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)

__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>