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.