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I have an interesting dynamical systems problem that has had me stumped for a few hours now, so I'm hoping I can get some help. The problem is concerned with flows on the torus. The model is given by $\dot {\theta}_1=\omega_1$ and $\dot {\theta}_2=\omega_2$, where $\theta_{1,2}$ are the phases of the oscillators and $\omega_{1,2}$ are the natural frequencies. The problem is to show if $\frac{\omega_1}{\omega_2}$ is irrational, then every trajectory is dense. The only idea I've come up with is contradiction, since it seems clear that the trajectories would have to be dense since they're quasiperiodic. Any help would be greatly appreciated. Thanks

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    $\begingroup$ all you really need to know is that, given some irrational number $\gamma,$ expressions of the form $ m + n \gamma$ are dense in the real numbers. Here $m,n$ are integers, allowed positive or negative or zero. $\endgroup$ – Will Jagy Dec 4 '15 at 2:31
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This classical result actually has a nice history to it, but is in effect a version of the Poincaré-Bendixson for the 2-torus, which basically says that on the 2-torus, the only $\omega-$sets are a singleton, a periodic orbit, or the whole 2-torus. This is a nice treatment of a general version on the $n-$torus.

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  • $\begingroup$ Why heteroclinic and homoclinic contours are excluded from consideration? They also could be $\omega$-limit sets. $\endgroup$ – Evgeny Dec 4 '15 at 20:46
  • $\begingroup$ I don't know if I believe a homo-/hetero-clinic orbit could be a $\omega-$limit set? Aren't the $\omega-$sets of these orbits the singleton stationary point that connects them? Can a limit cycle be enclosed by such an orbit? $\endgroup$ – charlestoncrabb Dec 4 '15 at 23:27
  • $\begingroup$ If you have multiple saddles and there are connections between them, the union of these connections with saddle equilibria forms (by definition) heteroclinic contour. The whole contour could be a limit set. It is stated in Poincare-Bendixson theorem for planar flows. $\endgroup$ – Evgeny Dec 5 '15 at 4:56

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