  Floquet's Theorem is a celebrated result in the theory of ordinary
differential equations. Essentially, the theorem states that, when studying a
linear differential system with $T$-periodic coefficients, we can apply a,
possibly complex, $T$-periodic change of variables that transforms it into a
linear system with constant coefficients. In this paper, we explore further the
question of the nature of this change of variables. We state necessary and
sufficient conditions for it to be real and $T$-periodic. Failing those
conditions, we prove that we can still find a real change of variables that is
``partially" $T$-periodic and ``partially" $T$-antiperiodic. We also present
applications of this new form of Floquet's Theorem to the study of the behavior
of solutions of nonlinear differential systems near periodic orbits.
