Due April 6

Please do not postpone the assignment until the last minute! 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. Please finish reading Chapter 7, particularly sections 7.6 and 7.7. Everyone should read sections 8.1 and 8.3; graduate students should spend some time with the proof of the Poincare-Bendixson Theorem in section 8.3. We will then move on to Chapter 9: Chaos in Differential Equations.

   Please do the following problems.

2. Verifying a Lyapunov function. Please do problem T7.13.
3. Finding a Lyapunov function. Please do problem T7.14.
4. Geometry of a Lyapunov function. For the previous problem (T7.14), draw a good three-dimensional sketch of the surface z=E(x,y), a part of the trajectory of the system through the point (2,2), the gradient , the angle between and . Evaluate in radians. Explain geometrically why (0,0) is globally asymptotically stable.
5. *Finding a Lyapunov function. Please do problem 7.9.
6. Potential functions. Please do problem 7.10.
7. limit sets. Please do problem T8.2.
8. *Closure of limit sets. Please do problem T8.3.
9. Finding limit sets. Please do problem 8.2.

Just For Fun

A set of dominoes all having the same thickness, 1 inch wide, and 2 inches long. The dominoes are stacked on top of each other with their long edges aligned so that each domino overhangs the one beneath it. If there are n dominoes in the stack, what is the largest distance that the top domino can be made to overhang the bottom domino? How many dominos can be stacked altogether before the stack topples?