Floquet theory

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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 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 $y = Q - 1 (t) x$ with $Q (t + 2 T) = Q (t)$ 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 $φ(t)$ is called fundamental matrix solution if all columns are linearly independent solutions. It is called a principal fundamental matrix at $t 0$ if $φ(t 0)$ is the identity. Because of existence and uniqueness of the solutions there is a principal fundamental matrix $Φ(t 0) = φ(t)φ - 1 (t 0)$ for each $t 0$ . The solution of the linear differential equation with the initial condition $x (0) = x 0$ is $x (t) = φ(t)φ - 1 (0) x 0$ where $φ(t)$ is any fundamental matrix solution.

[edit] Floquet's theorem

If $φ(t)$ 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 $t \in \mathbb{R}$ ,

φ(t + T) = φ(t)φ - 1 (0)φ(T).

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

e T B = φ - 1 (0)φ(T)

there is a periodic (period $T$ ) matrix function $t \to P(t)$ such that

φ(t) = P (t) e t B

for all $t \in \mathbb{R}$ .

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

φ(t) = Q (t) e t R

for all $t \in \mathbb{R}$ .

[edit] Consequences and applications

This mapping $φ(t) = Q (t) e t R$ 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 $φ(t) = P (t) e t B$ is called a Floquet normal form for the fundamental matrix $φ(t)$ .

The eigenvalues of $e T B$ 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 $μ$ such that $e μ T$ 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 to the study of dynamical systems
Floquet theory shows stability in Hill's equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field.

[edit] References

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).