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 x ˙ = A ( t ) x , {\displaystyle {\dot {x}}=A(t)x,} with x ∈ R n {\displaystyle x\in {R^{n}}} and A ( t ) ∈ R n × n {\displaystyle \displaystyle A(t)\in {R^{n\times n}}} being a piecewise continuous periodic function with period T {\displaystyle T} 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 y = Q − 1 ( t ) x {\displaystyle \displaystyle y=Q^{-1}(t)x} with Q ( t + 2 T ) = Q ( t ) {\displaystyle \displaystyle Q(t+2T)=Q(t)} 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 ϕ ( t ) {\displaystyle \phi \,(t)} is called a fundamental matrix solution if the columns form a basis of the solution set. A matrix Φ ( t ) {\displaystyle \Phi (t)} is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists t 0 {\displaystyle t_{0}} such that Φ ( t 0 ) {\displaystyle \Phi (t_{0})} is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using Φ ( t ) = ϕ ( t ) ϕ − 1 ( t 0 ) {\displaystyle \Phi (t)=\phi \,(t){\phi \,}^{-1}(t_{0})} . The solution of the linear differential equation with the initial condition x ( 0 ) = x 0 {\displaystyle x(0)=x_{0}} is x ( t ) = ϕ ( t ) ϕ − 1 ( 0 ) x 0 {\displaystyle x(t)=\phi \,(t){\phi \,}^{-1}(0)x_{0}} where ϕ ( t ) {\displaystyle \phi \,(t)} is any fundamental matrix solution.