CDS 140a Winter 2013 Homework 4 - Revision history

Macmardg at 03:09, 4 February 2013

2013-02-04T03:09:43Z

← Older revision		Revision as of 03:09, 4 February 2013
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	Use this result to compute the stable manifold for the system of problem 2 and 3 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 2 and 3 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 $h(x)$.

	'''Hint:''' to show $S$ is an invariant manifold, 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.		'''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>
	<li> '''Perko, Section 2.7, Problem 6'''		<li> '''Perko, Section 2.7, Problem 6'''

Macmardg at 16:00, 30 January 2013

2013-01-30T16:00:09Z

← Older revision		Revision as of 16:00, 30 January 2013
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	~~<!--~~
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	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, D. MacMartin		\| instructor = R. Murray, D. MacMartin
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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).

	~~(Not yet edited from 2011)~~

	<ol>		<ol>

Macmardg at 15:58, 30 January 2013

2013-01-30T15:58:52Z

← Older revision		Revision as of 15:58, 30 January 2013
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Macmardg at 03:19, 30 January 2013

2013-01-30T03:19:52Z

@@ Line 58: / Line 58: @@
 </li>
-<li> Compute the stable manifold for the system of problem 2 and 3 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 $h(x)$.
+<li> Prove that if
 </li>
 <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
@@ Line 71: / Line 93: @@
 </li>
 <font color=blue>And any two of the following:</font>
 <li>
@@ Line 95: / Line 119: @@
 Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
 </li>
-<li> '''Perko, Section 2.7, Problem 6'''
 <li>
 '''Perko, Section 2.8, Problem 1'''
@@ Line 135: / Line 150: @@
 W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
 </amsmath></center>
 </li>
 </ol>

Macmardg at 20:53, 27 January 2013

2013-01-27T20:53:47Z

@@ Line 20: / Line 20: @@
 Write the system
 <center><amsmath>
-\dot{x}_1=x_1+6x_2+x_1x_2,\qquad
+\aligned
-\dot{x}_2=4x_1+3x_2-x_1^2
+\dot{x}_1&=x_1+6x_2+x_1x_2,\\
 </amsmath></center>
 in the form
@@ Line 37: / Line 39: @@
 Find the first three successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, and $u^{(3)}(t,a)$ for
 <center><amsmath>
-\dot{x}_1=-x_1,\qquad
+\aligned
-\dot{x}_2=x_2+x_1^2
+\dot{x}_1&=-x_1,\\
 </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 $S$.
@@ Line 54: / Line 58: @@
 </li>
-<li> Repeat problem 2 and 3 above using Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.
+<li> Compute the stable manifold for the system of problem 2 and 3 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 $h(x)$.
 </li>
@@ Line 65: / Line 69: @@
 (b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}
 </amsmath></center>
 </li>
 </ol>
@@ Line 71: / Line 142: @@
 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.

Macmardg at 05:43, 27 January 2013

2013-01-27T05:43:57Z

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 <li> Repeat problem 2 and 3 above using Taylor series for $h(x)$ to define $S=\{(x_1,x_2)|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.
 </li>
 </ol>

Macmardg at 05:11, 27 January 2013

2013-01-27T05:11:58Z

← Older revision		Revision as of 05:11, 27 January 2013
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	(Not yet edited from 2011)		(Not yet edited from 2011)

			<ol>
			<li>'''Perko, Section 2.7, problem 1'''
			Write the system
			<center><amsmath>
			\dot{x}_1=x_1+6x_2+x_1x_2,\qquad
			\dot{x}_2=4x_1+3x_2-x_1^2
			</amsmath></center>
			in the form
			<center><amsmath>
			\dot{y}=By+G(y)
			</amsmath></center>
			where
			<center><amsmath>
			B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}
			</amsmath></center>
			with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.
			</li>

			<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>
			\dot{x}_1=-x_1,\qquad
			\dot{x}_2=x_2+x_1^2
			</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'''
			Solve the system in Problem 2 and show that $S$ and $U$ 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> Repeat problem 2 and 3 above using Taylor series for $h(x)$ to define $S=\{(x_1,x_2)\|x_2=h(x)\}$ and matching coefficients to solve for $h(x)$.
			</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.

Macmardg at 21:04, 25 January 2013

2013-01-25T21:04:23Z

← Older revision		Revision as of 21:04, 25 January 2013
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	{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}		{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}

	~~(Content forthcoming~~...)
			{{CDS homework
			\| instructor = R. Murray, D. MacMartin
			\| course = ACM 101/AM 125b/CDS 140a
			\| semester = Winter 2013
			\| title = Problem Set #3
			\| issued = 29 Jan 2013 (Tue)
			\| due = 5 Feb 2013 (Tue)
			}} __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
			this homework set (including reading).

			(Not yet edited from 2011)

Macmardg at 21:02, 25 January 2013

2013-01-25T21:02:05Z

← Older revision		Revision as of 21:02, 25 January 2013
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	(Content forthcoming...)		(Content forthcoming...)



			:1. '''Perko, Section 2.7, problem 1''' <br>
			:2. '''Perko, Section 2.7, problem 2'''<br>
			:3. '''Perko, Section 2.7, problem 3 '''<br>
			:4, 5. Choose any 2 of the following:
			::* '''Perko, Section 2.7, problem 4'''
			::* '''Perko, Section 2.7, problem 5'''
			::* '''Perko, Section 2.7, problem 6'''
			::* '''Perko, Section 2.8, problem 1'''
			:6. '''Perko, Section 2.9, problem 2(a)(b)'''

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2013-01-22T21:59:27Z

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