Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 32C / e / Solution

No. Consider two uncoupled oscillators with nonzero amplitudes:The orbit is bounded. If it had a period $T>0$, then ${\omega }_{1}T$ and ${\omega }_{2}T$ would both be integer multiples of $2\pi$, forcing ${\omega }_1/{\omega }_2\in \mathbb{Q}$. Choosing an irrational frequency ratio gives an incommensurate harmonic-oscillator orbit, which is quasiperiodic on an invariant torus and not periodic.