# Using Poincare Bendixson's Theorem to prove a periodic Orbit

1. Apr 17, 2013

### bagram

1. The problem statement, all variables and given/known data
The System is
$x˙ = 2x -y - x(x^2+y^2)$
$y˙ = 5x - 2y(x^2+y^2)$

Using a trapping region to show there is a periodic orbit

2. Relevant equations
Use Poincare Bendixson's Theroem

3. The attempt at a solution
I tried constructing 2 Lyapunov type functions to show that DV/dt>0 and DV/dt<0 which show that in one region the flow is going into one region while the other is going out of the region. I could show that there is a region where the flow goes in if I use $V=x^2+(ay^2/2)$ but I can't figure another function to use to show the flow leaves a subspace. Is there another way to attempt this question?

Last edited: Apr 17, 2013