Regard the ordinary differential equation
$$ \dot a(t) = z(t) a(t) $$
where $a(t)$ and $z(t)$ are matrix valued such that $z$ is periodic ($z(t+2\pi)=z(t)$). Then it is well-known (Floquet theory), that the solution can be written in the form $$ a(t)=P(t)\exp(tX) $$ with $P(t)$ a periodic matrix with $P(0)=E_n$.
If now additionally $z(t)$ has a finite Fourier expansion, is it true that $P(t)$ also has a finite Fourier expansion?
This is even not true for the most simple case $\dot a(t)=\sin(t)a(t)$ with the solution $a(t)=a(0)e^{1-\cos(t)}$. The expansion $$ e^{-\cos(t)}=\sum\frac{(-\cos(t))^k}{k!} $$ has components in every harmonic frequency.