CDS 140a Winter 2014 Homework 6

R. Murray, D. MacMartin	Issued: 11 Feb 2014 (Tue)
ACM 101b/AM 125b/CDS 140a, Winter 2014 (PDF)	Due: 19 Feb 2014 (Wed) @ noon

__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 $\phi_t$ 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: Consider the system <amsmath>

    \aligned
      \dot x &= -y + x(1-x^2 - y^2)(4 - x^2 - y^2) \\
      \dot y &= x + y(1-x^2 - y^2)(4 - x^2 - y^2) \\
      \dot z  &= z.
    \endaligned
   

   </amsmath>

   (a) Show that there are two periodic orbits $\Gamma_1$ and $\Gamma_2$ in the $x, y$ plane and determine their stability.

   (b) Show that there are two invariant cylinders for this system given by $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$.

   (c) Describe $W^s(\Gamma_j)$ and $W^u(\Gamma_j)$, $j = 1,2$, for the full system (in ${\mathbb R}^3$).