Floquet's Theorem

Theorem
Let $\mathbf A \left({t}\right)$ be a continuous matrix function with period $T$.

Let $\Phi \left({t}\right)$ be a fundamental matrix of the Floquet system $\mathbf x' = \mathbf A \left({t}\right) \mathbf x$.

Then $\Phi \left({t + T}\right)$ is also a fundamental matrix.

Moreover, there exists:
 * A nonsingular, continuously differentiable matrix function $\mathbf P \left({t}\right)$ with period $T$
 * A constant (possibly complex) matrix $\mathbf B$ such that:
 * $\Phi \left({t}\right) = \mathbf P \left({t}\right) e^{\mathbf Bt}$

Also see

 * General Fundamental Matrix