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:\dot{x} = A(t) x,\,

with \displaystyle A(t) a continuous periodic function with period T.

The main theorem of Floquet theory, Floquet's theorem (named after Gaston Floquet ), gives a Canonical Form for each Fundamental Matrix Solution of this common Linear System . It gives a Coordinate Change \displaystyle y=Q^{-1}(t)x with \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 (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 \phi(t) is called a fundamental matrix solution if all columns are linearly independent solutions. It is called a principal fundamental matrix at t_0 if \phi(t_0) is the identity. Because of existence and uniqueness of the solutions there is a principal fundamental matrix \Phi(t_0)=\phi(t)\phi^{-1}(t_0) for each t_0. The solution of the linear differential equation with the initial condition x(0)=x_0 is x(t)=\phi(t)\phi^{-1}(0)x_0 where \phi(t) is any fundamental matrix solution.


FLOQUET'S THEOREM

If \phi (t) is a fundamental matrix solution of the periodic system \dot{x}= A(t) x, with A(t) a periodic function with period T then, for all t \in \mathbb{R},

: \phi(t+T)=\phi(t) \phi^{-1}(0) \phi (T).

In addition, for each matrix B (possibly complex) such that:

:e^{TB}=\phi^{-1}(0) \phi (T),

there is a periodic (period T) matrix function t o P(t) such that

:\phi (t) = P(t)e^{tB} for all t \in \mathbb{R}.

Also, there is a ''real'' matrix R and a real periodic (period 2T) matrix function t o Q(t) such that

:\phi (t) = Q(t)e^{tR} for all t \in \mathbb{R}.


CONSEQUENCES AND APPLICATIONS

This mapping \phi (t) = Q(t)e^{tR} gives rise to a time-dependent change of coordinates ( y = Q^{-1}(t) x), under which our original system becomes a linear system with real constant coefficients \dot{y} = R y. Since Q(t) is continuous and periodic it must be bounded. Thus the stability of the zero solution for y(t) and x(t) is determined by the eigenvalues of R.

The representation \phi (t) = P(t)e^{tB} is called a ''Floquet normal form'' for the fundamental matrix \phi (t).

The Eigenvalue s of e^{TB} are called the Characteristic Multiplier s of the system. They are also the eigenvalues of the (linear) Poincaré maps x(t) o x(t+T). A Floquet exponent (sometimes called a characteristic exponent), is a complex \mu such that e^{\mu T} is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since e^{(\mu + rac{2 \pi i k}{T})T}=e^{\mu T}, where k 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.



FLOQUET'S THEOREM APPLIED TO MATHIEU EQUATION


Mathieu's equation is related to the wave equation for the elliptic cylinder.

Given a \in \mathbb{R}, q \in \mathbb{C}, the Mathieu Equation is given by

: rac {d^2 y} {dw^2} +(a-2q \cos 2w )y=0

The Mathieu equation is a linear second-order differential equation with periodic coefficients.

One of the most powerful results of Mathieu's functions is the Floquet's Theorem 2 .
It states that periodic solutions of Mathieu equation for any pair (''a'', ''q'') can be expressed in the form

: y(w)=F_{
u}(w)=e^{jw
u} P(w) or

: y(w)=F_{
u}(-w)=e^{-jw
u} P(-w),

where
u is a constant depending on ''a'' and ''q'' and ''P''(.) is \pi -periodic in ''w''.

The constant
u is called the ''characteristic exponent''.

If
u is an integer, then F_{
u}(w) and F_{
u}(-w) are linear dependent solutions. Furthermore,

y(w+k \pi) =e^{j
u k \pi}y(w) or y(w+k \pi) =e^{-j
u k \pi}y(w) , for the solution F_{
u}(w) or F_{
u}(-w), respectively.