From 05e18af169c5a32fa44c16d62bee37035513b756 Mon Sep 17 00:00:00 2001
From: Mitchell Welch <mwelch8@Mitchells-MBP-3.gateway>
Date: Thu, 11 Jun 2020 20:16:18 +1000
Subject: [PATCH] Adding another notebook
---
Lorenz_2.ipynb | 170 +++++++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 170 insertions(+)
create mode 100644 Lorenz_2.ipynb
diff --git a/Lorenz_2.ipynb b/Lorenz_2.ipynb
new file mode 100644
index 0000000..d828e58
--- /dev/null
+++ b/Lorenz_2.ipynb
@@ -0,0 +1,170 @@
+{
+ "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": 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 $y(t)$');"
+ ]
+ }
+ ],
+ "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": "3.7.6"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
--
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