CDS 140a Winter 2013 Homework 7: Difference between revisions
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<li> '''Perko, Section 3.2, problem 5''': | <li> '''Perko, Section 3.2, problem 5''':<br> | ||
(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 $\phi_t$ of $\dot x = f(x)$ is the $\alpha$- or $\omega$-limit set of a trajectory of $\dot x = f(x)$. | (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 $\phi_t$ of $\dot x = f(x)$ is the $\alpha$- or $\omega$-limit set of a trajectory of $\dot x = f(x)$. | ||
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(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 $A$. | (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 $A$. | ||
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(c) Is the cylinder in Example 3 of Section 3.2 an attractor for the system in that example? | (c) Is the cylinder in Example 3 of Section 3.2 an attractor for the system in that example? | ||
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Revision as of 01:04, 17 February 2013
| 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).
- 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 $\phi_t$ of $\dot x = f(x)$ 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:
- Perko, Section 3.4, problem 1:
- Perko, Section 3.4, problem 3:
- Perko, Section 3.5, problem 1:
- Perko, Section 3.5, problem 5:
- Perko, Section 3.9, problem 4a:
