CDS 140a Winter 2015 Homework 6 - Revision history

Murray at 22:06, 22 February 2015

2015-02-22T22:06:57Z

← Older revision		Revision as of 22:06, 22 February 2015
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	<ol>		<ol>
			<!-- 2015 comment: this problem is a bit too simple. Not much insight -->
	<li> '''Perko, Section 3.4, problem 1''': Show that $\gamma(t) = (2 \cos 2t, \sin 2t)$ is a periodic solution of the system		<li> '''Perko, Section 3.4, problem 1''': Show that $\gamma(t) = (2 \cos 2t, \sin 2t)$ is a periodic solution of the system
	<center><amsmath>		<center><amsmath>
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	</li>		</li>

			<!-- 2015 comment: this problem is slightly confusing if you think about it, and doesn't really make much sense without part b in the tet -->
	<li> '''Perko, Section 3.4, problem 3a''': Solve the linear system		<li> '''Perko, Section 3.4, problem 3a''': Solve the linear system
	<center><amsmath>		<center><amsmath>

Murray at 23:48, 10 February 2015

2015-02-10T23:48:51Z

← Older revision		Revision as of 23:48, 10 February 2015
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	<li> '''Perko, Section 3.4, problem 3a''': Solve the linear system		<li> '''Perko, Section 3.4, problem 3a''': Solve the linear system
	<center><amsmath>		<center><amsmath>
	\dot x = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}		\dot x = \begin{bmatrix} a & -b \\ b & a \end{bmatrix} x
	</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 at any 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>

Murray at 03:02, 10 February 2015

2015-02-10T03:02:57Z

← Older revision		Revision as of 03:02, 10 February 2015
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	\| pdf = cds140-wi15_hw6.pdf		\| pdf = cds140-wi15_hw6.pdf
	}} __MATHJAX__		}} __MATHJAX__
	{{warning\|This problem set is under construction. This banner will be removed when the problems are finalized.}}		<!-- {{warning\|This problem set is under construction. This banner will be removed when the problems are finalized.}} -->

	'''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		'''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

Murray at 22:57, 8 February 2015

2015-02-08T22:57:09Z

← Older revision		Revision as of 22:57, 8 February 2015
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	</amsmath></center>		</amsmath></center>
	must cross the vertical lines $x = \pm 1$.		must cross the vertical lines $x = \pm 1$.
	* Hint: you can use the fact (shown in Perko) that a limit cycle exists and that it is unique.		* Hint: you can use the fact (shown in Perko, Section 3.8) that a limit cycle exists for the van der Pol equation and that it is unique.
	</li>		</li>
	</ol>		</ol>

Murray: Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #6 | issued = 9 Feb 2015 | due = 18 Feb 2015 at 12:30 pm
In ..."

2015-02-08T22:53:24Z

Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #6 | issued = 9 Feb 2015 | due = 18 Feb 2015 at 12:30 pm<br>In ..."

New page

{{CDS homework
| instructor = R. Murray
| course = CDS 140
| semester = Winter 2015
| title = Problem Set #6
| issued = 9 Feb 2015
| due = 18 Feb 2015 at 12:30 pm<br>In class or to box across 107 STL
| pdf = cds140-wi15_hw6.pdf
}} __MATHJAX__
{{warning|This problem set is under construction. This banner will be removed when the problems are finalized.}}

'''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).

<ol>
<li> '''Perko, Section 3.4, problem 1''': Show that $\gamma(t) = (2 \cos 2t, \sin 2t)$ is a periodic solution of the system
<center><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></center>
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.
</li>

<li> '''Perko, Section 3.4, problem 3a''': Solve the linear system
<center><amsmath>
\dot x = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}
</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 $d(-x) = -d(x)$.
</li>


<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> '''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
<center><amsmath>
\det \Phi(t) = \exp \int_0^t \text{trace} A(s) ds,
</amsmath></center>
to show that if $m_j = e^{\lambda_j T}$, $j = 1, \dots, n$ are the characteristic multipliers of $\gamma(t)$ then
<center><amsmath>
\sum_{j=1}^n m_j = \text{trace} \Phi(T)
</amsmath></center>
and
<center><amsmath>
\prod_{j=1}^n m_j = \exp \int_0^T \text{trace} A(t)\, dt.
</amsmath></center>
* Hint: recall 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.
</li>


<li> '''Perko, Section 3.9, problem 4a''':
Show that the limit cycle of the van der Pol equation
<center><amsmath>
\aligned
\dot x &= y + x - x^3/3 \\
\dot y &= -x
\endaligned
</amsmath></center>
must cross the vertical lines $x = \pm 1$.
* Hint: you can use the fact (shown in Perko) that a limit cycle exists and that it is unique.
</li>
</ol>

CDS 140a Winter 2015 Homework 6 - Revision history

Murray at 22:06, 22 February 2015

Murray at 23:48, 10 February 2015

Murray at 03:02, 10 February 2015

Murray at 22:57, 8 February 2015

Murray: Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #6 | issued = 9 Feb 2015 | due = 18 Feb 2015 at 12:30 pmIn ..."

Murray: Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #6 | issued = 9 Feb 2015 | due = 18 Feb 2015 at 12:30 pm
In ..."