CDS 140a Winter 2014 Homework 5 - Revision history

Murray at 02:25, 25 January 2015

2015-01-25T02:25:22Z

← Older revision		Revision as of 02:25, 25 January 2015
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	'''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
	this homework set (including reading).		this homework set (including reading).

			<!-- 2014 TA comments: As in last year, there were people who wanted to have their work checked with the center manifold calculations, the coefficients for their
			approximation, (problem 4), this year I happened to have my worked out solutions and a students solutions from last year to compare this student's calculations with, and the student's calculations compared with one of them, but basically I did tell the students their forms might look different depending on how they transformed the system, and how they simplified their algebra, etc. -->

	<ol>		<ol>
			<!-- 2014 TA comments: there were a few questions on how to actually show that the circles lie in planes. The students were told they could tell by the phase portrait, and they were also shown an algebraic way to show the circles lie in places. -->
	<li>'''Perko, Section 2.9, problem 3'''		<li>'''Perko, Section 2.9, problem 3'''
	Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system		Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system
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	Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system.		Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system.
	</li>		</li>

			<!-- 2014 TA comments: there was some unclearness as to whether this Lyapunov function could show whether or not this system was asymptotically stable. For sure it could be shown to be stable. Picking a=... and b=... (or the opposite a=..., b=...), one could obtain Vdot = ... The problem was that if y=..., and x=..., Vdot=... still, even though the system was not necessarily at an equilibrium point. We did not have a good course solution for this problem from last year (we had half of an unlabeled solution that looking back does not correct). -->
	<li>		<li>
	Determine the stability of the system		Determine the stability of the system
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	<!-- Requires use of Lasalle; not covered in 2014 -->		<!-- Requires use of Lasalle; not covered in 2014 -->
			<!-- 2014 TA comments: Some of the students mentioned they couldn't use Gronwell's inequality because of a negative constant instead of positive. They were told they could do the problem without Gronwell's inequality, but if they could use Gronwell's inequality that would be good too. -->
	<li>		<li>
	Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.		Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.
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	</li>		</li>

			<!-- 2014 TA comments: some people tried to use the transform a1=..., and a2=..., instead of a1=.. and a2=.... The students were told them they are free to try the problem with that transform, but they were also told them that for sure they could come up with the answer we are looking for if they used the second transform cited here. It seemed people who used the first transform cited here became stuck at a certain point, I didn't know if it was because they didn't know how to do the problem, or if it was due to the form of the tranform they used (The first form cited here). I said go ahead and use any transform you would like, but try doing the problem and going through the process with the second transform cited here, and they were given more specific instructions on how to complete the problem using that transform (which they could then go and try to apply to the system tranformed with the first transform). In summary, it might save some students time if they were given the second transform to work with from the start, with me not knowing now if their other transforms are going to work to give the solutions that are expected. Some had questions about whether they could say that a stable manifold and unstable manifold didn't exist. Since there were not any nonzero eigenvalues for this system, they were told everything was on the center manifold, which didn't rule out their being stable and unstable parts of the center manifold, it was just that the usual theorems with the nonzero eigenvalues didn't apply here, so they needed to use a center manifold approximation, which in this case was to be a linear approximation. -->
	<li>'''Perko, Section 2.12, problem 2'''		<li>'''Perko, Section 2.12, problem 2'''
	Use Theorem 1 [Center Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system		Use Theorem 1 [Center Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system

Murray at 22:28, 15 February 2014

2014-02-15T22:28:16Z

← Older revision		Revision as of 22:28, 15 February 2014
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	{{CDS homework		{{CDS homework
	\| instructor = R. Murray~~, D. MacMartin~~		\| instructor = R. Murray
	\| course = ACM 101b/AM 125b/CDS 140a		\| course = ACM 101b/AM 125b/CDS 140a
	\| semester = Winter 2014		\| semester = Winter 2014

Murray at 23:45, 12 February 2014

2014-02-12T23:45:10Z

← Older revision		Revision as of 23:45, 12 February 2014
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	Motivated by the first equation, try a Lyapunov function of the form $V(x,y)=\alpha x^6+\beta y^2$. Is the origin asymptotically stable? Is the origin globally asymptotically stable?		Motivated by the first equation, try a Lyapunov function of the form $V(x,y)=\alpha x^6+\beta y^2$. Is the origin asymptotically stable? Is the origin globally asymptotically stable?
	</li>		</li>

			<!-- Requires use of Lasalle; not covered in 2014 -->
	<li>		<li>
	Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.		Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.

Murray at 00:01, 10 February 2014

2014-02-10T00:01:39Z

← Older revision		Revision as of 00:01, 10 February 2014
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	\| semester = Winter 2014		\| semester = Winter 2014
	\| title = Problem Set #5		\| title = Problem Set #5
	\| issued = 4 Feb ~~2013~~ (Tue)		\| issued = 4 Feb 2014 (Tue)
	\| due = 12 Feb ~~2013~~ (Wed) @ noon		\| due = 12 Feb 2014 (Wed) @ noon
	\| pdf = cds140-wi14_hw5.pdf		\| pdf = cds140-wi14_hw5.pdf
	}} __MATHJAX__		}} __MATHJAX__

Murray at 16:39, 5 February 2014

2014-02-05T16:39:47Z

← Older revision		Revision as of 16:39, 5 February 2014
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	\| issued = 4 Feb 2013 (Tue)		\| issued = 4 Feb 2013 (Tue)
	\| due = 12 Feb 2013 (Wed) @ noon		\| due = 12 Feb 2013 (Wed) @ noon
			\| pdf = cds140-wi14_hw5.pdf
	}} __MATHJAX__		}} __MATHJAX__

