Consider the 2-D system $\dot{x}=F(x)$ and let $S\subset\mathbb{R}^2$ be compact s.t.
- (1) $S$ contains no equilibria, or, contains exactly one equilibrium point $x_e$ s.t. all the eigenvalues of $D F(x_e)$ have positive real parts;
- (2) $S$ is positively invariant.
Then $S$ contains a closed orbit.
I believe above tells us how to use Poincaré-Bendixson Theorem, but I have some difficulty in proving it. If $S$ contains no equilibria, its easy to see the corresponding $\omega$ limit set satisfies the Theorem' s condition. If $S$ contains a unstable one equilibrium point as it says, the flow can not stop at the point, the proposition seems natural but I do not know how to write a valid proof. Also what happens when we have two equilibrium points?
Appreciate any help!