CDS 140a Winter 2013 Homework 7

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

__MATHJAX__

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

3. Perko, Section 3.4, problem 1: Show that $\gamma(t) = (2 \cos 2t, \sin 2t)$ is a periodic solution of the system <amsmath>

    \aligned
      \dot x &= -4y + x\left(1-\frac{x^2}{4} - y^2\right) \\
      \dot x &= x + y\left(1-\frac{x^2}{4} - y^2\right) \\
    \endaligned
   

   </amsmath>

   that lies on the ellipse $(x/2)^2 + y^2 = 1$ (i.e., $\gamma(t)$ represents a cycle $\Gamma$ of this system). Then use the corollary to Theorem 2 in Section 3.4 to show that $\Gamma$ is a stable limit cycle.

4. Perko, Section 3.4, problem 3a: Solve the linear system <amsmath>

    \dot x = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}
   

   </amsmath>

   and show that any at point $(x_0, 0)$ on the $x$-axis, the Poincare map for the focus at the origin is given by $P(x_0) = x_0 \exp(2 \pi a / |b|)$. For $d(x) = P(x) - x$, compute $d'(0)$ and show that $d(-x) = -d(x)$.

5. Perko, Section 3.5, problem 1:
6. Perko, Section 3.5, problem 5:
7. Perko, Section 3.9, problem 4a: