I'm trying to classify periodic matrices $A(t)$ where its Floquet decomposition has a finite number of harmonies.
Let $$\dot{x}_1=A(t)x_1 $$
By Floquet Theory, the transition matrix is:$$P(t)e^{(t-t_0)R} P^{-1} (t_0)$$
The equation that connects those 2 is:$$A(t)P(t)=\dot{P}(t)+P(t)R$$
Let's declare matrix $A(t)\in\Bbb R^{2\times 2}$ matrix with a finite number of harmonies, for example:
$$A(t)=\begin{pmatrix}-\cos(2wt)&-w-\sin(2wt)\\ w-\sin(wt)& \cos(2wt)\end{pmatrix}$$
Can I find a specific property about matrix $A(t)$ that can assure me a matrix $P(t)$ with a finite number of harmonies?
For this example,$$P(t)=\begin{pmatrix}-\sin(wt)&\cos(wt)\\ \cos(wt)& \sin(wt)\end{pmatrix}, \quad R=\begin{pmatrix}1&0\\ 0& -1\end{pmatrix}$$
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