any discrete dynamical system. This notebook demonstrates how to do this with two additional models that come included in the Pynamical module (the *Cubic Map* and the *Singer Map*) as well as with one additional model defined in this notebook (the *Mandelbrot Map*). Any other models can be created and easily plugged-in, as demonstrated below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numba import jit\n",
    "from pynamical import simulate, cubic_map, singer_map, bifurcation_plot, phase_diagram, phase_diagram_3d\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cubic map\n",
    "\n",
    "First we visualize the behavior of the *Cubic Map*. To do this, our code is nearly identical to the previous [examples using the Logistic Map](pynamical-demo-logistic-model.ipynb), with the exception that we are now passing `model=cubic_map` into our `simulate` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=cubic_map, num_gens=100, rate_min=1, rate_max=4, num_rates=1000, num_discard=100)\n",
    "bifurcation_plot(pops, title='Cubic Map Bifurcation Diagram', xmin=1, xmax=4, save=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=cubic_map, num_gens=1000, rate_min=3.99, num_rates=1, num_discard=100)\n",
    "phase_diagram(pops, xmin=-1, xmax=1, ymin=-1, ymax=1, save=False, title='Cubic Map 2D Phase Diagram')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=cubic_map, num_gens=1000, rate_min=3.99, num_rates=1, num_discard=100)\n",
    "phase_diagram_3d(pops, xmin=-1, xmax=1, ymin=-1, ymax=1, zmin=-1, zmax=1, save=False, title='Cubic Map 3D Phase Diagram')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=cubic_map, num_gens=3000, rate_min=3.5, num_rates=30, num_discard=100)\n",
    "phase_diagram_3d(pops, xmin=-1, xmax=1, ymin=-1, ymax=1, zmin=-1, zmax=1, save=False, alpha=0.2, color='viridis',\n",
    "                 azim=330, title='Cubic Map 3D Phase Diagram, r = 3.5 to 4.0')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Singer Map\n",
    "\n",
    "Next we visualize the behavior of the *Singer Map*. We pass `model=singer_map` into our `simulate` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=singer_map, num_gens=100, rate_min=0.9, rate_max=1.08, num_rates=1000, num_discard=100)\n",
    "bifurcation_plot(pops, title='Singer Map Bifurcation Diagram', xmin=0.9, xmax=1.08, save=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=singer_map, num_gens=1000, rate_min=1.07, num_rates=1, num_discard=100)\n",
    "phase_diagram(pops, title='Singer Map 2D Phase Diagram, r = 1.07', save=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=singer_map, num_gens=1000, rate_min=1.07, num_rates=1, num_discard=100)\n",
    "phase_diagram_3d(pops, title='Singer Map 3D Phase Diagram, r = 1.07', save=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=singer_map, num_gens=3000, rate_min=1.04, rate_max=1.07, num_rates=30, num_discard=100)\n",
    "phase_diagram_3d(pops, save=False, alpha=0.2, color='viridis', title='Singer Map 3D Phase Diagram, r = 1.04 to 1.07')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ...Or define your own model\n",
    "\n",
    "Here we define the equation for the *Mandelbrot Map* and then visualize its behavior (use jit to compile it for fast simulation)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "@jit(nopython=True)\n",
    "def mandelbrot_map(pop, rate):\n",
    "    return pop ** 2 + rate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=mandelbrot_map, num_gens=100, rate_min=-2, rate_max=0, num_rates=1000, num_discard=100)\n",
    "bifurcation_plot(pops, title='Mandelbrot Map Bifurcation Diagram', xmin=-2, xmax=0, ymin=-2, ymax=2, save=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=mandelbrot_map, num_gens=1000, rate_min=-1.99, num_rates=1, num_discard=100)\n",
    "phase_diagram(pops, title='Mandelbrot Map 2D Phase Diagram', xmin=-2, xmax=2, ymin=-2, ymax=2, save=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=mandelbrot_map, num_gens=1000, rate_min=-1.99, num_rates=1, num_discard=100)\n",
    "phase_diagram_3d(pops, title='Mandelbrot Map 3D Phase Diagram', xmin=-2, xmax=2, ymin=-2, ymax=2, zmin=-2, zmax=2, save=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pops = simulate(model=mandelbrot_map, num_gens=3000, rate_min=-1.99, rate_max=-1.5, num_rates=30, num_discard=100)\n",
    "phase_diagram_3d(pops, title='Mandelbrot Map 3D Phase Diagram', alpha=0.2, color='viridis', color_reverse=True,\n",
    "                 xmin=-2, xmax=2, ymin=-2, ymax=2, zmin=-2, zmax=2, save=False)"
   ]
  },
  {
   "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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