View source for CDS 140a Winter 2013 Homework 4

{{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).

<ol>
<li>'''Perko, Section 2.7, problem 1'''
Write the system
<center><amsmath>
\aligned
\dot{x}_1&=x_1+6x_2+x_1x_2,\\
\dot{x}_2&=4x_1+3x_2-x_1^2
\endaligned
</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>
\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'''
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> 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> '''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>

<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.