Floquet's Theorem/Proof 2

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


Proof

Let $S\left({t}\right)=\Phi \left({t + T}\right){\Phi \left({T}\right)}^{-1}$, then we have that:

\(\displaystyle \frac {\mathrm d} {\mathrm d t} \left({S \left({t}\right)}\right)\) \(=\) \(\displaystyle \Phi' \left({t + T}\right) {\Phi \left({T}\right)}^{-1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \mathbf A \left({t + T}\right) \Phi \left({t + T}\right) {\Phi \left({T}\right)}^{-1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \mathbf A \left({t}\right) S \left({t}\right)\) $\quad$ $\quad$

So $S\left({t}\right)$ is a fundamental matrix and $S\left({0}\right)=Id$ then $S\left({t}\right)= \Phi\left({t}\right)$ which it means that $\Phi \left({t + T}\right)=\Phi \left({t}\right)\Phi \left({T}\right)$.

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

$\Phi \left({T}\right) = e^{\mathbf BT}$


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

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

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

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

$\blacksquare$


Source of Name

This entry was named for Achille Marie Gaston Floquet.