Math 4791/5791 - Problem Set 4
Fall 1998

Due October 24 (one week from today)

1. Be sure you have read Chapter 5 in its entirety. It gives a very good survey of a topic that often consumes many more pages.
2. Detailed analysis of a linear system. Please do problem 5.1.9 of the text.
3. Some terminology. Read 5.1.10 for some useful terminology.
4. Complex eigenvalues. Consider the system

   

   where a and b are real numbers.

   1. Analyze this system in the manner described in Problem 5.2.2 in which the eigenvalues and eigenvectors are determined explicitly. Note that when the eigenvalues are complex they appear in conjugate pairs. Eigenvectors associated with a pair of complex eigenvalues are also complex conjugates of each other. Finally, it helps to use the fact that if x is a complex quantity, then .
   2. *Analyze this system by letting z(t)=x(t)+iy(t) and . Show that the system can be reduced to a single first order ODE in z.
   3. In both cases show that (i) if a=0 the trajectories in the phase place are circles, (ii) if a>0 the trajectories are spirals that expand, and (iii) if a<0 the trajectories are spirals that contract on the origin.
5. Classification of systems. Please do problems 5.2.3 and 5.2.8.
6. *A Pursuit Problem. A dog walks north from a crossroads at 1 mile per hour. The dog's master begins one mile east of the crossroads and walks at all times directly at the dog with a speed of s>1 miles per hour. When and where does the master overtake the dog? Please get started on this one immediately. To first goal is to find an ODE (second order nonlinear) that describes the dog's path in the plane. The second goal is to solve it!