I am trying to find the Floquet multipliers of $x'=f(t)A_ox$ where f(t) is a scalar T-periodic function and $A_o$ is a constant matrix with real distinct eigenvalues.
I know that the floquet multipliers are the eigenvalues of L where the fundamental matrix $X(t)=e^{Lt}$.
I guess my first problem is that I don't know how to find the fundamental matrix of $x'=f(t)A_ox$. It doesn't fit any of the standard forms I have learned.
Thanks for any help you can give.
This is a time-varying linear system of the form $$\dot{x}=A(t)x$$ with the special structure $$A(t):=f(t)A_0$$ and $f(t)$ a $T$-periodic function. Due to the commutativity property $$A(t)A(s)=A(s)A(t)$$ the fundamendal matrix which is given by the Peano-Baker series takes the special form $$\Phi(t,t_0)=\exp\left(\int_{t_0}^tf(s)A_0ds\right)$$ For a $T$-periodic $A(t)$ the Floquet multipliers are the eigenvalues of the matrix $\Phi(T,0)$ and since the eigenvalues of $A_0$ are distinct the multipliers are $$\exp\left(\lambda_i(A_0)\int_0^Tf(s)ds\right)$$ with $\lambda_i(A_0)$ the $i$-th eigenvalue of $A_0$.