How to check the following system on existence of periodic solutions?

Question

Let's have following system of DE: $$ \begin{cases} \dot{x} = y(1+x - y^{2}) \\ \dot{y} = x(1+y-x^{2})\end{cases}, \quad x, y \geqslant 0 $$ How to check whether this system contains periodic solutions?

Answer 1

HINT:

Try some software that sketches the phase portrait of the system. For instance this link. There may not be any closed curves.

Another online phase plotter here.

Answer 2

First you should use Bendixson's Criterion to even see if it can have periodic solutions at all. Bendixon's Criterion states that if you take the divergence of this system and if the divergence never changes signs on a region $D$, and there are no holes in $D$ then there cannot be any periodic solutions in $D$. So in this case $$\frac{\partial x'}{\partial x}+\frac{\partial y'}{\partial y} = y + x$$ is the divergence. The actual system is easy to check for holes so that should be practice for you.

Now if you know there can exist periodic solutions. We can use the Poincare-Bendixson theorem to find out whether a region around the hole contains a periodic orbit. The way we do this is we take the dot product of the system with a normal vector to the outer hole and the inner hole, and if they have the same sign there is no periodic solution, otherwise there is. This is only if you're doing explicitly the outer normal or inner normal on both.

The general idea here is if you have a solution that is between these two 'holes', then it will always stay in between them because whenever it gets close to one of the boundaries it gets pushed off back in to the region.