CDS 140a Winter 2013 Homework 4

R. Murray, D. MacMartin  Issued: 29 Jan 2013 (Tue) 
ACM 101/AM 125b/CDS 140a, Winter 2013  Due: 5 Feb 2013 (Tue) 
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)
 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_2x_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 $y_2$.
 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 $S$.
 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 $E^u$.
 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 $h(x)$.
 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_1^2x_2^21\\2x_2\end{bmatrix}
</amsmath><amsmath> (b)\quad\begin{bmatrix}x_2x_1^2+2\\2x_2^22x_1x_2\end{bmatrix}
</amsmath> 
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
<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>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.
 Perko, Section 2.7, problem 5
Solve the system and show that
<amsmath> S:\,\,x_3=\frac{1}{3}x_2^2\frac{1}{6}x_1^2x_2\frac{1}{30}x_1^4
</amsmath>and
<amsmath> U:\,\,x_1=x_2=0
</amsmath>Note that these formulae actually determine the global stable and unstable manifolds $W^s(0)$ and $W^u(0)$ respectively.
 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)\\,xy
</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 xy
</amsmath> 
Perko, Section 2.8, Problem 1
Solve the system
<amsmath> \aligned \dot{y}_1&=y_1\\ \dot{y}_2&=y_2+z^2\\ \dot{z}&=z \endaligned
</amsmath>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
<amsmath> H=\int_0^1L^{s}H_0T^sds.
</amsmath>Use the homemorphism $H$ to find the stable and unstable manifolds
<amsmath> W^s(0)=H^{1}(E^s)\quad {\mathrm{and}}\quad W^u(0)=H^{1}(E^u)
</amsmath>for this system.
HINT: You should find
<amsmath>\aligned H(y_1,y_2,z)&=(y_1,y_2z^2/3,z)^T\\ W^s(0)&=\{x\in{\mathbb{R}}z=0\} \endaligned
</amsmath>and
<amsmath> W^u(0)=\{x\in{\mathbb{R}}y_1=0,y_2=z^2/3\}.
</amsmath>
And any two of the following:
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.