Floquet theory

Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to linear differential equations of the form

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

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

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 ( \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. A matrix \(\Phi(t) \) is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists \( t_0 \) such that \( \Phi(t_0) \) is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using \( \Phi(t)=\phi\,(t){\phi\,}^{-1}(t_0) \). 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

Let \dot{x}= A(t) x be a linear first order differential equation, where x(t) is a column vector of length n and A(t) an n \times n periodic matrix with period T (that is A(t + T) = A(t) for all real values of t). Let \phi\, (t) be a fundamental matrix solution of this differential equation. Then, for all t \in \mathbb{R},

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

Here

\( \phi^{-1}(0) \phi (T)\ \)

is known as the monodromy matrix. 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 \( t \mapsto P(t) \) such that

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

Also, there is a real matrix R and a real periodic (period-2T) matrix function \( t \mapsto Q(t) \) such that

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

In the above B, P, Q and R are n \times n matrices.

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 eigenvalues of \( e^{TB} \) are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps \( x(t) \to x(t+T)\). 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 \( e^{(\mu + \frac{2 \pi i k}{T})T}=e^{\mu T}\), 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 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.

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

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

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 [1, 2]. It states that periodic solutions of Mathieu equation for any pair (a, q) can be expressed in the form

\( y(w)=F_{\nu}(w)=e^{iw \nu} P(w) \, \)

or

\( y(w)=F_{\nu}(-w)=e^{-iw \nu} P(-w) \, \)

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

The constant \nu is called the characteristic exponent.

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

\( y(w+k \pi) =e^{i \nu k \pi}y(w)\text{ or }y(w+k \pi) =e^{-i \nu k \pi}y(w), \, \)

for the solution \( F_{\nu}(w) or F_{\nu}(-w) \), respectively.

We assume that the pair (a, q) is such that \( | \cosh (i \nu \pi) | <1 \) so that the solution y(w) is bounded on the real axis. General solution of Mathieu's equation ( \(q \in \mathbb{R}, \nu \)non-integer) is the form

\( y(w) =c_1 e^{i w \nu}P(w)+ c_2e^{-i w \nu}P(-w), \, \)

where \( c_1 \) and \( c_2 \) 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 [3]:

If \( a( \nu +2p,q) \) and \( a( \nu +2s,q) \) are simple roots of

\( \cos(\pi\nu) - y(\pi = 0) = 0, \, \)

then:

\( \int_0^\pi F_{\nu+2p} (w) F_{\nu+2s}(-w) \, dw = 0,\qquad p \ne s, \)

i.e.,

\( \langle F_{\nu +2p} (w),F_{\nu +2s} (w)\rangle = 0, \qquad p \ne s, \)

where <.,.> denotes an inner product defined from 0 to π.

Floquet theory

References

C. Chicone. Ordinary Differential Equations with Applications. Springer-Verlag, New York 1999.
Ekeland, Ivar (1990). "One". Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. 19. Berlin: Springer-Verlag. pp. x+247. ISBN 3-540-50613-6. MR1051888.
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. Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society.
M.S.P. Eastham, "The Spectral Theory of Periodic Differential Equations", Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973.