Floquet theory

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

with a piecewise continuous periodic function with period 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 with that transforms the periodic system to a traditional linear system with constant, real coefficients.

In solid-state physics, the analogous result is known as Bloch's theorem.

Note that the solutions of the linear differential equation form a vector space. A matrix is called a fundamental matrix solution if all columns are linearly independent solutions. A matrix is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists such that is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using . The solution of the linear differential equation with the initial condition is where is any fundamental matrix solution.

Floquet's theorem[edit]

Let be a linear first order differential equation, where is a column vector of length and an periodic matrix with period (that is for all real values of ). Let be a fundamental matrix solution of this differential equation. Then, for all ,

Here

is known as the monodromy matrix. In addition, for each matrix (possibly complex) such that

there is a periodic (period ) matrix function such that

Also, there is a real matrix and a real periodic (period-) matrix function such that

In the above , , and are matrices.

Consequences and applications[edit]

This mapping gives rise to a time-dependent change of coordinates (), under which our original system becomes a linear system with real constant coefficients . Since is continuous and periodic it must be bounded. Thus the stability of the zero solution for and is determined by the eigenvalues of .

The representation is called a Floquet normal form for the fundamental matrix .

The eigenvalues of are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps . A Floquet exponent (sometimes called a characteristic exponent), is a complex such that is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since , where 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.

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