I was studying an example of a coupled oscillator the other day, namely two identical masses attached to three springs, the lateral ones of which with the same elastic constant, when I came across the claim that:
“If the ratio of the normal modes periods“ is rational then the system moves periodically, that is, $(x_1(t), x_2(t))$ is a periodic function, where $x_1$, $x_2$ refer to the positions of the masses.”
Having a rational ratio of periods, $T_1/T_2=n/m$, means that the periods are $T_1=nT_0$ and $T_2=mT_0$ for some integers $n$, $m$. We can suppose that $n$ and $m$ have no common integer factor (we could cancel it out if there were such a factor). In this case the motion will exactly repeat for the first time after time $T_3=nm T_0$ because $T_3= mT_1=nT_2$. The motion is therefore periodic with period $T_3$.