with a piecewise continuous periodic function with period .
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 with that transforms the periodic system to a traditional linear system with constant, real coefficients.
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. A matrix is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists such that is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using . The solution of the linear differential equation with the initial condition is where is any fundamental matrix solution.
Let be a linear first order differential equation, where is a column vector of length and an periodic matrix with period (that is for all real values of ). Let be a fundamental matrix solution of this differential equation. Then, for all ,
is known as the monodromy matrix. In addition, for each matrix (possibly complex) such that
there is a periodic (period ) matrix function such that
Also, there is a real matrix and a real periodic (period-) matrix function such that
In the above , , and are matrices.
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 .
The 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 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 dynamical systems.
- Floquet theory shows stability in Hill differential equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field.
- Bond softening and bond hardening in intense laser fields can be described in terms of solutions obtained from the Floquet theorem.
Floquet's theorem applied to Mathieu equation
Mathieu's equation is related to the wave equation for the elliptic cylinder.
Given , the Mathieu equation is 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 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,
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
where <·,·> denotes an inner product defined from 0 to π.
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- Floquet, Gaston (1883), "Sur les équations différentielles linéaires à coefficients périodiques", Annales de l'École Normale Supérieure 12: 47–88
- Krasnosel'skii, M.A. (1968), The Operator of Translation along the Trajectories of Differential Equations, Providence: American Mathematical Society, Translation of Mathematical Monographs, 19, 294p.
- W. Magnus, S. Winkler. Hill's Equation, Dover-Phoenix Editions, ISBN 0-486-49565-5.
- N.W. McLachlan, Theory and Application of Mathieu Functions, New York: Dover, 1964.
- Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
- M.S.P. Eastham, "The Spectral Theory of Periodic Differential Equations", Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973. ISBN 978-0-7011-1936-2.