Question on nonlinear dynamics

[JuanYsimura]

Homework Statement

Consider the system x' + x = F(t), where F is smooth, T-periodic function. IS it true that the system necessarily has a stable T-periodic solution x(t)? If so, Prove it; If not, find an F that provides a counterexample.

Homework Equations

x' + x = F(t)

The Attempt at a Solution

So, I think this system necessarily have a T-periodic solution x(t). To prove it, I let y = t so i can eliminate the time parameter from the equation so then I get a 2D system

x' = F(y) - x
y' = 1

SO, then I plan to find a poincare map so that showing that x(t) is T-periodic.

MY question is, before I go further, is this a good approach to the problem?

thanks.

 

[lanedance]

could you consider a Fourier expansion?