Problem Set 6 is the original assignment for challenge problems from the text book. Undergraduates must do one challenge problem and graduate students must do two challenge problems. It appears to me that the easiest challenge problems, partly because they use material that we have covered in class, are #7 and #8. I will provide some additional guidance on these problems. If you have completed other challenge problems or are making good progress others, then carry on ! You needn't solve #7 and/or #8.

1. Challenge problem 7. This is a wonderful problem that combines many ideas that we have studied throughout the semester. As an educational experience, I recommend this problem highest of all. Here are a few other hints.
   • Draw pictures every step of the way. This is a very geometrical proof.
   • In Step 3, make a rough sketch of the graph of T. It must be one-to-one and have a fixed point. Explain why it must be an increasing function.
   • In Steps 5 -11, Figure 7.25 is essential. Sketch your own version of it twice: once with r<q and once with r>q.
   • Steps 7 -9 deal with the case r>q.
   • In Step 8, notice that the dependence on appears through the z in the denominator.
   • In Step 11, the goal is to show that the limit cycle is attracting (equivalently, that the fixed point of T is stable). If the book's hint helps, then use it. You need to show that the trajectory moves outward if and the trajectory moves inward if .
2. Challenge problem 8. The mathematics in this problem is not difficult. However, if you choose this problem, you must provide detailed graphs (preferably computer generated) to show exactly how the torus arises and what it looks like.
   • In Step 4, make a good sketch of the 2-d phase plane. Equation 8.6 is an autonomous ODE that cannot be satisfied by a solution of the form given.
   • In Step 6, make good sketches of the phase plane in 2-d with several different values of c.
   • The whole problem really amounts to Step 8. You need to draw sketeches showing how trajectories pierce slices through the slab . First show what a typical slice looks like in the case that c is rational. Then show several slices in the case that c is irrational. Finally show clearly how the torus arises.