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  {
   "cell_type": "markdown",
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   "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": null,
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t, x_t = w.result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xyz_avg = x_t.mean(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(xyz_avg[:,0])\n",
    "plt.title('Average $x(t)$');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(xyz_avg[:,1])\n",
    "plt.title('Average $y(t)$');"
   ]
  }
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