- Floquet theory
Floquet theory is a branch of the theory of
ordinary differential equationsrelating to the class of solutions to linear differential equations of the form,
with a continuous periodic function with period .
The main theorem of Floquet theory, Floquet's theorem (named after
Gaston Floquet), gives a canonical formfor each fundamental matrix solution of this common linear system. It gives a coordinate change with that transforms the periodic systemto a traditional linear system with constant, real coefficients.
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 is called a fundamental matrix solution if all columns are linearly independent solutions. It is called a principal fundamental matrix at if is the identity. Because of existence and uniqueness of the solutions there is a principal fundamental matrix for each . The solution of the linear differential equation with the initial condition is where is any fundamental matrix solution.
If is a fundamental matrix solution of the periodic system , with a periodic function with period then, for all ,
In addition, for each matrix (possibly complex) such that:
there is a periodic (period ) matrix function such that
: for all .
Also, there is a "real" matrix and a real periodic (period ) matrix function such that
: for all .
Consequences and applications
This mapping gives rise to a time-dependent change of coordinates (), under which our original system becomes a linear system with real constant coefficients . Since is continuous and periodic it must be bounded. Thus the stability of the zero solution for and is determined by the eigenvalues of .
The representation is called a "Floquet normal form" for the fundamental matrix .
eigenvalues of 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 such that is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since , 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 theory is very important for the study of
* Floquet theory shows stability in
Hill's equation(introduced by George William Hill) approximating the motion of the moonas a harmonic oscillatorin a periodic gravitational field.
Floquet's theorem applied to Mathieu equation
Mathieu's equation is related to the wave equation for the elliptic cylinder.
Given , the
Mathieu equationis given by
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 [1, 2] . It states that periodic solutions of Mathieu equation for any pair ("a", "q") can be expressed in the form
where is a constant depending on "a" and "q" and "P"(.) is -periodic in "w".
The constant is called the "characteristic exponent".
If is an integer, then and are linear dependent solutions. Furthermore,
or , for the solution or , respectively.
We assume that the pair ("a", "q") is such that so that the solution is bounded on the real axis. General solution of Mathieu's equation (, non-integer) is the form
where and are arbitrary constants.
All bounded solutions --those of fractional as well as integral order-- are described by an infinite series of
harmonic oscillations whose amplitudes decrease with increasing frequency.
Another very important property of Mathieu's functions is the orthogonality  :
If and are simple roots of – y() = 0, then:
: , , i.e.,
, , where <.,.> denote an
inner productdefined from 0 to .
*Chicone, Carmen. "Ordinary Differential Equations with Applications." Springer-Verlag, New York 1999
* Gaston Floquet, "Sur les équations différentielles linéaires à coefficients périodiques," "Ann. École Norm. Sup." 12, 47-88 (1883). http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1883_2_12_/ASENS_1883_2_12__47_0/ASENS_1883_2_12__47_0.pdf
*N.W. McLachlan, Theory and Application of Mathieu Functions, New York: Dover, 1964.
* Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, http://www.mat.univie.ac.at/~gerald/ftp/book-ode/
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