# ODE with finite Fourier expansion periodic coefficients

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.