Floquet's Theorem

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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:

$\Phi \left({t}\right) = \mathbf P \left({t}\right) e^{\mathbf Bt}$

We assume the two hypotheses of the theorem.

We have that:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \frac {\mathrm d} {\mathrm d t} \left({\Phi \left({t + T}\right)}\right)$	$=$	$\displaystyle \Phi' \left({t + T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \mathbf A \left({t + T}\right) \Phi \left({t + T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \mathbf A \left({t}\right) \Phi \left({t + T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

So the first implication of the theorem holds, i.e. that $\Phi \left({t + T}\right)$ is a fundamental matrix.

Because $\Phi \left({t}\right)$ and $\Phi \left({t + T}\right)$ are both fundamental matrices, there must exist some matrix $\mathbf C$ such that:

$\Phi \left({t + T}\right) = \Phi \left({t}\right) \mathbf C$

Hence by the existence of the matrix logarithm, there exists a matrix $\mathbf B$ such that:

$\mathbf C = e^{\mathbf BT}$

Defining $\mathbf P \left({t}\right) = \Phi \left({t}\right) e^{-\mathbf B t}$, it follows that:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \mathbf P \left({t + T}\right)$	$=$	$\displaystyle \Phi \left({t + T}\right) e^{-\mathbf B t - \mathbf B T}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \Phi \left({t}\right) C e^{-\mathbf B T} e^{-\mathbf B t}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \Phi \left({t}\right) e^{-\mathbf B t}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \mathbf P \left({t}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

and hence $\mathbf P \left({t}\right)$ is periodic with period $T$.

As $\Phi \left({t}\right) = \mathbf P \left({t}\right) e^{\mathbf B t}$, the second implication also holds.

$\blacksquare$

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This entry was named for Gaston Floquet.

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \frac {\mathrm d} {\mathrm d t} \left({\Phi \left({t + T}\right)}\right)\)	\(=\)	\(\displaystyle \Phi' \left({t + T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \mathbf A \left({t + T}\right) \Phi \left({t + T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \mathbf A \left({t}\right) \Phi \left({t + T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathbf P \left({t + T}\right)\)	\(=\)	\(\displaystyle \Phi \left({t + T}\right) e^{-\mathbf B t - \mathbf B T}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \Phi \left({t}\right) C e^{-\mathbf B T} e^{-\mathbf B t}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \Phi \left({t}\right) e^{-\mathbf B t}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \mathbf P \left({t}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)