Liouville formula for a fundamental matrix ($\det \Phi (t)=\exp \left({\int }_0^t\mathrm{tr}A(s)ds\right)$)

If $\dot{\Phi }=A(t)\Phi$ and $\Phi (0)=I$, differentiation of the determinant gives $\det \Phi (t)=\exp ({\int }_0^t\mathrm{tr}A(s)ds)$. For a planar periodic orbit this gives the nontrivial Floquet multiplier, since the tangent multiplier is one.