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__

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 ${\mathbb R}^3$).

  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 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.

  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: 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 $\gamma(t)$?

  6. 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>
  7. 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 $x = \pm 1$.