A quasiperiodic orbit on a torus has angular coordinates modulo . It is periodic exactly when the nonzero frequencies are commensurate; rationally independent frequencies produce a dense orbit.
For with irrational frequency ratio and nonzero amplitudes, . The phase trajectory is dense on the two-torus, so this bound is a supremum. The required exact phase alignment may never occur at finite time. Thus an exam's maximum-amplitude estimate can be a limiting bound rather than an attained displacement; finite observation intervals have smaller maxima.
For two uncoupled harmonic oscillators with nonzero amplitudes and irrational frequency ratio , 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.
