{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["# Homework 5.1: Ruling out oscillations (30 pts)\n", "\n", "*This problem is based on problem 3.6 of [Del Vecchio and Murray](http://www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page).*\n", "\n", "\n", "
"]}, {"cell_type": "markdown", "metadata": {}, "source": ["An important result from dynamical systems known as the **Bendixon criterion** makes it possible to rule out sustained oscillations (defined as **orbits**, closed curves in the phase plane on which trajectories remain after entering). \n", "\n", "Consider a dynamical system\n", "\n", "\\begin{align}\n", "&\\frac{\\mathrm{d}x}{\\mathrm{d}t} = f(x, y)\\\\[1em]\n", "&\\frac{\\mathrm{d}y}{\\mathrm{d}t} = g(x,y).\n", "\\end{align}\n", "\n", "In a simply connected region $D$ of the $x$-$y$ plane, if the quantity\n", "\n", "\\begin{align}\n", "\\frac{\\partial f}{\\partial x} + \\frac{\\partial g}{\\partial y}\n", "\\end{align}\n", "\n", "is nonzero and does not change sign on $D$, then the dynamical system has no orbits entirely in $D$. To illustrate the power of this, I really like the following exercise from [Del Vecchio and Murray's book](http://www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page) in which they ask you to eliminate circuits below that cannot have sustained oscillations by using Bendixson's criterion. (Image below, (c) Princeton University Press.)\n", "\n", "\n", " \n", "\n", "\n", "
\n", "\n", "Identify the circuits that cannot have sustained oscillations by Bendixson's criterion."]}, {"cell_type": "markdown", "metadata": {}, "source": ["
"]}], "metadata": {"anaconda-cloud": {}, "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": "3.7.7"}}, "nbformat": 4, "nbformat_minor": 4}