CDS 140a Winter 2013 Homework 7

WARNING: This homework set is still being written. Do not start working on these problems until this banner is removed.

R. Murray, D. MacMartin	Issued: 19 Feb 2013 (Tue)
ACM 101/AM 125b/CDS 140a, Winter 2013	Due: 5 Mar 2013 (Tue)

__MATHJAX__

Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading).

1. Perko, Section 3.2, problem 5: (a) According to the corollary of Theorem 2 (in Section 3.2), every $\omega$-limit set is an invariant set of the flow $\phi_t$ of $\dot x = f(x)$. Give an example to show that not every set invariant with respect to the flow $\phi_t$ of $\dot x = f(x)$ is the $\alpha$- or $\omega$-limit set of a trajectory of $\dot x = f(x)$. (b) Any stable limit cycle $\Gamma$ is an attracting set and $\Gamma$ is the $\omega$-limit set of every trajectory in a neighborhood of $\Gamma$. Give an example to show that not every attracting set $A$ is the $\omega$-limit set of a trajectory in a neighborhood of $A$. (c) Is the cylinder in Example 3 of Section 3.2 an attractor for the system in that example?
2. Perko, Section 3.3, problem 8:
3. Perko, Section 3.4, problem 1:
4. Perko, Section 3.4, problem 3:
5. Perko, Section 3.5, problem 1:
6. Perko, Section 3.5, problem 5:
7. Perko, Section 3.9, problem 4a: