CDS 140a Winter 2014 Homework 3 - Revision history

Murray at 23:40, 19 January 2015

2015-01-19T23:40:39Z

← Older revision		Revision as of 23:40, 19 January 2015
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	<!-- 2014 TA comments:		<!-- 2014 TA comments:
	A lot of students also had questions about problem		* A lot of students also had questions about problem 4 I believe it was, kind of wondering if the solution was really as easy as it looked, and the students were told that yes, it was that easy.
	4 I believe it was, kind of wondering if the solution was really as easy
	as it looked, and the students were told that yes, it was that easy.		* On problem 4, many students got the prinicipal axes wrong, and they were also frequently confused about the ... [rest of text missing]
	-->		-->
	<li> '''Perko, Section 2.5, problem 4''':		<li> '''Perko, Section 2.5, problem 4''':
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	<!-- 2014 TA comments:		<!-- 2014 TA comments:
	Students didn't seem to understand what they were supposed to do.		* Students didn't seem to understand what they were supposed to do. They were showed an example of a 1-d system with a pitchfork bifurcation, and also a bifurcation plot of a 2-d system with a pitchfork bifurcation to help get intuition about what they were actually looking for, so they could apply the concepts to the 3-d system. I didn't see that they had to classify the equilibrium points because that was in the Note at the end of the problem, and I didn't realize that Notes contained further instruction, so I told a few students they only needed to find mu where the single equilibrium split into several equilibriums. So we decided to grade lightly for this problem. The students were also showed how to do the stability analysis for the example of 1-d system (and qualitatively for the 2-d system with the phase portrait), in hopes it would give them more intuition about why this problem was important, (not realizing that the Note at the end of the problem asked the students to actually classify the equilibrium points).
	They were showed an example of a 1-d system with a pitchfork bifurcation,
	and also a bifurcation plot of a 2-d system with a pitchfork bifurcation		* On problem 6, students not performing stability analysis on the equilibrium points for mu > 1 as required.
	to help get intuition about what they were actually looking for, so they
	could apply the concepts to the 3-d system. I didn't see that they had to
	classify the equilibrium points because that was in the Note at the end of
	the problem, and I didn't realize that Notes contained further
	instruction, so I told a few students they only needed to find mu where
	the single equilibrium split into several equilibriums. So we decided to
	grade lightly for this problem. The students were also showed how to do
	the stability analysis for the example of 1-d system (and qualitatively
	for the 2-d system with the phase portrait), in hopes it would give them
	more intuition about why this problem was important, (not realizing that
	the Note at the end of the problem asked the students to actually classify
	the equilibrium points).
	-->		-->
	<li> '''Perko, Section 2.6, problem 2''':		<li> '''Perko, Section 2.6, problem 2''':

Murray at 00:02, 10 February 2014

2014-02-10T00:02:51Z

← Older revision		Revision as of 00:02, 10 February 2014
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	\| title = Problem Set #3		\| title = Problem Set #3
	\| issued = 21 Jan 2014 (Tue)		\| issued = 21 Jan 2014 (Tue)
	\| due = 29 Jan ~~2011~~ (Wed) @ noon <br>Turn in to box outside Steele House		\| due = 29 Jan 2014 (Wed) @ noon <br>Turn in to box outside Steele House
	\| pdf = cds140-wi14_hw3.pdf		\| pdf = cds140-wi14_hw3.pdf
	}} __MATHJAX__		}} __MATHJAX__

Murray at 00:41, 3 February 2014

2014-02-03T00:41:45Z

@@ Line 28: / Line 28: @@
 </li>
 <li> '''Perko, Section 2.2, problem 3''' <font color=blue>(if you have not had ACM 125a or CDS 201)</font>: Use the method of successive approximations to show that if $f(x,t)$ is continuous in $t$ in some interval containing $t = 0$ and continuously differentiable in $x$ for all $x$ in some open set $E \subset {\mathbb R}^n$ containing $x_0$, then there exists $a > 0$ such that the initial value problem
 <center><amsmath>
@@ Line 35: / Line 40: @@
 * Note: this problem is very similar to the case we worked out in class (and that you can find in Perko), but now $f(x,t)$ depends on the time $t$.  If you get stuck, there is a hint in Perko (not transcribed here).
 <li> '''Perko, Section 2.3, problem 1''':
 Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem
@@ Line 51: / Line 62: @@
 </li>
 <li> '''Perko, Section 2.5, problem 4''':
 Sketch the flow of the linear system
@@ Line 67: / Line 83: @@
 </li>
 <li> '''Perko, Section 2.6, problem 2''':
 Classify the equilibrium points of the Lorenz equation

Murray at 17:56, 22 January 2014

2014-01-22T17:56:03Z

← Older revision		Revision as of 17:56, 22 January 2014
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	\| issued = 21 Jan 2014 (Tue)		\| issued = 21 Jan 2014 (Tue)
	\| due = 29 Jan 2011 (Wed) @ noon <br>Turn in to box outside Steele House		\| due = 29 Jan 2011 (Wed) @ noon <br>Turn in to box outside Steele House
			\| pdf = cds140-wi14_hw3.pdf
	}} __MATHJAX__		}} __MATHJAX__
	~~{{warning\|This homework set is still under preparation. This banner will be removed when it is 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
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	</amsmath></center>		</amsmath></center>
	for $\mu > 0$. At what value of the parameter $\mu$ do two new equilibrium points "bifurcate" from the equilibrium point at the origin?		for $\mu > 0$. At what value of the parameter $\mu$ do two new equilibrium points "bifurcate" from the equilibrium point at the origin?
			* Note: the number and/or stability type of equilibrium points will change depending on the value of $\mu$. Make sure to classify the equilibrium points for different ranges of $\mu$ and not just one value of $\mu$. If you get stuck, there are some hints in problem 1(e) of Perko.
	</li>		</li>

Murray at 17:52, 22 January 2014

2014-01-22T17:52:23Z

← Older revision		Revision as of 17:52, 22 January 2014
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	</amsmath></center>		</amsmath></center>
	has a unique solution $x(t)$ on the interval $[-a, a]$.		has a unique solution $x(t)$ on the interval $[-a, a]$.
	* Note: this problem is very similar to the case we worked out in class (and that you can find in Perko), but now $f(x,t)$ depends on the time $t$.		* Note: this problem is very similar to the case we worked out in class (and that you can find in Perko), but now $f(x,t)$ depends on the time $t$. If you get stuck, there is a hint in Perko (not transcribed here).

