Given an autonomous ode $\dot{x}=f(x)$ in $\mathbb{R}^n$ possessing a period-p time-periodic solution $\bar x(t)$, one can use the so-called variational equation about $\bar x$ to study its stability. This equation satisfies \begin{align} \dot{y}(t)=A(t)y \end{align}, where $A(t)=D_xf(\bar x(t))$.
Now the typical definition of Floquet multipliers $\lambda$ is that they are the e.values of the time-p flow map U(p) of the variational equation, i.e., of the matrix/operator U(p) where U(t) itself satisfies the variational equation, with $U(0)=I_{n\times n}$. The Floquet exponents $\mu$ are then defined as $\frac{1}{p}\log{\lambda}$.
An alternative definition can apparently be given by starting with the exponents instead. Here one defines $\mu$ as the eigenvalues of the operator $L[x](t)=(\frac{d}{dt}-A(t))x(t)$ on the space of p-periodic functions. And then the multipliers are defined analogously as $\lambda=e^{-\mu p}$.
I am looking for a proof that both the definitions are the same (modulo some signs if need be).
