Homework 4 - Due Nov 7th

Due Nov 7, 2019 by 5p.m.
Points 20

1. Problems in Lynch: Section 4.5 -- 5
(First, find nullclines and critical points and classify critical points. Next, mark the nullclines and fixed points on the the first quadrant. Determine the sign of dx/dt and dy/dt in each region enclosed by nullclines. Use all of this information to sketch a few orbits. For example, parts of stable and unstable manifolds, etc. Finally, interpolate other trajectories and finish the phase portraits. Check your work using a computer algebra system. You should be able to sketch phase portraits without the aid of a computer at the exam. So use the computer algebra system only to check your work!)

2. Write each of these in polar coordinates and determine whether the origin is a stable focus, unstable focus or a centre.

(a) $\dot{x}=x-y,\ \dot{y}=x+y$

(b) $\dot{x}=-y+xy^2,\ \dot{y}=x+y^3$

3. Consider the non-linear system

$\begin{pmatrix}\dot{x} \\ \dot{y} \\ \dot{z}\end{pmatrix} = \begin{pmatrix}-x\\ -y+x^2 \\ z + x^2\end{pmatrix}.$

(a) Find its linearization at $(0,0,0).$

(b) Next, consider the continuous map $H: \mathbb{R}^3 \to \mathbb{R}^3$ given by

$H :\begin{pmatrix}x \\ y \\ z\end{pmatrix} \mapsto \begin{pmatrix}x\\ y+x^2 \\ z + x^2/3\end{pmatrix}.$

Show that it has a continuous inverse $H^{-1}$ .

(c) Further, show that under the non-linear coordinate transform given by $H$ , the original non-linear system gets transformed into its linearization at $(0,0,0).$

(This is an example of Hartman-Grobman theorem in action.)

4. Consider the system

$\begin{pmatrix}\dot{x} \\ \dot{y} \\ \dot{z}\end{pmatrix} = \begin{pmatrix}-y-xy^2+z^2-x^3\\ x+z^3-y^3 \\ -xz-zx^2-yz^2-z^5\end{pmatrix}.$

(a) Use the Lyapunov function $V(x,y,z)=x^2+y^2+z^2$ to show that the origin is asymptotically stable equilibrium point of the system.

(b) Show that the trajectories of the linearization near the origin lie on on circles parallel to the $xy-$ plane, and hence, the origin is stable but not asymptotically stable for the linearization.