Math 4/5027 - Nonlinear Dynamics and Chaos
Problem Set #1 - Fall 1998

You will typically have two weeks for each assignment. Please do not postpone the assignment until the last minute! All assignments will require reading, contemplation, and then adequate time for working problems. Remember you also need to identify a Challenge problem (one for undergraduates and two for graduate students) to work before the end of the semester. * marks problems for graduate students (extra credit for undergraduates).

1. Reading. We will work through most of Chapter 1 in class, so you will want to read the entire chapter (at least once) as we go.
2. Finding period-2 orbits. Do problem T1.5.
3. Period-1 and -2 for the logistic map. Do problem T1.6.
4. The a=4 case. Do problems T1.8 and T1.9.
5. Itineraries.* Do problem T1.14(a), but find the entire interval LLR.
6. Behavior at a fixed point. Do exercise 1.2(a) (at the end of the chapter).
7. Period-3 orbits.* Do exercise 1.4.
8. Nonperiodic points.* Do exercise 1.8.
9. Another map. Do exercise 1.9.
10. Explicit formula.* Do exercise 1.15. Extra-extra credit: derive this formula!
11. A Sleep Length Model. Suppose the sequence describes the amount of sleep you get on nights . After many nights of observation you notice that
    • if you get less sleep on night n than night n-1, then you will sleep proportionately longer on night n+1 than on night n (as you try to catch up on sleep);
    • if you get more sleep on night n than night n-1, then you will sleep proportionately less on night n+1 than on night n (because you have accumulated excess sleep).

    Find a linear second-order difference equation with one parameter that describes this situation. Then solve the difference equation. Describe your steady state sleep pattern as .