CDS 140a Winter 2015 Homework 3 - Revision history

Murray at 06:15, 22 January 2015

2015-01-22T06:15:51Z

← Older revision		Revision as of 06:15, 22 January 2015
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	\| pdf = cds140-wi15_hw3.pdf		\| pdf = cds140-wi15_hw3.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
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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 00:14, 20 January 2015

2015-01-20T00:14:25Z

← Older revision		Revision as of 00:14, 20 January 2015
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	<ol>		<ol>
	<!-- 2014 TA comments: A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1. -->		<!-- 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, 3'''		<li>'''Perko, Section 2.7, problem 2, 3'''<br>
	(a) Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for		(a) Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
	<center><amsmath>		<center><amsmath>

Murray at 00:13, 20 January 2015

2015-01-20T00:13:56Z

← Older revision		Revision as of 00:13, 20 January 2015
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	<ol>		<ol>
	<!-- 2014 TA comments: A lot of people also were confused about solving for the unstable manifold by simply letting xdot = - f(x) on #1. -->		<!-- 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, 3'''
	Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for		(a) Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
	<center><amsmath>		<center><amsmath>
	\aligned		\aligned
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	</amsmath></center>		</amsmath></center>
	and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.		and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.

			(b) Sketch $S$, $U$, $E^s$ and $E^u$.
	</li>		</li>

Murray at 00:10, 20 January 2015

2015-01-20T00:10:41Z

@@ Line 42: / Line 42: @@
 * 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 $\mathbb R^2$.)
 </li>
@@ Line 67: / Line 56: @@
 is transformed into a linear system using this transformation, i.e., if $y = H(x)$, show that $\dot y = A x$ for an appropriate $A$.  Use this transformation to compute and sketch the stable and unstable manifolds for the nonlinear system.
 * Note: since the nonlinear system can be transformed into a linear one, it follows that $S = H^{-1}(E^s)$ and $U = H^{-1}(E^u)$.
 </li>

Murray at 00:03, 20 January 2015

2015-01-20T00:03:57Z

← Older revision		Revision as of 00:03, 20 January 2015
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	</li>		</li>

	~~<li> '''Perko, Section 2.7, Problem 6'''~~
	~~Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that~~
	~~<center><amsmath>~~
	~~\|F(x)-F(y)\|\leq\\|DF(\xi)\\|\,\|x-y\|~~
	~~</amsmath></center>~~
	(cf. Theorem 4.19 and the proof of Theorem 9.19 in [R]) to prove that if $F\in C^1(E)$ and $F(0)=DF(0)=0$ then given any $\epsilon>0$ there exists a $\delta>0$ such that for all $x, y\in N_\delta(0)$ we have
	~~<center><amsmath>~~
	~~\|F(x)-F(y)\|<\epsilon \|x-y\|~~
	~~</amsmath></center>~~
	~~</li>~~

	<li>'''Perko, Section 2.9, problem 2(a)(b)'''		<li>'''Perko, Section 2.9, problem 2(a)(b)'''
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	</li>		</li>

	~~<font color=blue>And any two of the following:</font>~~
	<li>		<li>
	'''Perko, Section 2.7, problem 4'''		'''Perko, Section 2.7, problem 4'''
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	Show that $u^{(3)}(t,a)=u^{(4)}(t,a)=\cdots$ and hence $u(t,a)=u^{(3)}(t,a)$. Find $S$ and $U$ for this problem.		Show that $u^{(3)}(t,a)=u^{(4)}(t,a)=\cdots$ and hence $u(t,a)=u^{(3)}(t,a)$. Find $S$ and $U$ for this problem.
	</li>		</li>

	<li> '''Perko, Section 2.7, problem 5'''		<li> '''Perko, Section 2.7, problem 5'''
	Solve the system and show that		Solve the system and show that

Murray at 23:59, 19 January 2015

2015-01-19T23:59:16Z

@@ Line 14: / Line 14: @@
 <ol>
 <!-- 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'''
@@ Line 43: / Line 24: @@
 </amsmath></center>
 and use $u^{(3)}(t,a)$ to approximate $S$ near the origin.  Also approximate the unstable manifold $U$ near the origin for this system.  Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$.  Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.
 </li>
@@ Line 70: / Line 39: @@
 g(u,h(u))=Dh(u)f(u,h(u))
 </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 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 ${\Bbb R^n}$, 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 $\mathbb R^2$.)