Murray at 16:37, 5 February 2014

2014-02-05T16:37:41Z

← Older revision		Revision as of 16:37, 5 February 2014
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	\| due = 12 Feb 2013 (Wed) @ noon		\| due = 12 Feb 2013 (Wed) @ noon
	}} __MATHJAX__		}} __MATHJAX__

	~~{{warning\|This homework set is under preparation. 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 16:36, 5 February 2014

2014-02-05T16:36:20Z

← Older revision		Revision as of 16:36, 5 February 2014
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	<center><amsmath>k_1\\|x\\|^2\leq V(x)\leq k_2\\|x\\|^2,\quad \dot V(x)\leq -k_3\\|x\\|^2		<center><amsmath>k_1\\|x\\|^2\leq V(x)\leq k_2\\|x\\|^2,\quad \dot V(x)\leq -k_3\\|x\\|^2
	</amsmath></center>		</amsmath></center>
	for positive constants $k_1$, $k_2$ and $k_3$, then an equilibrium point at the origin is exponentially stable		for positive constants $k_1$, $k_2$ and $k_3$, then the equilibrium point at the origin is exponentially stable.
	<!-- Added hint for 2014 -->		<!-- Added hint for 2014 -->
	* Hint: you can use Gronwall's inequality from Section 2.3 of Perko.		* Hint: you can use Gronwall's inequality from Section 2.3 of Perko.
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	<li>'''Perko, Section 2.12, problem 2'''		<li>'''Perko, Section 2.12, problem 2'''
	Use Theorem 1 [~~Centre~~ Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system		Use Theorem 1 [Center Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system
	<center><amsmath>		<center><amsmath>
	\aligned		\aligned

Murray at 16:34, 5 February 2014

2014-02-05T16:34:13Z

← Older revision		Revision as of 16:34, 5 February 2014
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	</li>		</li>
	<li>		<li>
	An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.		Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.
	~~Prove that~~ an equilibrium point $~~x_0~~=0$ ~~is exponentially stable~~ if there is a function $V(x)$ satisfying
			Let $\dot x = f(x)$ be a dynamical system with an equilibrium point at $x_e = 0$. Show that if there is a function $V(x)$ satisfying
	<center><amsmath>k_1\\|x\\|^2\leq V(x)\leq k_2\\|x\\|^2,\quad \dot V(x)\leq -k_3\\|x\\|^2		<center><amsmath>k_1\\|x\\|^2\leq V(x)\leq k_2\\|x\\|^2,\quad \dot V(x)\leq -k_3\\|x\\|^2
	</amsmath></center>		</amsmath></center>
	for positive constants $k_1$, $k_2$ and $k_3$.		for positive constants $k_1$, $k_2$ and $k_3$, then an equilibrium point at the origin is exponentially stable
	<!-- Added hint for 2014 -->		<!-- Added hint for 2014 -->
	* Hint: you can use Gronwall's inequality from Section 2.3 of Perko.		* Hint: you can use Gronwall's inequality from Section 2.3 of Perko.

Murray at 21:38, 1 February 2014

2014-02-01T21:38:10Z

← Older revision		Revision as of 21:38, 1 February 2014
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	</amsmath></center>		</amsmath></center>
	for positive constants $k_1$, $k_2$ and $k_3$.		for positive constants $k_1$, $k_2$ and $k_3$.
			<!-- Added hint for 2014 -->
	* Hint: you can use Gronwall's inequality from Section 2.3 of Perko.		* Hint: you can use Gronwall's inequality from Section 2.3 of Perko.
	</li>		</li>
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	</li>		</li>

	<!-- ~~This~~ problem was a bit tricky to grade. consider putting a different problem in next year? ~~-->~~		<!-- Deleted in 2014: tricky to grade and not clear what it is trying to show
			<!-- 2013 note: this problem was a bit tricky to grade. consider putting a different problem in next year?
	<li> Consider the following system		<li> Consider the following system
	<center><amsmath>\aligned		<center><amsmath>\aligned
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	\endaligned</amsmath></center>		\endaligned</amsmath></center>
	with stable manifold $W^s=\{(x,y):x=0\}$		with stable manifold $W^s=\{(x,y):x=0\}$
	<ol>		<ol style="a">
	<li> Show that the Taylor series approximation to the centre manifold gives $W^c=\{(x,y):y=0\}$.</li>		<li> Show that the Taylor series approximation to the centre manifold gives $W^c=\{(x,y):y=0\}$.</li>
	<li> Show that the set $\{(x,y):y=h(x)\}$ with		<li> Show that the set $\{(x,y):y=h(x)\}$ with
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	</ol>		</ol>
	</li>		</li>
			-->
	</ol>		</ol>

Murray at 21:25, 1 February 2014

2014-02-01T21:25:07Z

← Older revision		Revision as of 21:25, 1 February 2014
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	</amsmath></center>		</amsmath></center>
	for positive constants $k_1$, $k_2$ and $k_3$.		for positive constants $k_1$, $k_2$ and $k_3$.
			* Hint: you can use Gronwall's inequality from Section 2.3 of Perko.
	</li>		</li>

	<li>'''Perko, Section 2.12, problem 2'''		<li>'''Perko, Section 2.12, problem 2'''
	Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system		Use Theorem 1 [Centre Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system