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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.

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closed as off-topic by John B, Xander Henderson, Aweygan, Chris Custer, user99914 Apr 21 '18 at 1:56

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    $\begingroup$ Possible duplicate of Elementary properties of gradient systems $\endgroup$ – John B Apr 20 '18 at 21:27
  • $\begingroup$ Do have a look at that other question. By the way, sometimes we really need to try a bit more. $\endgroup$ – John B Apr 20 '18 at 21:28