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.
Read more about Floquet Theory: Floquet's Theorem, Consequences and Applications, Floquet's Theorem Applied To Mathieu Equation
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