Floquet theory

Given a system in which the forces are periodic—such as a pendulum under a periodic driving force, or an oscillating circuit driven by alternating current—the overall behavior of the system is not necessarily fully periodic. For instance, consider a child being pushed on a swing: although the motion is driven by regular, periodic pushes, the swing can gradually reach greater heights while still oscillating to and fro. This results in a combination of underlying periodicity and growth.

Floquet theory provides a way to analyze such systems. Its essential insight is similar to the swing example: the solution can be decomposed into two parts—a periodic component (reflecting the repeated motion) and an exponential factor (reflecting growth, decay, or neutral stability). This decomposition allows for the analysis of long-term behavior and stability in time-periodic systems.

Formally, Floquet theory is a branch of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form

x

˙

=

A

(

t

)

x

,

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

with

x

∈

R

n

{\displaystyle x\in {R^{n}}}

and

A

(

t

)

∈

R

n

×

n

{\displaystyle \displaystyle A(t)\in {R^{n\times n}}}

being a periodic function with period

T

{\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

y

=

Q

−

1

(

t

)

x

{\displaystyle \displaystyle y=Q^{-1}(t)x}

with

Q

(

t

+

2

T

)

=

Q

(

t

)

{\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.