floquet

Floquet refers to a mathematical framework for analyzing linear systems with periodic coefficients, named after Gaston Floquet. The theory provides a structured way to understand how solutions behave when the governing parameters repeat in time.

In its classical form for a continuous-time linear system x′(t) = A(t)x(t) with A(t+T) = A(t), Floquet's theorem states that every solution can be written as x(t) = P(t) e^{Rt}, where P(t) is T-periodic and R is a constant matrix. Equivalently, the state after one period is x(T) = M x(0), with M = Φ(T) the monodromy (or fundamental) matrix. The eigenvalues of M, known as Floquet multipliers, determine stability: all multipliers inside the unit circle imply asymptotic decay; any multiplier outside implies growth; multipliers on the unit circle indicate neutral stability. Extensions cover discrete-time systems and various numerical approaches for computing the periodic factor P(t) and R.

In physics, Floquet theory is used to treat quantum systems with time-periodic Hamiltonians H(t+T) = H(t). Solutions of the Schrödinger equation can be written as |ψ(t)> = e^{-iεt/ħ} |φ(t)>, with |φ(t+T)> = |φ(t)>, where ε are quasi-energies defined modulo 2πħ/T. The one-period evolution U(T,0) yields a Floquet spectrum e^{-iεT/ħ}. This formalism supports high-frequency expansions and Floquet engineering, whereby periodic driving induces effective static properties, enabling control over materials, cold atoms, and photonic structures.

Overall, Floquet theory provides a unifying lens for periodic time dependence across mathematics, engineering, and physics.