CDS 140a Winter 2014 Homework 4 - Revision history

Murray at 00:15, 20 January 2015

2015-01-20T00:15:06Z

← Older revision		Revision as of 00:15, 20 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 based on office hours:~~
	* There were a ton of questions about problem 3 (manifold problem). A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1. There were surprisingly not that many questions about #5 (Hartman-Grobman successive approximations).

	* Also, I told a couple of students that since their successive approximations converged exactly in problem 1, that they could simply state in problem 2 that their approximations to the stable and unstable manifold were just in fact exact.

	* However, if you wanted them to do #2 by actually solving the system of differential equations analytically and then figuring out the stable and unstable manifolds by inspection, then maybe you could email the class and make that clear.
	~~---~~
	* The year I took the class we solved problem 2, (or 2.7.3), by solving the system of diff eqns, and then we looked at the limit at t goes to infinite, and then solved for the integration constants/initial conditions in terms each other, which then gave the stable/unstable manifolds (see the beginning of chapter 2.7 for an example in the book that does this).

	* It seemed people weren't clear that the unstable manifold can be found by letting t go to -t and then solving for the stable manifold, or as Anandh said here, solving for -xdot = f(x), or xdot=-f(x) for an autonomous system.

	* Most people didn't have any idea on how to solve for problem 3, or why finding the normal vector to the tangent of the stable manifold, or to the vector field, is important for this problem. Perhaps for future years the hint for R^2, about the normal vector being orthogonal to the vector field could be explained as meaning that anytime a trajectory starts off on the stable manifold, it stays on the stable manifold since there is no verticle component to "lift" the trajectory off of the stable manifold.
	~~-->~~

	<ol>		<ol>
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	-->		-->

	~~<!-- 2014 TA comments: A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1. -->~~
	<li>'''Perko, Section 2.7, problem 2'''		<li>'''Perko, Section 2.7, problem 2'''
	Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for		Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
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	</li>		</li>

	<!-- TA comment: I told a couple of students that since their successive approximations converged exactly in problem 1, that they could simply state in problem 2 that their approximations to the stable and unstable manifold were just in fact exact. if you wanted them to do #2 by actually solving the system of differential equations analytically and then figuring out the stable and unstable manifolds by inspection, then maybe you could email the class and make that clear. -->
	<li>'''Perko, Section 2.7, problem 3'''		<li>'''Perko, Section 2.7, problem 3'''
	Show that $S$ and $U$ for the previous problem are given by		Show that $S$ and $U$ for the previous problem are given by
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	<!-- Lots of people didn't see the part about using this result to compute the stable manifold for the problem above -->		<!-- Lots of people didn't see the part about using this result to compute the stable manifold for the problem above -->
	<!-- Most people didn't have any idea on how to solve for problem 3, or why finding the normal vector to the tangent of the stable manifold, or to the vector field, is important for this problem. Perhaps for future years the hint for R^2, about the normal vector being orthogonal to the vector field could be explained as meaning that anytime a trajectory starts off on the stable manifold, it stays on the stable manifold since there is no verticle component to "lift" the trajectory off of the stable manifold. -->
	<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if		<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if
	<center><amsmath>		<center><amsmath>

Murray at 23:55, 19 January 2015

2015-01-19T23:55:08Z

← Older revision		Revision as of 23:55, 19 January 2015
Line 11:		Line 11:
	'''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 based on office hours:
			* There were a ton of questions about problem 3 (manifold problem). A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1. There were surprisingly not that many questions about #5 (Hartman-Grobman successive approximations).

			* Also, I told a couple of students that since their successive approximations converged exactly in problem 1, that they could simply state in problem 2 that their approximations to the stable and unstable manifold were just in fact exact.

			* However, if you wanted them to do #2 by actually solving the system of differential equations analytically and then figuring out the stable and unstable manifolds by inspection, then maybe you could email the class and make that clear.
			---
			* The year I took the class we solved problem 2, (or 2.7.3), by solving the system of diff eqns, and then we looked at the limit at t goes to infinite, and then solved for the integration constants/initial conditions in terms each other, which then gave the stable/unstable manifolds (see the beginning of chapter 2.7 for an example in the book that does this).

			* It seemed people weren't clear that the unstable manifold can be found by letting t go to -t and then solving for the stable manifold, or as Anandh said here, solving for -xdot = f(x), or xdot=-f(x) for an autonomous system.

