Floquet theory

${\displaystyle \dot{x}=A(t)x,}$

with ${\displaystyle x\in R^n}$ and ${\displaystyle {\displaystyle A(t)\in R^{n\times n}}}$ being a piecewise continuous periodic function with period ${\displaystyle T}$ and defines the state of the stability of solutions.

The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet (1883), gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change ${\displaystyle {\displaystyle y=Q^{-1}(t)x}}$ with ${\displaystyle {\displaystyle Q(t+2T)=Q(t)}}$ that transforms the periodic system to a traditional linear system with constant, real coefficients.

When applied to physical systems with periodic potentials, such as crystals in condensed matter physics, the result is known as Bloch's theorem.

Note that the solutions of the linear differential equation form a vector space. A matrix ${\displaystyle \varphi (t)}$ is called a fundamental matrix solution if the columns form a basis of the solution set. A matrix ${\displaystyle \mathrm{\Phi }(t)}$ is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists ${\displaystyle t_0}$ such that ${\displaystyle \mathrm{\Phi }(t_0)}$ is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using ${\displaystyle \mathrm{\Phi }(t)=\varphi (t){\varphi }^{-1}(t_0)}$. The solution of the linear differential equation with the initial condition ${\displaystyle x(0)=x_0}$ is ${\displaystyle x(t)=\varphi (t){\varphi }^{-1}(0)x_0}$ where ${\displaystyle \varphi (t)}$ is any fundamental matrix solution.

Floquet's theorem

Let ${\displaystyle \dot{x}=A(t)x}$ be a linear first order differential equation, where ${\displaystyle x(t)}$ is a column vector of length ${\displaystyle n}$ and ${\displaystyle A(t)}$ an ${\displaystyle n\times n}$ periodic matrix with period ${\displaystyle T}$ (that is ${\displaystyle A(t+T)=A(t)}$ for all real values of ${\displaystyle t}$). Let ${\displaystyle \varphi (t)}$ be a fundamental matrix solution of this differential equation. Then, for all ${\displaystyle t\in \mathbb{R}}$,

${\displaystyle \varphi (t+T)=\varphi (t){\varphi }^{-1}(0)\varphi (T).}$

Here

${\displaystyle {\varphi }^{-1}(0)\varphi (T)}$

is known as the monodromy matrix. In addition, for each matrix ${\displaystyle B}$ (possibly complex) such that

${\displaystyle e^{TB}={\varphi }^{-1}(0)\varphi (T),}$

there is a periodic (period ${\displaystyle T}$) matrix function ${\displaystyle t\mapsto P(t)}$ such that

${\displaystyle \varphi (t)=P(t)e^{tB}\text{ for all }t\in \mathbb{R}.}$

Also, there is a real matrix ${\displaystyle R}$ and a real periodic (period-${\displaystyle 2T}$) matrix function ${\displaystyle t\mapsto Q(t)}$ such that

${\displaystyle \varphi (t)=Q(t)e^{tR}\text{ for all }t\in \mathbb{R}.}$

In the above ${\displaystyle B}$, ${\displaystyle P}$, ${\displaystyle Q}$ and ${\displaystyle R}$ are ${\displaystyle n\times n}$ matrices.