CDS 140a Winter 2013 Homework 1 - Revision history

Murray at 18:49, 15 January 2013

2013-01-15T18:49:18Z

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	\| due = 15 Jan 2013		\| due = 15 Jan 2013
	}} __MATHJAX__		}} __MATHJAX__

			<!--
			For future years:
			* Many of the students in the class have ''not'' had ACM 125a/CDS 101. They have not done proofs, Cauchy convergence, etc.
			* Would be worth having Friday recitation provide background on relevant ACM 125a topics and proof techniques
			* Think about whether we want to require CDS 201 for CDS 140 (but then aero students may never take the class)
			-->

	'''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 01:54, 13 January 2013

2013-01-13T01:54:12Z

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	</li>		</li>

	<li> (Based on Perko, Section 1.4, Exercise 6) Let $E \subset {\mathbb R}^n$ an invariant of $A:{\mathbb R}^n \to {\mathbb R}^n$ (i.e., for all $x \in E$, $A x \in E$). Show that if $x(t)$ is the solution of the initial value problem		<li> (Based on Perko, Section 1.4, Exercise 6) Let $E \subset {\mathbb R}^n$ an invariant subspace of $A:{\mathbb R}^n \to {\mathbb R}^n$ (i.e., for all $x \in E$, $A x \in E$). Show that if $x(t)$ is the solution of the initial value problem
	\begin{align}		\begin{align}
	\dot x &= A x &\qquad x(0) &= x_0		\dot x &= A x &\qquad x(0) &= x_0

Murray at 19:49, 27 December 2012

2012-12-27T19:49:34Z

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	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, D. MacMartin		\| instructor = R. Murray, D. MacMartin
	\| course = ACM 101/AM ~~125a~~/CDS 140a		\| course = ACM 101/AM 125b/CDS 140a
	\| semester = Winter 2013		\| semester = Winter 2013
	\| title = Problem Set #1		\| title = Problem Set #1

Murray at 19:49, 27 December 2012

2012-12-27T19:49:00Z

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	</li>		</li>

	<li> '''Perko, Section 1.5, Exercise 4:'''		<li> '''Perko, Section 1.5, Exercise 1:''' Use the theorem in Seciton 1.5 to determine if the linear system $\dot x = A x$ has a saddle, node, focus or center at the origin and determine the stability of each node or focus:
			{\| style="margin: 1em auto 1em auto"
			\|-
			\| (a) $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
			\| width=10% \|
			\| (b) $A = \begin{bmatrix} \lambda & -2 \\ 1 & \lambda \end{bmatrix}$
			\| width=10% \|
			\| (c) $A = \begin{bmatrix} \lambda & 2 \\ 1 & \lambda \end{bmatrix}$
			\|}
			(If your answer depends on the value of a parameter, make sure to describe all possible cases.)
	</li>		</li>

	<li> '''Perko, Section 1.5, Exercise 9:'''		<li> '''Perko, Section 1.6, Exercise 2:''' Solve the initial value problem
	~~</li>~~		\begin{align}
			\dot x &= A x &\qquad x(0) &= x_0
	~~<li> '''Perko,~~ Section 1.6~~, Exercise 2:'''~~		\end{align}
			with
			\begin{align}
			A &= \begin{bmatrix} 0 & -2 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & -2 \end{bmatrix}.
			\end{align}
			Determine the stable and unstable subspaces and sketch the (3D) phase portrait. (Hint: see Figure 1 in Section 1.6 for an example of a 3D phase portrait.)
	</li>		</li>
	</ol>		</ol>

Murray at 19:34, 27 December 2012

2012-12-27T19:34:39Z

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	</li>		</li>

	<li> '''Perko, Section 1.2, Exercise 6:'''		<li> '''Perko, Section 1.2, Exercise 6:''' Let the $n \times n$ matrix $A$ have real, distinct eigenvalues. Let $\phi(t, x)$ the the solution of the initial value problem
			\begin{align}
			\dot x &= A x &\qquad x(0) &= x_0.
			\end{align}
			Show that for each fixed $t \in {\mathbb R}$,
			\begin{align}
			\lim_{y_0 \to x_0} \phi(t, y_0) = \phi(t, x_0).
			\end{align}
			This shows that the solution $\phi(t, x_0)$ is a continuous function of the initial condition.
	</li>		</li>
	<li> ~~'''~~Perko, Section 1.3, Exercise 5, ~~parts~~ a, c, e~~:'''~~		<li> (Based on Perko, Section 1.3, Exercise 5, 6)
			<br> (a) For each matrix below, find the eigenvalues of $A$ and $e^A$:
			{\| style="margin: 1em auto 1em auto"
			\|-
			\| (i) $\begin{bmatrix} a & 0 \\ 0 & -b \end{bmatrix}$
			\| width=10% \|
			\| (ii) $\begin{bmatrix} 1 & 0 \\ a & 1 \end{bmatrix}$
			\| width=10% \|
			\| (iii) $\begin{bmatrix} 5 & -6 \\ 3 & -4 \end{bmatrix}$
			\|}

