My Floquet theory is little bit rusty so I have a trivial question. Does a monodromy matrix depend on the initial conditions? On the surface it should since it is given as $B=X^{-1}(0)X(T)$ where $T$ is periodicity of my system $\dot{x}=A(t)x$, $A(t+T)=A(t)\mbox{, } T>0$. However $B$ actually doesn't depend on the choice of the fundamental matrix do I am not sure that it depends on the initial condition?
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A monodromy matrix does not depend on the initial conditions. Incidentally, note that $B=X(t)^{-1}X(t+T)$ for all $t$. However a monodromy matrix does depend on the choice of fundamental matrix, although all monodromy matrices are conjugate.
