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 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 A ( t ) {displaystyle displaystyle A(t)} 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. In solid-state physics, the analogous 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 all columns are linearly independent solutions. 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. Let x ˙ = A ( t ) x {displaystyle {dot {x}}=A(t)x} be a linear first order differential equation,where x ( t ) {displaystyle x(t)} is a column vector of length n {displaystyle n} and A ( t ) {displaystyle A(t)} an n × n {displaystyle n imes n} periodic matrix with period T {displaystyle T} (that is A ( t + T ) = A ( t ) {displaystyle A(t+T)=A(t)} for all real values of t {displaystyle t} ). Let ϕ ( t ) {displaystyle phi ,(t)} be a fundamental matrix solution of this differential equation. Then, for all t ∈ R {displaystyle tin mathbb {R} } ,