CDS 140a Winter 2013 Homework 7 - Revision history

Murray at 16:33, 3 March 2013

2013-03-03T16:33:00Z

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	and		and
	<center><amsmath>		<center><amsmath>
	\prod_{j-1}^n m_j = \exp \int_0^T \text{trace} A(t)\, dt.		\prod_{j=1}^n m_j = \exp \int_0^T \text{trace} A(t)\, dt.
	</amsmath></center>		</amsmath></center>
	</li>		</li>

Murray at 16:32, 3 March 2013

2013-03-03T16:32:15Z

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	to show that if $m_j = e^{\lambda_j T}$, $j = 1, \dots, n$ are the characteristic multipliers of $\gamma(t)$ then		to show that if $m_j = e^{\lambda_j T}$, $j = 1, \dots, n$ are the characteristic multipliers of $\gamma(t)$ then
	<center><amsmath>		<center><amsmath>
	\sum_{j-1}^n m_j = \text{trace} \Phi(T)		\sum_{j=1}^n m_j = \text{trace} \Phi(T)
	</amsmath></center>		</amsmath></center>
	and		and

Murray at 22:53, 2 March 2013

2013-03-02T22:53:23Z

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	(b) Show that there are two invariant cylinders for this system given by $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$.		(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) $W^s(\Gamma_j)$ and $W^u(\Gamma_j)$, $j = 1,2$, for the full system (in ${\mathbb R}^3$.		(c) Describe $W^s(\Gamma_j)$ and $W^u(\Gamma_j)$, $j = 1,2$, for the full system (in ${\mathbb R}^3$).
	</li>		</li>

Murray at 18:53, 19 February 2013

2013-02-19T18:53:48Z

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	~~{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}~~


	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, D. MacMartin		\| instructor = R. Murray, D. MacMartin

Murray at 18:52, 19 February 2013

2013-02-19T18:52:13Z

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	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>
	(a) Show that ~~tehre~~ are two periodic orbits $\Gamma_1$ and $\Gamma_2$ in the $x, y$ plane and determine their stability.		(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 $x^2 + y^2 = 4$.		(b) Show that there are two invariant cylinders for this system given by $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$.

Murray at 02:13, 17 February 2013

2013-02-17T02:13:58Z

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	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>
	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 $\gamma(t)$?		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 $\gamma(t)$?
	</li>		</li>

Murray at 02:12, 17 February 2013

2013-02-17T02:12:30Z

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	<li> '''Perko, Section 3.2, problem 5''':<br>		<li> '''Perko, Section 3.2, problem 5''':<br>

	<p> (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 $\dot x = f(x)$.</p>		<p> (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 $\dot x = f(x)$.</p>

	<p> (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 $A$. </p>		<p> (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 $A$. </p>

Murray at 02:12, 17 February 2013

2013-02-17T02:12:00Z

← Older revision		Revision as of 02:12, 17 February 2013
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	and		and
	<center><amsmath>		<center><amsmath>
	\prod_{j-1}^n m_j = \exp \int_0^T \text{trace}\, dt.		\prod_{j-1}^n m_j = \exp \int_0^T \text{trace} A(t)\, dt.
	</amsmath></center>		</amsmath></center>
	</li>		</li>
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	\dot x &= y + x - x^3/3 \\		\dot x &= y + x - x^3/3 \\
	\dot y &= -x		\dot y &= -x
			\endaligned
	</amsmath></center>		</amsmath></center>
	must cross the vertical lines $x = \pm 1$.		must cross the vertical lines $x = \pm 1$.
	</li>		</li>
	</ol>		</ol>

Murray at 02:11, 17 February 2013

2013-02-17T02:11:00Z

@@ Line 68: / Line 68: @@
 </li>
-<li> '''Perko, Section 3.5, problem 5''':
+<li> '''Perko, Section 3.5, problem 5a''':
 </li>
 <li> '''Perko, Section 3.9, problem 4a''':
 </li>
 </ol>

Murray at 01:21, 17 February 2013

2013-02-17T01:21:27Z

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	\aligned		\aligned
	\dot x &= -4y + x\left(1-\frac{x^2}{4} - y^2\right) \\		\dot x &= -4y + x\left(1-\frac{x^2}{4} - y^2\right) \\
	\dot x &= x + y\left(1-\frac{x^2}{4} - y^2\right) \\		\dot y &= x + y\left(1-\frac{x^2}{4} - y^2\right) \\
	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>
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	\dot x = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}		\dot x = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}
	</amsmath></center>		</amsmath></center>
	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 $d(-x) = -d(x)$.		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 $d(-x) = -d(x)$.
	</li>		</li>

	<li> '''Perko, Section 3.5, problem 1''':		<li> '''Perko, Section 3.5, problem 1''': Show that the nonlinear system
			<center><amsmath>
			\aligned
			\dot x &= -y + x z^2 \\
			\dot y &= x + y z^2 \\
			\dot z &= -z (x^2 + y^2) \\
			\endaligned
			</amsmath></center>
			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 $\gamma(t)$?
	</li>		</li>