$$ y(t) = \begin{bmatrix} cos\sqrt\omega & -sin\sqrt3\omega & 0 & 0 \\ sin\sqrt3\omega & cos\sqrt3\omega & 0 & 0 \\ 0 & 0 & cos\omega & -sin\omega \\ 0 & 0 & sin\omega & cos\omega \end{bmatrix} y(0)$$
QUESTION: Suppose these two systems have frequencies $$\omega_1 \text{ and } \omega_2$$ Show that the general solution is periodic only if the ratio of the frequencies is a rational number.
INTIAL APPROACH: So, for $\omega_1=\sqrt3\omega$ it's clear that the period is $t=\frac{2\pi}{\sqrt3\omega}$ and for $\omega_2=\omega$ the period is $t=\frac{2\pi}{\omega}$
But where will a rational number come from this?
If $\dfrac{\omega_1}{\omega_2}=\dfrac pq$ where $(p,q)=1$, then $\dfrac{T_1}{T_2}=\dfrac{\omega_2}{\omega_1}=\dfrac qp$, which means there exist a time $T=pT_1=qT_2$ such that both two periodic motions are return to origins.
On the contrary, if the ratio isn't a rational number, such $T$ will never happen.