Floquet theory

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 ${\displaystyle \phi \,(t)}$ be a fundamental matrix solution of this differential equation. Then, for all ${\displaystyle t\in \mathbb {R} }$,

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

Here

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

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

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

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

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

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

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

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

This mapping ${\displaystyle \phi \,(t)=Q(t)e^{tR}}$ gives rise to a time-dependent change of coordinates (${\displaystyle y=Q^{-1}(t)x}$), under which our original system becomes a linear system with real constant coefficients ${\displaystyle {\dot {y}}=Ry}$. Since ${\displaystyle Q(t)}$ is continuous and periodic it must be bounded. Thus the stability of the zero solution for ${\displaystyle y(t)}$ and ${\displaystyle x(t)}$ is determined by the eigenvalues of ${\displaystyle R}$.

The representation ${\displaystyle \phi \,(t)=P(t)e^{tB}}$ is called a Floquet normal form for the fundamental matrix ${\displaystyle \phi \,(t)}$.

The eigenvalues of ${\displaystyle e^{TB}}$ are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps ${\displaystyle x(t)\to x(t+T)}$. A Floquet exponent (sometimes called a characteristic exponent), is a complex ${\displaystyle \mu }$ such that ${\displaystyle e^{\mu T}}$ is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since ${\displaystyle e^{(\mu +{\frac {2\pi ik}{T}})T}=e^{\mu T}}$, where ${\displaystyle k}$ is an integer. The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if the Lyapunov exponents are nonpositive and unstable otherwise.

• Floquet theory is very important for the study of dynamical systems, such as the Mathieu equation.
• Floquet theory shows stability in Hill differential equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field.
• Bond softening and bond hardening in intense laser fields can be described in terms of solutions obtained from the Floquet theorem.

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