Homework 2 - Due Oct 3rd

Due Oct 3, 2019 by 5p.m.
Points 20

Problems in red will be graded.

Problems in Lynch, Section 3.7 -- 1 (1 (a) only), 2 (a), (f)
Problems in Lynch, Section 8.6 -- 1
(a) Show that the following sets of vector functions are linearly independent.
(i) $\underline{x}^1(t)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},\ \underline{x}^2(t)=\begin{bmatrix} t \\ 0 \end{bmatrix}$

(ii) $\underline{x}^1(t)=\begin{bmatrix} e^{2t} \\ e^{-t} \\ 0 \end{bmatrix},\ \underline{x}^2(t)=\begin{bmatrix} 0\\ e^{-t} \\ e^{4t}\end{bmatrix}$,\ \underline{x}^3(t)=\begin{bmatrix} e^{2t}\\ 0 \\ e^{4t} \end{bmatrix}$

(iii) $\underline{x}^{1}(t)=\begin{bmatrix} e^{2t} \\ t^2 \end{bmatrix},\ \underline{x}^{2}(t)=\begin{bmatrix} 0 \\ e^t \end{bmatrix}$
(b) In each of the cases above, calculate the Wronskian corresponding to the matrix $M(t)=\begin{bmatrix} \underline{x}^1(t)& \underline{x}^2(t) & \dots &\underline{x}^k(t) \end{bmatrix}$ .
(c) Suppose $M(t)=\begin{bmatrix} \underline{x}^1(t)& \underline{x}^2(t) & \dots &\underline{x}^k(t) \end{bmatrix}$ is indeed a matrix solution of $\dot{\underline{x}}=A\underline{x}$ . Show that in this case, $W(0)=0 \implies W(t)=0,\ t\in \mathbb{R}$ without using the Louville's formula.
Note: This problem shows that we cannot use the Wronskian being non-zero as a way to define linear independence in general situations.
Consider the linear system

$\dot{x}=ax - by,\ \dot{y}=bx + ay.$

(a) Show that these equations reduce to

$\dot{r}=ar,\ \dot{\theta}=b$

in polar coordinates.

(b) Solve these equations with the initial conditions $r(0) = r_0$ and $\theta(0) = \theta_0$ .
(c) Sketch the phase portrait using the solutions you obtained.
(a) Solve the system

$\dot{\underline{x}}= \begin{pmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{pmatrix} \underline{x}$

(b) Find the stable, unstable and center subspaces, i.e., $E^s, E^u$ and $E^c$ .
(c) Show that each of these are invariant under the flow, i.e., for each $j= c, s$ and $u$ , if $\underline{x}_0 \in E^j$ , then $e^{At}x_0 \in E^j$ for all $t$ .
(a) Show that

$\Phi(t)=\begin{pmatrix} e^{-2t}\cos t & - \sin t \\ e^{-2t}\sin t & \cos t \end{pmatrix}$

is a fundamental matrix solution of

$\dot{\underline{x}}= \begin{pmatrix} -2\cos^2 t & - 1- \sin 2t \\ 1-\sin 2t & -2 \sin^2 t \end{pmatrix}\underline{x}.$

(b) Determine whether the fixed point $(0,0)$ is stable or not.

(c) In general, if $A(t)$ is periodic matrix of period $T$ then the fundamental matrix solution of $\dot{\underline{x}}= A(t)\underline{x}$ takes the form $P(t)e^{Bt}$ where $P(t)$ is a matrix of period $2T$ and $B$ is a constant matrix (Floquet Theorem).

Determine $P(t)$ and $B$ corresponding to the given $\Phi(t)$ .