Irrational flow on torus

Question

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

Answer 1

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.