Homework 1
Math 323, Fall 2014
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 12
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
PlotRange -> {0,1}, Mesh -> All, and ImageSize -> 800 options to format the
output.
2. (a) Repeat question 1 for the functions f (x) = sin(kx), using several different values
of k between 1 and 2.5.
(b) Estimate the 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. (a) 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.
(b) Determine the period of the attracting cycle in each case.
4. (a) Make a graph showing the orbit of 0.5 under the function f (x) = sin(3x). Does
the orbit appear to be chaotic?
(b) 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.