Homework 1

The goal of this homework assignment is just to get you started with using the Mathematica computer algebra system to analyze dynamical systems.

Due Date: Friday, September 11

Instructions: Feel free to work together with one or two other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with. You can turn in your homework assignment by e-mailing me your Mathematica notebook, which should include your graphs as well as answers to the questions.

1. Use the NestList and ListLinePlot commands to make a graph showing the first 100 terms of the orbit of \(0.5\) under the function \(f(x) = \sin x\). (That is, make a graph with \(n\) on the horizontal axis and \(f^n(0.5)\) on the vertical axis.)

   Please use the following options to format the output:

   • PlotRange -> {0,1}
   • Mesh -> All
   • ImageSize -> 800
2. 1. Repeat question 1 for the functions \(f(x) = \sin(k x)\), using several different values of \(k\) between 1 and 2.5.
   2. Estimate the value of \(k\) at which the positive fixed point makes the transition from attracting to repelling. Your answer must be correct to within \(0.05\)
3. 1. Make graphs showing the orbit of \(0.5\) under the function \(f(x) = \sin(kx)\) for \(k = 2.5\), \(k = 2.63\), \(k = 2.69\), and \(k=2.71\).
   2. Determine the period of the attracting cycle in each case.
4. 1. Make a graph showing the orbit of \(0.5\) under the function \(f(x) = \sin(3x)\). Does the orbit appear to be chaotic?
   2. Make a single graph showing the orbits of \(0.5\) and \(0.500001\) on the same axes. Does this system exhibit sensitive dependence on initial conditions? Explain.