			(b) Show that if $x$ is the eigenvector of $A$ corresponding to the eigenvalue $\lambda$, then $x$ is also an eigenvector of $e^A$ corresponding to the eigenvalue $e^\lambda$.
			<br>
			(c) If $A = P \text{diag} [\lambda_j] P^{-1}$, use Corollary 1 in Section 1.3 to show that
			\begin{align}
			\det\, e^A = e^{\text{trace}\, A}
			\end{align}
	</li>		</li>
	<li> '''Perko, Section 1.3, Exercise ~~6 (for part a, use 5a, 5c, 5e)~~:'''
			<li> '''Perko, Section 1.4, Exercise 4:''' Given
			\begin{align}
			A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 1 & 0 & -1 \end{bmatrix},
			\end{align}
			compute the $3 \times 3$ matrix $e^{At}$ and solve $\dot x = A x$.
	</li>		</li>
	<li> ~~'''~~Perko, Section 1.4, Exercise 4:~~'''~~
	~~</li>~~		<li> (Based on Perko, Section 1.4, Exercise 6) Let $E \subset {\mathbb R}^n$ an invariant of $A:{\mathbb R}^n \to {\mathbb R}^n$ (i.e., for all $x \in E$, $A x \in E$). Show that if $x(t)$ is the solution of the initial value problem
	~~<li> '''Perko~~, ~~Section 1~~.~~4, Exercise 6:'''~~		\begin{align}
			\dot x &= A x &\qquad x(0) &= x_0
			\end{align}
			with $x_0 \in E$, then $x(t) \in E$ for all $t \in {\mathbb R}$.
	</li>		</li>

	<li> '''Perko, Section 1.5, Exercise 4:'''		<li> '''Perko, Section 1.5, Exercise 4:'''
	</li>		</li>

	<li> '''Perko, Section 1.5, Exercise 9:'''		<li> '''Perko, Section 1.5, Exercise 9:'''
	</li>		</li>

	<li> '''Perko, Section 1.6, Exercise 2:'''		<li> '''Perko, Section 1.6, Exercise 2:'''
	</li>		</li>

Murray at 18:25, 27 December 2012

2012-12-27T18:25:23Z

@@ Line 18: / Line 18: @@
 where $a$ is a constant.  Sketch the phase portraits for $a = -1$, $a = 0$ and $a = 1$ and notice the qualitative structure of the phase portrait is the same for all $a < 0$ as well as for all $a > 0$, but that it changes at the parameter value $a = 0$, called a ''bifurcation value''.
 </li>
 <li> '''Perko, Section 1.1, Exercise 6:'''
 </li>
 <li> '''Perko, Section 1.2, Exercise 6:'''
 </li>

Murray at 18:12, 27 December 2012

2012-12-27T18:12:46Z

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	<hr>		<hr>
	Notes:		Notes:
	* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.		* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko (third edition) as the definitive source of the problem statement.
	* There are a number of problems that can be solved using MATLAB. If you just give the answer with no explanation (or say "via MATLAB"), the TAs will take off points. Instead, you should show how the solutions can be worked out by hand, along the lines of what is done in the text book. It is fine to check everything with MATLAB.		* There are a number of problems that can be solved using MATLAB. If you just give the answer with no explanation (or say "via MATLAB"), the TAs will take off points. Instead, you should show how the solutions can be worked out by hand, along the lines of what is done in the text book. It is fine to check everything with MATLAB.
	* For numerical calculations, it is OK to use MATLAB to invert a matrix. But you should not use it to compute the matrix exponential and just put down the answer. Instead, show how to get the matrix exponential into a form in which the calculation can be done by hand (similar to what was done in lecture on 6 Jan) and then carry out the computation.		* For numerical calculations, it is OK to use MATLAB to invert a matrix. But you should not use it to compute the matrix exponential and just put down the answer. Instead, show how to get the matrix exponential into a form in which the calculation can be done by hand (similar to what was done in lecture on 6 Jan) and then carry out the computation.
	* For phase portraits, you should generate the diagram by hand and make sure to label any important features. Describe why the portrait looks as it does based on the relevant properties of the dynamical system (eg, eigenvalues of the A matrix).		* For phase portraits, you should generate the diagram by hand and make sure to label any important features. Describe why the portrait looks as it does based on the relevant properties of the dynamical system (eg, eigenvalues of the A matrix).
	* For the final exam, you will ''not'' be allowed to use MATLAB, so make sure you understand what you are computing and drawing!		* For the final exam, you will ''not'' be allowed to use MATLAB, so make sure you understand what you are computing and drawing!

