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).
- 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 $y_2$.
- 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 $E^u$.
- 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 $\mathbb R^2$.)
- 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> - 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>
