Invariant torus

An invariant torus is a subset homeomorphic to a torus and preserved by a flow. In action-angle variables, an integrable Hamiltonian flow moves linearly in the angular coordinates of each regular invariant torus.

A quasiperiodic orbit on a torus has angular coordinates $\theta (t)=\theta (0)+\omega t$ modulo $2\pi$. It is periodic exactly when the nonzero frequencies are commensurate; rationally independent frequencies produce a dense orbit.

For two uncoupled harmonic oscillators with nonzero amplitudes and irrational frequency ratio ${\omega }_1/{\omega }_2$, the orbit is bounded but not periodic. It winds quasiperiodically on a two-dimensional invariant torus; a common period would force the frequency ratio to be rational.