Using Poincare Bendixson's Theorem to prove a periodic Orbit

[bagram]

Homework Statement

The System is
[itex]x˙ = 2x -y - x(x^2+y^2)[/itex]
[itex]y˙ = 5x - 2y(x^2+y^2)[/itex]

Using a trapping region to show there is a periodic orbit

Homework Equations

Use Poincare Bendixson's Theroem

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 [itex]V=x^2+(ay^2/2)[/itex] but I can't figure another function to use to show the flow leaves a subspace. Is there another way to attempt this question?

 

[mighty2000]

I would suggest using Poincare-Bendixson's theorem to prove the existence of a periodic orbit in the given system.

According to Poincare-Bendixson's theorem, if a dynamical system has a trapping region and the vector field is continuous and non-zero within the region, then the system must exhibit a periodic orbit.

In this case, we can define the trapping region as the set of points where x^2 + y^2 < 1. This region is bounded and the vector field is continuous and non-zero within this region.

Next, we can show that the vector field is pointing inward at some points within the trapping region and outward at other points. This can be done by analyzing the sign of the derivatives of x˙ and y˙ with respect to x and y within the trapping region.

If we can show that there exists a point within the trapping region where the vector field is pointing inward and another point where it is pointing outward, then we can conclude that there must exist a periodic orbit within the trapping region.

Therefore, by using Poincare-Bendixson's theorem, we can prove the existence of a periodic orbit in the given system.