= Solution
\b[No.] Consider two uncoupled oscillators with nonzero amplitudes:
$$
q_i(t)=A_i\cos(\omega_it+\phi_i),
\qquad
p_i(t)=-A_i\omega_i\sin(\omega_it+\phi_i),
\qquad i=1,2.
$$
The orbit is bounded. If it had a <periodic orbit>[period] $T>0$, then $\omega_1T$ and $\omega_2T$ 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.
Back to article page