			* Most people didn't have any idea on how to solve for problem 3, or why finding the normal vector to the tangent of the stable manifold, or to the vector field, is important for this problem. Perhaps for future years the hint for R^2, about the normal vector being orthogonal to the vector field could be explained as meaning that anytime a trajectory starts off on the stable manifold, it stays on the stable manifold since there is no verticle component to "lift" the trajectory off of the stable manifold.
			-->

	<ol>		<ol>
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	-->		-->

			<!-- 2014 TA comments: A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1. -->
	<li>'''Perko, Section 2.7, problem 2'''		<li>'''Perko, Section 2.7, problem 2'''
	Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for		Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
Line 45:		Line 60:
	</li>		</li>

			<!-- TA comment: I told a couple of students that since their successive approximations converged exactly in problem 1, that they could simply state in problem 2 that their approximations to the stable and unstable manifold were just in fact exact. if you wanted them to do #2 by actually solving the system of differential equations analytically and then figuring out the stable and unstable manifolds by inspection, then maybe you could email the class and make that clear. -->
	<li>'''Perko, Section 2.7, problem 3'''		<li>'''Perko, Section 2.7, problem 3'''
	Show that $S$ and $U$ for the previous problem are given by		Show that $S$ and $U$ for the previous problem are given by
Line 57:		Line 73:

	<!-- Lots of people didn't see the part about using this result to compute the stable manifold for the problem above -->		<!-- Lots of people didn't see the part about using this result to compute the stable manifold for the problem above -->
			<!-- Most people didn't have any idea on how to solve for problem 3, or why finding the normal vector to the tangent of the stable manifold, or to the vector field, is important for this problem. Perhaps for future years the hint for R^2, about the normal vector being orthogonal to the vector field could be explained as meaning that anytime a trajectory starts off on the stable manifold, it stays on the stable manifold since there is no verticle component to "lift" the trajectory off of the stable manifold. -->
	<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if		<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if
	<center><amsmath>		<center><amsmath>

Murray at 22:27, 15 February 2014

2014-02-15T22:27:44Z

← Older revision		Revision as of 22:27, 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 23:38, 12 February 2014

2014-02-12T23:38:39Z

← Older revision		Revision as of 23:38, 12 February 2014
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	</li>		</li>

			<!-- Lots of people didn't see the part about using this result to compute the stable manifold for the problem above -->
	<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if		<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if
	<center><amsmath>		<center><amsmath>

Murray at 06:11, 29 January 2014

2014-01-29T06:11:09Z

← Older revision		Revision as of 06:11, 29 January 2014
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	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>
	and show that the successive approximations $\~~Phi_k~~\rightarrow\Phi$ and $\~~Psi_k~~\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find		and show that the successive approximations $\Phi^(k)\rightarrow\Phi$ and $\Psi^(k)\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find
	<center><amsmath>		<center><amsmath>
	H (x) =\int_0^1 e^{-As} H_0 \bigl(\phi_s(x) \bigr) ds,		H (x) =\int_0^1 e^{-As} H_0 \bigl(\phi_s(x) \bigr) ds,

Murray at 06:10, 29 January 2014

2014-01-29T06:10:02Z

← Older revision		Revision as of 06:10, 29 January 2014
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	\| issued = 28 Jan 2014 (Tue)		\| issued = 28 Jan 2014 (Tue)
	\| due = 4 Feb 2014 (Wed) @ noon <br>Turn in to box outside Steele House		\| due = 4 Feb 2014 (Wed) @ noon <br>Turn in to box outside Steele House
			\| pdf = cds140-wi14_hw4.pdf
	}} __MATHJAX__		}} __MATHJAX__

Murray at 06:09, 29 January 2014

2014-01-29T06:09:02Z

← Older revision		Revision as of 06:09, 29 January 2014
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	and show that the successive approximations $\Phi_k\rightarrow\Phi$ and $\Psi_k\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find		and show that the successive approximations $\Phi_k\rightarrow\Phi$ and $\Psi_k\rightarrow\Psi$ as $k\rightarrow\infty$ for all $x=(y_1,y_2,z)\in{\mathbb R}^3$. Define $H_0=(\Phi,\Psi)^T$ and use this homeomorphism to find
	<center><amsmath>		<center><amsmath>
	H=\int_0^1L^{-s}~~H_0T^sds.~~		H (x) =\int_0^1 e^{-As} H_0 \bigl(\phi_s(x) \bigr) ds,
	</amsmath></center>		</amsmath></center>
	Use the homemorphism $H$ to find the stable and unstable manifolds		where $A$ is the linearization of the nonlinear dynamics at the origin and $\phi_t(x)$ is the flow of the full system. Use the homemorphism $H$ to find the stable and unstable manifolds
	<center><amsmath>		<center><amsmath>
	W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)		W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)

