CDS 140a Winter 2014 Homework 4

R. Murray, D. MacMartin	Issued: 28 Jan 2014 (Tue)
ACM 101b/AM 125b/CDS 140a, Winter 2014	Due: 4 Feb 2014 (Wed) @ noon Turn in to box outside Steele House

__MATHJAX__

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1. 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 $S$.

2. Perko, Section 2.7, problem 3 Show that $S$ and $U$ for the previous problem 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 $E^u$.

3. 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 $\mathbb R^2$.)

4. 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>
5. 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>