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Let $X$ be a non-vanishing vector field on the unit square $I^2$ in $\mathbb{R}^2$. I would like to show that every integral curve to $X$ exits the unit square in finite time.

This fact is used in a paper I am reading, in which the author says "(assume there is an integral curve that does not exit the unit square,) it would approach asymptotically some simple closed curve in $I^2$. In the interior of this curve the vector field would have to have a singularity." This does seem reasonable, since integral curves cannot cross themselves (unless they are simply closed curves, but as the author has noted this would result in a contradiction in the interior of the closed curve) so they should have no place to go except wrapping around. However I cannot make this idea rigorous at all.

Any help is appreciated!

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    $\begingroup$ Do you know Poincaré-Bendixson theorem? If some trajectory stays in $I^2$, then its $\omega$-limit set is non-empty. Due to Poincaré-Bendixson theorem there are three options for $\omega$-limit set: either it is a limit cycle, an equilibrium, or heteroclinic cycle/homoclinic loop. Last two options are instantly ruled out by the fact that vector field is non-vanishing on $I^2$. The presence of limit cycle implies that there is an equilibrium point in the domain enclosed by limit cycle, hence this case is also ruled out. $\endgroup$
    – Evgeny
    Jun 29, 2018 at 5:11
  • $\begingroup$ @Evgeny No I didn't, but that's exactly the thing I needed. Thanks! $\endgroup$
    – Chi Cheuk Tsang
    Jun 29, 2018 at 6:47

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Poincare-Hopf Theorem. The turning number on the boundary of the disk is ...what?

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  • $\begingroup$ Can you please elaborate a bit more? The turning number of the boundary of a disk is 1 (or -1) but I can only see how that proves there can't be any simply closed integral curves, instead of showing integral curves must asymptotically approach a simply closed curve. By Morse Index Theorem do you mean the Index theorem in Morse theory? On which space and function should I consider the Morse theory of? $\endgroup$
    – Chi Cheuk Tsang
    Jun 29, 2018 at 3:24
  • $\begingroup$ Sorry -- I thought that the part you were having trouble with was the existence of a singularity in the interior. And as for the morse index theorem... I screwed up and meant the Poincare-Hopf theorem. I shouldn't type answers late at night. I've edited to fix this last bit. $\endgroup$
    – John Hughes
    Jun 29, 2018 at 20:50

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