Murray at 23:56, 19 January 2015

2015-01-19T23:56:50Z

← Older revision		Revision as of 23:56, 19 January 2015
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	<ol>		<ol>
	~~<!-- Removed in 2014; problem seems very trivial; what's the point?~~
	<li>'''Perko, Section 2.7, problem 1'''		<li>'''Perko, Section 2.7, problem 1'''
	Write the system		Write the system
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	with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.		with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.
	</li>		</li>
	~~-->~~

			<!-- 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>
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	</li>		</li>

	~~<!--~~
	<li> '''Perko, Section 2.7, Problem 6'''		<li> '''Perko, Section 2.7, Problem 6'''
	Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that		Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin. Use the fact that if $F\in C^1(E)$ then for all $x, y\in N_\delta(0)\subset E$ there exists a $\xi\in N_\delta(0)$ such that
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	</amsmath></center>		</amsmath></center>
	</li>		</li>

	<li>'''Perko, Section 2.9, problem 2(a)(b)'''		<li>'''Perko, Section 2.9, problem 2(a)(b)'''
	Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by		Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by
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	</amsmath></center>		</amsmath></center>
	</li>		</li>
	~~-->~~

	<!-- New for 2014 -->		<!-- New for 2014 -->
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	</li>		</li>

	~~<!--~~
	<font color=blue>And any two of the following:</font>		<font color=blue>And any two of the following:</font>
	<li>		<li>
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	Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.		Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
	</li>		</li>
	~~-->~~

	<li>		<li>

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2015-01-19T23:28:14Z

Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #3 | issued = 19 Jan 2015 | due = 28 Jan 2015 at 12:30 pm<br>In..."

New page

{{CDS homework
| instructor = R. Murray
| course = CDS 140
| semester = Winter 2015
| title = Problem Set #3
| issued = 19 Jan 2015
| due = 28 Jan 2015 at 12:30 pm<br>In class or to box across 107 STL
| pdf = cds140-wi15_hw3.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 2.7, problem 2'''
Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
<center><amsmath>
\aligned
\dot{x}_1&=-x_1,\\
\dot{x}_2&=x_2+x_1^2
\endaligned
</amsmath></center>
and use $u^{(3)}(t,a)$ to approximate $S$ near the origin. Also approximate the unstable manifold $U$ near the origin for this system. Note that $u^{(2)}(t,a)=u^{(3)}(t,a)$ and therefore $u^{(j+1)}(t,a)=u^{(j)}(t,a)$ for $j\geq 2$. Thus $u(t,a)=u^{(2)}(t,a)$ which gives the exact function defining $S$.
</li>

<li>'''Perko, Section 2.7, problem 3'''
Show that $S$ and $U$ for the previous problem are given by
<center><amsmath>
S:\,x_2=-\frac{x_1^2}{3}
</amsmath></center>
<center><amsmath>
U:\,x_1=0
</amsmath></center>
Sketch $S$, $U$, $E^s$ and $E^u$.
</li>


<li> Consider a dynamical system with $x = (u, v) \in {\Bbb R}^n$. For the case $n = 2$, prove that if
<center><amsmath>
\aligned
\dot{u}&=f(u,v),\qquad u\in\mathbb{R}^k\\
\dot{v}&=g(u,v),\qquad v\in\mathbb{R}^{n-k}
\endaligned
</amsmath></center>
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>
g(u,h(u))=Dh(u)f(u,h(u))
</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 $h(x)$.
* 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 $\mathbb R^2$.)
</li>




<li>'''Perko, Section 2.6, problem 3'''
Show that the continuous map $H:{\Bbb R}^3 \to {\Bbb R}^3$ defined by
<center><amsmath>
\begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix} = H(x) = \begin{bmatrix} x_1 \\ x_2 + x_1^2 \\ x_3 + x_1^2 / 3 \end{bmatrix}
</amsmath></center>
has a continuous inverse $H^{-1}:{\Bbb R}^3 \to {\Bbb R}^3$ and that the nonlinear system
<center><amsmath>
\frac{dx}{dt} = \begin{bmatrix} -x_1 \\ -x_2 + x_1^2 \\ x_3 + x_1^2 \end{bmatrix}
</amsmath></center>
is transformed into a linear system using this transformation, i.e., if $y = H(x)$, show that $\dot y = A x$ for an appropriate $A$. Use this transformation to compute and sketch the stable and unstable manifolds for the nonlinear system.
* Note: since the nonlinear system can be transformed into a linear one, it follows that $S = H^{-1}(E^s)$ and $U = H^{-1}(E^u)$.
</li>



<li>
'''Perko, Section 2.8, Problem 1'''
Solve the system
<center><amsmath>
\aligned
\dot{y}_1&=-y_1\\
\dot{y}_2&=-y_2+z^2\\
\dot{z}&=z
\endaligned
</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
<center><amsmath>
H (x) =\int_0^1 e^{-As} H_0 \bigl(\phi_s(x) \bigr) ds,
</amsmath></center>
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>
W^s(0)=H^{-1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{-1}(E^u)
</amsmath></center>
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
<center><amsmath>
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>
</li>
</ol>

<hr>
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.