Difference between revisions of "CDS 140a Winter 2013 Homework 4"

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{{warning|This homework set is still being written.  Do not start working on these problems until this banner is removed.}}
 
 
 
 
{{CDS homework
 
{{CDS homework
 
  | instructor = R. Murray, D. MacMartin
 
  | instructor = R. Murray, D. MacMartin
Line 13: Line 10:
 
'''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>
Line 20: Line 15:
 
Write the system
 
Write the system
 
<center><amsmath>
 
<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,\\
 +
\dot{x}_2&=4x_1+3x_2-x_1^2
 +
\endaligned
 
</amsmath></center>
 
</amsmath></center>
 
in the form  
 
in the form  
Line 37: Line 34:
 
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
 
<center><amsmath>
 
<center><amsmath>
\dot{x}_1=-x_1,\qquad
+
\aligned
\dot{x}_2=x_2+x_1^2
+
\dot{x}_1&=-x_1,\\
 +
\dot{x}_2&=x_2+x_1^2
 +
\endaligned
 
</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 $S$.
Line 54: Line 53:
 
</li>
 
</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> Prove that if
 +
<center><amsmath>
 +
\aligned
 +
\dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\
 +
\dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m
 +
\endaligned
 +
</amsmath></center>
 +
then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if
 +
<center><amsmath>
 +
g(x,h(x))=Dh(x)f(x,h(x))
 +
</amsmath></center>
 +
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:''' 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'''
 +
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)'''
 +
Determine the stability of the equilibrium points of the system (1) with $f(x)$ given by
 +
<center><amsmath>
 +
(a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}
 +
</amsmath></center>
 +
<center><amsmath>
 +
(b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}
 +
</amsmath></center>
 +
</li>
 +
 +
 +
<!--
 +
<font color=blue>And any two of the following:</font>
 +
<li>
 +
'''Perko, Section 2.7, problem 4'''
 +
Find the first four successive approximations $u^{(1)}(t,a)$, $u^{(2)}(t,a)$, $u^{(3)}(t,a)$, and $u^{(4)}(t,a)$ for the system
 +
<center><amsmath>
 +
\aligned
 +
\dot{x}_1&=-x_1\\
 +
\dot{x}_2&=-x_2+x_1^2\\
 +
\dot{x}_3&=x_3+x_2^2
 +
\endaligned
 +
</amsmath></center>
 +
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> '''Perko, Section 2.7, problem 5'''
 +
Solve the system and show that
 +
<center><amsmath>
 +
S:\,\,x_3=-\frac{1}{3}x_2^2-\frac{1}{6}x_1^2x_2-\frac{1}{30}x_1^4
 +
</amsmath></center>
 +
and
 +
<center><amsmath>
 +
U:\,\,x_1=x_2=0
 +
</amsmath></center>
 +
Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
 +
</li>
 +
 +
<li>
 +
'''Perko, Section 2.8, 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=\int_0^1L^{-s}H_0T^sds.
 +
</amsmath></center>
 +
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.
 +
 +
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>
 +
W^u(0)=\{x\in{\mathbb{R}}|y_1=0,y_2=z^2/3\}.
 +
</amsmath></center>
 +
</li>
 +
</!--
 
</ol>
 
</ol>
  
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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.
 
 
: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)'''
 

Latest revision as of 03:09, 4 February 2013

R. Murray, D. MacMartin Issued: 29 Jan 2013 (Tue)
ACM 101/AM 125b/CDS 140a, Winter 2013 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).

  1. Perko, Section 2.7, problem 1 Write the system
    <amsmath>

    \aligned \dot{x}_1&=x_1+6x_2+x_1x_2,\\ \dot{x}_2&=4x_1+3x_2-x_1^2 \endaligned

    </amsmath>

    in the form

    <amsmath>

    \dot{y}=By+G(y)

    </amsmath>

    where

    <amsmath>

    B=\begin{bmatrix}\lambda_1&0\\0&\lambda_2\end{bmatrix}

    </amsmath>

    with $\lambda_1<0$, $\lambda_2>0$ and $G(y)$ is quadratic in $y_1$ and $y_2$.

  2. 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
    <amsmath>

    \aligned \dot{x}_1&=-x_1,\\ \dot{x}_2&=x_2+x_1^2 \endaligned

    </amsmath>

    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 $S$.

  3. Perko, Section 2.7, problem 3 Solve the system in Problem 2 and show that $S$ and $U$ are given by
    <amsmath>

    S:\,x_2=-\frac{x_1^2}{3}

    </amsmath>
    <amsmath>

    U:\,x_1=0

    </amsmath>

    Sketch $S$, $U$, $E^s$ and $E^u$.

  4. Prove that if
    <amsmath>

    \aligned \dot{x}&=f(x,y),\qquad x\in\mathbb{R}^k\\ \dot{y}&=g(x,y),\qquad g\in\mathbb{R}^m \endaligned

    </amsmath>

    then the manifold $S=\{(x,y)\in\mathbb R^k\times\mathbb R^m|y=h(x)\}$ is an invariant manifold of the system if

    <amsmath>

    g(x,h(x))=Dh(x)f(x,h(x))

    </amsmath>

    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: 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 $\mathbb R^2$.)

  5. 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
    <amsmath>

    |F(x)-F(y)|\leq\|DF(\xi)\|\,|x-y|

    </amsmath>

    (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

    <amsmath>

    |F(x)-F(y)|<\epsilon |x-y|

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

    (a)\quad\begin{bmatrix}x_1^2-x_2^2-1\\2x_2\end{bmatrix}

    </amsmath>
    <amsmath>

    (b)\quad\begin{bmatrix}x_2-x_1^2+2\\2x_2^2-2x_1x_2\end{bmatrix}

    </amsmath>