Dynamical Systems     MATH 374

Homework Assignment #2

Due Thursday, Feb. 4, 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.

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 3 (pp. 26 - 28)
Problems:   3, 5, 7 (b, c, e, h), 8, 11, 12, 13, 14

Note: For problem #12, delete the phrase "Can you" in the directions. Using your answer to problem #14, explain why the doubling function has a period n cycle for any natural number n.

Chapter 4 (pp. 34 - 35)
Problems:   1a, 1g, 2a, 4a, 4e, 4f, 6, 7 (a, b, c, d, e, g)

Note: For problem #6, change F(x) = x^2 - 1.1 to the function F(x) = x^2 - 2. For problems #7e and #7g, describe the difference in orbits between "a" being positive and "a" being negative.