Here's my problem.
Let $X=-\nabla f$, with $f\in C^r$, $r\geq 2$. Then $X$ has no periodic orbits. Show the $\omega$ - limit is either a single point or an infinite set.
My idea is to use Poincaré-Bendixson theorem to prove this, but i'm not sure how to apply it cause' I don't have much information about the singularities.