Floquet theory

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 formula_2 a piecewise continuous periodic function with period formula_3 and defines the state of the stability of solutions.
The main theorem of Floquet theory, Floquet's theorem, due to , gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change formula_4 with formula_5 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. A matrix formula_6 is called a "fundamental matrix solution" if all columns are linearly independent solutions. A matrix formula_7 is called a "principal fundamental matrix solution" if all columns are linearly independent solutions and there exists formula_8 such that formula_9 is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using formula_10. The solution of the linear differential equation with the initial condition formula_11 is formula_12 where formula_13 is any fundamental matrix solution.
Floquet's theorem.
Let formula_14 be a linear first order differential equation,
where formula_15 is a column vector of length formula_16 and formula_17 an formula_18 periodic matrix with period formula_3 (that is formula_20 for all real values of formula_21). Let formula_22 be a fundamental matrix solution of this differential equation. Then, for all formula_23,
Here
is known as the monodromy matrix.
In addition, for each matrix formula_26 (possibly complex) such that
there is a periodic (period formula_3) matrix function formula_29 such that
Also, there is a "real" matrix formula_31 and a "real" periodic (period-formula_32) matrix function formula_33 such that
In the above formula_26, formula_36, formula_37 and formula_31 are formula_18 matrices.
Consequences and applications.
This mapping formula_40 gives rise to a time-dependent change of coordinates (formula_41), under which our original system becomes a linear system with real constant coefficients formula_42. Since formula_43 is continuous and periodic it must be bounded. Thus the stability of the zero solution for formula_44 and formula_15 is determined by the eigenvalues of formula_31.
The representation formula_47 is called a "Floquet normal form" for the fundamental matrix formula_48.
The eigenvalues of formula_49 are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps formula_50. A "Floquet exponent" (sometimes called a characteristic exponent), is a complex formula_51 such that formula_52 is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since formula_53, where formula_54 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.