CDS 140a Winter 2013 Homework 7
R. Murray, D. MacMartin | Issued: 19 Feb 2013 (Tue) |
ACM 101/AM 125b/CDS 140a, Winter 2013 | Due: 5 Mar 2013 (Tue) |
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).
- 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?
- 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 ${\mathbb R}^3$).
- 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.
- 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 $d(-x) = -d(x)$.
- 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 $\gamma(t)$?
- 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> - 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 $x = \pm 1$.