Assume $\dot x(t)=v(x,t)$ is a $T$-periodic, with respect to $t$ dynamical system. That is: $x\in\mathbb R^n,\quad t\in\mathbb R,\qquad v(x,t+T)=v(x,t)$.
Let $x_0$ be a smooth periodic solution in elementary functions. How can one use Mathematica to compute symbolically a fundamental matrix for the linearized system around $x_0$ and its monodromy matrix?