Solutions to the differential equation $my'' + ky = F \sin \omega t$ show resonance when the driving frequency $\omega$ equals the natural frequency $\sqrt{k/m}$. That is, solutions are unbounded when $\omega = \sqrt{k/m}$ and periodic for all other frequencies. It seems that when the sine function is replaced by a sawtooth function, there are more resonant frequencies. Numerical experiments [here](https://www.johndcook.com/blog/2020/09/19/sawtooth-waves/) with $m = k = F = 1$ seem to be resonant when $\omega = 1/n$ for any integer $n$. Are there any theoretical results that prove this conjecture?



Resonance arising when harmonic oscillator is excited using sawtooth
The sawtooth function $f$ has Fourier decomposition
$$
f(t) = \frac{1}{2}-\frac{1}{\pi}\sum\_{n=1}^\infty \frac{1}{n} \sin(n\omega t)
$$
Therefore, if $\omega=\frac{\omega\_0}{n}$, the $n$-th harmonic of $f$ will have angular frequency $n\omega=\omega\_0$, resulting in resonance.


