Floquet Theory

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

with a piecewise continuous periodic function with period .

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 (generalized to three dimensions) 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.