CDS 140a Winter 2015 Homework 6

R. Murray	Issued: 9 Feb 2015
CDS 140, Winter 2015 (PDF)	Due: 18 Feb 2015 at 12:30 pm In class or to box across 107 STL

__MATHJAX__

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1. 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 y &= 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.

2. 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 $d(-x) = -d(x)$.

3. Perko, Section 3.5, problem 1: Show that the nonlinear system <amsmath>

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

   </amsmath>

   has a periodic orbit $\gamma(t) = (\cos t, \sin t, 0)$. Find the linearization of this system about $\gamma(t)$, the fundamental matrix $\Phi(t)$ for the autonomous system that satisfies $\Phi(0) = I$, and the characteristic exponents and multipliers of $\gamma(t)$. What are the dimensions of the stable, unstable and center manifolds of $\gamma(t)$?

4. Perko, Section 3.5, problem 5a: Let $\Phi(t)$ be the fundamental matrix for $\dot x = A(t) x$ satisfying $\Phi(0) = I$. Use Liouville's theorem, which states that <amsmath>

    \det \Phi(t) = \exp \int_0^t \text{trace} A(s) ds,
   

   </amsmath>

   to show that if $m_j = e^{\lambda_j T}$, $j = 1, \dots, n$ are the characteristic multipliers of $\gamma(t)$ then

   <amsmath>

    \sum_{j=1}^n m_j = \text{trace} \Phi(T)
   

   </amsmath>

   and

   <amsmath>

    \prod_{j=1}^n m_j = \exp \int_0^T \text{trace} A(t)\, dt.
   

   </amsmath>
   • Hint: recall that the determinant of a matrix is equal to the product of its eigenvalues, and the trace of a matrix is equal to the sum of the eigenvalues.
5. Perko, Section 3.9, problem 4a: Show that the limit cycle of the van der Pol equation <amsmath>

    \aligned
      \dot x &= y + x - x^3/3 \\
      \dot y &= -x
    \endaligned
   

   </amsmath>

   must cross the vertical lines $x = \pm 1$.

   • Hint: you can use the fact (shown in Perko, Section 3.8) that a limit cycle exists for the van der Pol equation and that it is unique.