CDS 140a Winter 2014 Homework 7: Difference between revisions
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<!-- 2014 TA comments: People were confused on #3 about the fact that the dimensions of the stable, unstable, and center manifolds added up to more than n for a periodic orbit. --> | |||
<li> '''Perko, Section 3.5, problem 1''': Show that the nonlinear system | <li> '''Perko, Section 3.5, problem 1''': Show that the nonlinear system | ||
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<!-- 2014 TA comments: | |||
* People were also getting stuck on #4 because they did not know that the trace of a matrix is equal to the sum of the eigenvalues. | |||
* A possible hint for future years is 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. | |||
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<li> '''Perko, Section 3.5, problem 5a''': | <li> '''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 | Let $\Phi(t)$ be the fundamental matrix for $\dot x = A(t) x$ satisfying $\Phi(0) = I$. Use Liouville's theorem, which states that | ||
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<!-- 2014 TA comments: | |||
* There was some trouble on #5 regarding showing that the limit cycle crossed both x=+1 AND x=-1 and not just one or the other. | |||
* Perko had a fix for the last issue, in their solution mentioning the symmetry of the Van der Pol equation with respect to x to -x and y to -y. If we wanted to shorten these for the future, perhaps that could be suggested as a hint. | |||
* Also, some people, as in last year, thought that they had to prove the existence of the limit cycle (this is for problem 5). I just told them that for this equation it is known that there already exists one limit cycle and so they could use the fact that it is well known that it has a limit cycle in their proof. To shorten this the question could possibly be reworded so they understand that they could use the fact that the limit cycle exists and that there is only one for this particular system (referring to other sections in Perko if necessary to see that there exists a limit cycle and that there is only one limit cycle). | |||
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<li> '''Perko, Section 3.9, problem 4a''': | <li> '''Perko, Section 3.9, problem 4a''': | ||
Show that the limit cycle of the van der Pol equation | Show that the limit cycle of the van der Pol equation | ||
Latest revision as of 22:50, 8 February 2015
| R. Murray, D. MacMartin | Issued: 18 Feb 2014 (Tue) |
| ACM 101b/AM 125b/CDS 140a, Winter 2014 | Due: 16 Feb 2014 (Wed) @ noon |
__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).
- 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 $x = \pm 1$.
