Homework 5 - Due Nov 21st

Due Nov 21, 2019 by 5p.m.
Points 20

1. (a) Determine a center manifold for the rest point at the origin of the system

$\dot{x}=-xy,\ \dot{y}=-y+x^2-2y^2$

and a differential equation for the dynamics on this center manifold.

(b) Show there is an open set of initial conditions (containing the center manifold) that is attracted to the center manifold.

Hint: Look for the center manifold as a graph of a function of the form $y=h(x)=ax^2+bx^3+\dots$ . Why does the expected $h$ have $h(0)=0$ and $h'(0)=0$ ? The condition for invariance is $\dot{y}=h'(x)\dot{x}$ (Why?). Find the first few terms of the series expansion for $h$ , formulate a conjecture about the form of $h$ , and then find $h$ explicitly. Once $h$ is known, the differential equation for the flow on the center manifold is given by $\dot{x}=-h(x)x$ (Why?).

2. Problems in Lynch, Section 5.6 -- 1, 7 (b), (e) and 8 (a). (This will be graded in the next assignment. If you submitted you don't have to resubmit.)

3. Show that the planar two body problem given below can be written as a Hamiltonian system with two degrees of freedom.

$\ddot{x}=-\frac{x}{(x^2+y^2)^{3/2}}$