Floquet theory

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

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

with $\displaystyle A(t)$ a piecewise continuous periodic function with period $T$ and defines the state of the stability of solutions.

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 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.
• Bond softening and bond hardening in intense laser fields can be described in terms of solutions obtained from the Floquet theorem.

References

• C. Chicone. Ordinary Differential Equations with Applications. Springer-Verlag, New York 1999.
• , 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.
• 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.