Dynamical Systems     MATH 374

Homework Assignment #3

Due Thursday, Feb. 18, START of Class

Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the primary textbook by Devaney. Unless otherwise indicated, all parts of a problem (a), (b), etc. should be completed. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be YOUR OWN WORK.

Note: Please list the names of any students or faculty who you worked with on the assignment.

Life's Ups and Downs
Read Life's Ups and Downs, the third chapter of Gleick's book on Chaos.

This is an important chapter with some crucial ideas we will explore with more mathematical rigor than presented here. Who wrote the paper "Period Three Implies Chaos?" (careful here -- there are two authors.) This famous article is the first journal paper to use the word "chaos" in its title. What was the main result in the paper? Who also had proven a similar result 11 years earlier and what country was he from? Why was his work unknown to the authors of "Period Three Implies Chaos?"

Chapter 5 (pp. 50 - 51)
Problems:   1a, 1c, 1f, 2c, 2f, 4a, 4f, 5, 7

Note: For problem #4, you may want to use a graphing calculator or Maple to draw accurate graphs. For help with Maple you can type "?plot" to review how to plot functions in Maple.

Chapter 6 (pp. 67 - 68)
Problems:   1a, 1b, 1c, 3, 4, 5, 6, 7, 8

Note: For the directions to problem #1, ignore the "sketch the phase portrait" instruction. Instead, describe in words the changes that take place before, at and after the bifurcation.

Additional Problem:
Prove the Repelling Fixed Point Theorem as stated in class on 2/2. Use the proof of the Attracting Fixed Point Theorem as a guide.