	<li> '''Perko, Section 2.3, problem 1''':		<li> '''Perko, Section 2.3, problem 1''':
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	\dot \Phi = A \Phi, \qquad \Phi(0) = I.		\dot \Phi = A \Phi, \qquad \Phi(0) = I.
	</amsmath></center>		</amsmath></center>
			* Note: this problem works through the more general result for nonlinear systems (Corallary on page 83) for the special case of a linear system.
	</li>		</li>

Murray at 17:46, 22 January 2014

2014-01-22T17:46:47Z

← Older revision		Revision as of 17:46, 22 January 2014
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	</amsmath></center>		</amsmath></center>
	has a unique solution $x(t)$ on the interval $[-a, a]$.		has a unique solution $x(t)$ on the interval $[-a, a]$.
			* Note: this problem is very similar to the case we worked out in class (and that you can find in Perko), but now $f(x,t)$ depends on the time $t$.

	<li> '''Perko, Section 2.3, problem 1''':		<li> '''Perko, Section 2.3, problem 1''':

Murray at 17:44, 22 January 2014

2014-01-22T17:44:01Z

← Older revision		Revision as of 17:44, 22 January 2014
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	Classify the equilibrium points of the Lorenz equation		Classify the equilibrium points of the Lorenz equation
	<center><amsmath>		<center><amsmath>
	\frac{}{dt} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =		\frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =
	\begin{bmatrix} x_2 - x_1 \\ \mu x_1 - x_2 - x_1 x_3 \\ x_1 x_2 - x_3 \end{bmatrix}		\begin{bmatrix} x_2 - x_1 \\ \mu x_1 - x_2 - x_1 x_3 \\ x_1 x_2 - x_3 \end{bmatrix}
	</amsmath></center>		</amsmath></center>

Murray at 17:43, 22 January 2014

2014-01-22T17:43:35Z

← Older revision		Revision as of 17:43, 22 January 2014
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	</li>		</li>

			<li> '''Perko, Section 2.6, problem 2''':
			Classify the equilibrium points of the Lorenz equation
			<center><amsmath>
			\frac{}{dt} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =
			\begin{bmatrix} x_2 - x_1 \\ \mu x_1 - x_2 - x_1 x_3 \\ x_1 x_2 - x_3 \end{bmatrix}
			</amsmath></center>
			for $\mu > 0$. At what value of the parameter $\mu$ do two new equilibrium points "bifurcate" from the equilibrium point at the origin?
			</li>

			<!-- Dropped for 2014
	<li> Choose ''one'' of the following systems and determine all of the equilibrium points for		<li> Choose ''one'' of the following systems and determine all of the equilibrium points for
	the system, indicating whether each is a sync, source, or saddle. <br>		the system, indicating whether each is a sync, source, or saddle. <br>
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	</li>		</li>
	</ol>		</ol>
			-->

	<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 as the definitive source of the problem statement.

Murray at 04:23, 20 January 2014

2014-01-20T04:23:11Z

← Older revision		Revision as of 04:23, 20 January 2014
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	has a unique solution $x(t)$ on the interval $[-a, a]$.		has a unique solution $x(t)$ on the interval $[-a, a]$.

	~~<!-- Removted for 2014~~
	<li> '''Perko, Section 2.3, problem 1''':		<li> '''Perko, Section 2.3, problem 1''':
	Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem		Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem
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	</amsmath></center>		</amsmath></center>
	</li>		</li>
	~~-->~~

	<li> '''Perko, Section 2.5, problem 4''':		<li> '''Perko, Section 2.5, problem 4''':

Murray at 04:19, 20 January 2014

2014-01-20T04:19:30Z

← Older revision		Revision as of 04:19, 20 January 2014
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	has a unique solution $x(t)$ on the interval $[-a, a]$.		has a unique solution $x(t)$ on the interval $[-a, a]$.

			<!-- Removted for 2014
	<li> '''Perko, Section 2.3, problem 1''':		<li> '''Perko, Section 2.3, problem 1''':
	Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem		Use the fundamental theorem for linear systems in Chapter 1 of Perko to solve the initial value problem
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	</amsmath></center>		</amsmath></center>
	</li>		</li>
			-->

	<li> '''Perko, Section 2.5, problem 4''':		<li> '''Perko, Section 2.5, problem 4''':
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	<li> '''Perko, Section 2.5, problem 5''':		<li> '''Perko, Section 2.5, problem 5''':
	Determine the flow $\phi_t:{\mathbb R}^2 \to {\mathbb R}^2$ for the ~~linear~~ system		Determine the flow $\phi_t:{\mathbb R}^2 \to {\mathbb R}^2$ for the nonlinear system
	<center><amsmath>		<center><amsmath>
	\dot x = f(x) \quad\text{with}\quad f(x) = \begin{bmatrix} -x_1 \\ 2 x_2 + x_1^2 \end{bmatrix}		\dot x = f(x) \quad\text{with}\quad f(x) = \begin{bmatrix} -x_1 \\ 2 x_2 + x_1^2 \end{bmatrix}