CDS 140a Winter 2015 Homework 6

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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__

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 3.4, 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} x
    
    </amsmath>

    and show that at any 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.