Murray at 06:04, 29 January 2014

2014-01-29T06:04:47Z

← Older revision		Revision as of 06:04, 29 January 2014
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	</amsmath></center>		</amsmath></center>
	for this system.		for this system.
			* Note: this problem involves some simple but somewhat tedious computations. If you know how to use Mathematica or a similar program, you may wish to carry out the computations for the successive approximations using that software. However, make sure to show the results at each step of the calculation.
	~~HINT~~: You should find		* Hint: You should find
	~~<center><amsmath>\aligned~~
	~~H(y_1,y_2,z)&=(y_1,y_2-z^2/3,z)^T\\~~
	~~W^s(0)&=\{x\in{\mathbb{R}}\|z=0\}~~
	~~\endaligned~~
	~~</amsmath></center>~~
	~~and~~
	<center><amsmath>		<center><amsmath>
	W^u(0)=\{x\in{\mathbb{R}}\|y_1=0,y_2=z^2/3\}.		H(y_1,y_2,z) = \begin{bmatrix} y_1 \\ y_2-z^2/3 \\ z \end{bmatrix}, \qquad
			\aligned
			W^s(0) &=\{x\in{\mathbb{R}}\|z=0\}, \\
			W^u(0) &=\{x\in{\mathbb{R}}\|y_1=0,y_2=z^2/3\}.
			\endaligned
	</amsmath></center>		</amsmath></center>
	</li>		</li>

Murray at 05:59, 29 January 2014

2014-01-29T05:59:08Z

← Older revision		Revision as of 05:59, 29 January 2014
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	</li>		</li>

	<li> Consider a ~~planar~~ dynamical system with $(u, v) \in {\Bbb R}^2$. ~~Prove~~ that if		<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if
	<center><amsmath>		<center><amsmath>
	\aligned		\aligned
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	</amsmath></center>		</amsmath></center>
	Use this result to compute the stable manifold for the system of problem 1 and 2 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)\|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.		Use this result to compute the stable manifold for the system of problem 1 and 2 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)\|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.
	* Note: The result holds for ${\Bbb R^n}$ ~~and~~ you ~~can prove it for~~ the ~~general~~ case ~~if you prefer. If you prove the results for $\{Bbb R}^2$, you can take~~ $n = 2$ and $k=1$.		* Note: The result holds for ${\Bbb R^n}$, but you only need to consider the case $n = 2$ and $k = 1$ (although you are free to prove the more general case if you prefer).
	* Hint: One way to show $S$ is an invariant manifold in $\mathbb R^2$ is to show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point. (It is sufficient to prove the result for $\mathbb R^2$.)		* Hint: One way to show $S$ is an invariant manifold in $\mathbb R^2$ is to show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point. (It is sufficient to prove the result for $\mathbb R^2$.)
	</li>		</li>

Murray at 05:56, 29 January 2014

2014-01-29T05:56:32Z

← Older revision		Revision as of 05:56, 29 January 2014
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	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>
	then the manifold $S=\{(u,v)\in\mathbb R^k\times\mathbb R^m\|v=h(u)\}$ is an invariant manifold of the system if		then the manifold $S=\{(u,v)\in\mathbb R^k\times\mathbb R^{n-k}\|v=h(u)\}$ is an invariant manifold of the system if
	<center><amsmath>		<center><amsmath>
	g(u,h(u))=Dh(u)f(u,h(u))		g(u,h(u))=Dh(u)f(u,h(u))
	</amsmath></center>		</amsmath></center>
	Use this result to compute the stable manifold for the system of problem 1 and 2 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)\|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.		Use this result to compute the stable manifold for the system of problem 1 and 2 above using the Taylor series for $h(x)$ to define $S=\{(x_1,x_2)\|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.
			* Note: The result holds for ${\Bbb R^n}$ and you can prove it for the general case if you prefer. If you prove the results for $\{Bbb R}^2$, you can take $n = 2$ and $k=1$.
	* Hint: One way to show $S$ is an invariant manifold in $\mathbb R^2$ is to show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point. (It is sufficient to prove the result for $\mathbb R^2$.)		* Hint: One way to show $S$ is an invariant manifold in $\mathbb R^2$ is to show that the normal vector (orthogonal to the tangent to $S$ at $(x,h(x))$) is orthogonal to the vector field $(f,g)$ at that point. (It is sufficient to prove the result for $\mathbb R^2$.)
	* Note: The result holds for ${\Bbb R^n}$ and you can prove it for the general case if you prefer.
	</li>		</li>