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.

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 that we will explore later with more mathematical rigor. 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 the paper, "Period Three Implies Chaos?"

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 Problems:

1. Suppose that x₀ is a neutral fixed point with f'(x₀) = -1. Find necessary and sufficient conditions involving the second and third derivatives of f at x₀ that guarantee it is a weakly attracting fixed point.   Hint: Use f²(x) and its higher derivatives at x₀.
2. This problem concerns the cosine family C_k(x) = k cos(x).
   (a)Find the exact value of k, where 1 < k < 2, for which 0 is on a super-attracting period 2-cycle. Be sure to show that the cycle is super-attracting.

   (b) As k increases from 1, C_k(x) undergoes a period-doubling bifurcation for some value of k where 1 < k < 1.4. Use Maple to find this bifurcation value accurate to 14 decimal places.

   (c) As k increases from 1.4, C_k(x) undergoes another period-doubling bifurcation for some value of k where 1.4 < k < 1.9. Use Maple to find this bifurcation value accurate to 14 decimal places.