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
being 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
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.
Note that the solutions of the linear differential equation form a vector space.
is called a fundamental matrix solution if the columns form a basis of the solution set.
is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists
A principal fundamental matrix can be constructed from a fundamental matrix using
The solution of the linear differential equation with the initial condition
is any fundamental matrix solution.
be a linear first order differential equation, where
is a column vector of length
be a fundamental matrix solution of this differential equation.
, Here is known as the monodromy matrix.
In addition, for each matrix
and a real periodic (period-
gives rise to a time-dependent change of coordinates (
), under which our original system becomes a linear system with real constant coefficients
is continuous and periodic it must be bounded.
is called a Floquet normal form for the fundamental matrix
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
is a characteristic multiplier of the system.
Notice that Floquet exponents are not unique, since
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.