Homework 1
Math 323, Spring 2011
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: Thursday, February 3
Instructions: Follow the directions below. Feel free to work together with one or two other
students in the class. The assignment you turn in should include explicit answers to the
questions as well as a printout of your Mathematica notebook. If you prefer, you may e-mail
me the assignment as a LATEX, PDF, Word, or ASCII file together with your Mathematica
notebook.
1. Let f : [0, 1] → [0, 1] be the function f (x) = sin x. Use a computer to graph the orbits
of 0.2, 0.4, 0.6, 0.8, and 1 under the function f . (That is, make a graph with n on the
horizontal axis and f n (x) on the vertical axis for the given values of x.) What is the
long-term behavior of these orbits?
2. Do the same for the functions f (x) = sin(ωx), for several values of ω between 1 to 2.5.
How does the value of ω affect the behavior of the orbits? Where is the cutoff between
the two behaviors?
3. Graph orbits of f (x) = sin(ωx) for ω = 2.6 and ω = 2.63. How does the behavior
change between these two values of ω? Do the same for ω = 2.69 and ω = 2.71. How
does the behavior change in this case? (The difference here is subtle, but should be
noticeable.)
4. Graph some orbits of the function f (x) = sin(3x). Do the orbits look chaotic? If you
graph the orbits of nearby points such as 0.5 and 0.5001, you can see the sensitive
dependence on initial conditions.