Dynamical Systems     MATH 374

Homework Assignment #2

Due Thursday, Feb. 6, START of Class

Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to the primary course text by Robert 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 your work. A nonempty subset will be graded. You are encouraged to work on these problems with other classmates, and it is ok to use internet sources for help if it's absolutely necessary; however, the solutions you turn in should be YOUR OWN WORK and written in YOUR OWN WORDS.

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

Read Revolution, the second chapter of Gleick's book on Chaos.

Who is Stephen Smale and why did the National Science Foundation once cancel one of his grants? What discovery did Smale make, described by some as a "paradigm shift"? What was the geometric shape Smale used to model his theory?
Note: Robert Devaney's Ph.D. thesis advisor was Stephen Smale.

Chapter 4 (pp. 34 - 35)
Problems:   1a, 1g, 2a, 4e, 4f, 7 (a, b, c, d, e, g)

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

Note: For problem #4, you may want to use a graphing calculator or Maple to draw accurate graphs. For help with Maple, type ?plot at the command prompt to review how to plot one or more functions.

Additional Problems:

  1. Find all the fixed points and period-two points for F(x) = x2 - 2. Classify each point as attracting, repelling or neutral.
  2. Prove the Repelling Fixed Point Theorem as stated in class on 2/4. Use the proof of the Attracting Fixed Point Theorem as a guide.