Murray at 18:07, 27 December 2012

2012-12-27T18:07:54Z

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	\| issued = 8 Jan 2013		\| issued = 8 Jan 2013
	\| due = 15 Jan 2013		\| due = 15 Jan 2013
	}}		}} __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		'''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
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	\dot x_1 &= x_1 \\ x_2 &= a x_2		\dot x_1 &= x_1 \\ x_2 &= a x_2
	\end{align}		\end{align}
	where $a$ is a constant. Sketch the phase portraits for $a = -1$, $a = 0$ and $~~math \|~~ a = 1$ and notice the qualitative structure of the phase portrait is the same for all $a < 0$ as well as for all $a > 0$, but that it changes at the parameter value $a = 0$, called a ''bifurcation value''.		where $a$ is a constant. Sketch the phase portraits for $a = -1$, $a = 0$ and $a = 1$ and notice the qualitative structure of the phase portrait is the same for all $a < 0$ as well as for all $a > 0$, but that it changes at the parameter value $a = 0$, called a ''bifurcation value''.
	</li>		</li>
	<li> '''Perko, Section 1.1, Exercise 6:'''		<li> '''Perko, Section 1.1, Exercise 6:'''

Murray at 18:07, 27 December 2012

2012-12-27T18:07:15Z

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	<ol>		<ol>
	<li> '''Perko, Section 1.1, Exercise 3''': Find the general solution of the linear system		<li> '''Perko, Section 1.1, Exercise 3''': Find the general solution of the linear system
	{~~{displaymath \|~~		\begin{align}
	~~\aligned~~
	\dot x_1 &= x_1 \\ x_2 &= a x_2		\dot x_1 &= x_1 \\ x_2 &= a x_2
	\~~endaligned~~		\end{align}
	}}		where $a$ is a constant. Sketch the phase portraits for $a = -1$, $a = 0$ and $math \| a = 1$ and notice the qualitative structure of the phase portrait is the same for all $a < 0$ as well as for all $a > 0$, but that it changes at the parameter value $a = 0$, called a ''bifurcation value''.
	where ~~{{math \|~~ a}} is a constant. Sketch the phase portraits for ~~{{math \|~~ a = -1}}, ~~{{math \|~~ a = 0}} and {{math \| a = 1}} and notice the qualitative structure of the phase portrait is the same for all ~~{{math \|~~ a < 0}} as well as for all ~~{{math \|~~ a > 0}}, but that it changes at the parameter value ~~{{math \|~~ a = 0}}, called a ''bifurcation value''.
	</li>		</li>
	<li> '''Perko, Section 1.1, Exercise 6:'''		<li> '''Perko, Section 1.1, Exercise 6:'''

Murray at 17:52, 27 December 2012

2012-12-27T17:52:27Z

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	<ol>		<ol>
	<li> '''Perko, Section 1.1, Exercise 3''':		<li> '''Perko, Section 1.1, Exercise 3''': Find the general solution of the linear system
			{{displaymath \|
			\aligned
			\dot x_1 &= x_1 \\ x_2 &= a x_2
			\endaligned
			}}
			where {{math \| a}} is a constant. Sketch the phase portraits for {{math \| a = -1}}, {{math \| a = 0}} and {{math \| a = 1}} and notice the qualitative structure of the phase portrait is the same for all {{math \| a < 0}} as well as for all {{math \| a > 0}}, but that it changes at the parameter value {{math \| a = 0}}, called a ''bifurcation value''.
	</li>		</li>
	<li> '''Perko, Section 1.1, Exercise 6:'''		<li> '''Perko, Section 1.1, Exercise 6:'''