Floquet theory

A theory concerning the structure of the space of solutions, and the properties of solutions, of a linear system of differential equations with periodic coefficients

$$ \tag{1 } x ^ \prime = \ A ( t) x,\ \ t \in \mathbf R ,\ \ x \in \mathbf R ^ {n} ; $$

the matrix $ A ( t) $ is periodic in $ t $ with period $ \omega > 0 $ and is summable on every compact interval in $ \mathbf R $.

1) Every fundamental matrix $ X $ of the system (1) has a representation

$$ \tag{2 } X ( t) = F ( t) \ \mathop{\rm exp} ( tK), $$

called the Floquet representation (see [1]), where $ F ( t) $ is some $ \omega $- periodic matrix and $ K $ is some constant matrix. There is a basis $ x _ {1} \dots x _ {n} $ of the space of solutions of (1) such that $ K $ has Jordan form in this basis; this basis can be represented in the form

$$ x _ {i} = \ ( \psi _ {1i} \mathop{\rm exp} ( \alpha _ {i} t) \dots \psi _ {ni} \mathop{\rm exp} ( \alpha _ {i} t)), $$

where $ \psi _ {ki} $ are polynomials in $ t $ with $ \omega $- periodic coefficients, and the $ \alpha _ {i} $ are the characteristic exponents (cf. Characteristic exponent) of the system (1). Every component of a solution of (1) is a linear combination of functions of the form (of the Floquet solutions) $ \psi _ {ki} \mathop{\rm exp} ( \alpha _ {i} t) $. In the case when all the characteristic exponents are distinct (or if there are multiple ones among them, but they correspond to simple elementary divisors), the $ \psi _ {ki} $ are simply $ \omega $- periodic functions. The matrices $ F ( t) $ and $ K $ in the representation (2) are, generally speaking, complex valued. If one restricts oneself just to the real case, then $ F ( t) $ does not have to be $ \omega $- periodic, but must be $ 2 \omega $- periodic.

2) The system (1) can be reduced to a differential equation with a constant matrix, $ y ^ \prime = Ky $, by means of the Lyapunov transformation

$$ \tag{3 } x = F ( t) y, $$

where $ F ( t) $ and $ K $ are the matrices from the Floquet representation (2) (see [2]). The combination of representation (2) together with the substitution (3) is often called the Floquet–Lyapunov theorem.

3) Let $ \{ \alpha _ {1} \dots \alpha _ {l} \} $ be the spectrum of the matrix $ K $. For every $ \alpha \in \mathbf R $ such that $ \alpha \neq \mathop{\rm Re} \alpha _ {j} $, $ j = 1 \dots l $, in view of (2) the space $ \mathbf R ^ {n} $ splits into the direct sum of two subspaces $ S _ \alpha $ and $ U _ \alpha $

$$ ( \mathbf R ^ {n} = \ S _ \alpha + U _ \alpha ,\ \ S _ \alpha \cap U _ \alpha = \emptyset ) $$

such that

$$ \lim\limits _ {t \rightarrow + \infty } \ \mathop{\rm exp} (- \alpha t) V ( t) x ( 0) = 0 \ \iff \ \ x ( 0) \in S _ \alpha , $$

$$ \lim\limits _ {t \rightarrow - \infty } \mathop{\rm exp} (- \alpha t) V ( t) x ( 0) = 0 \ \iff \ x ( 0) \in U _ \alpha ; $$

here $ V ( t) $ is the fundamental matrix of (1) normalized at zero. This implies exponential dichotomy of (1) if $ \mathop{\rm Re} \alpha _ {j} \neq 0 $ for any $ j = 1 \dots l $.

References

[1]	G. Floquet, Ann. Sci. Ecole Norm. Sup. , 12 : 2 (1883) pp. 47–88
[2]	A.M. Lyapunov, "Problème général de la stabilité du mouvement" , Collected works , 2 , Princeton Univ. Press , Moscow-Leningrad (1956) pp. 7–263 (In Russian)
[3]	B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)
[4]	V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients" , Wiley (1975) (Translated from Russian)
[5]	J.L. Massera, J.J. Shäffer, "Linear differential equations and function spaces" , Acad. Press (1966)
[6]	N.P. Erugin, "Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients" , Acad. Press (1966) (Translated from Russian)

References

[a1]	J.K. Hale, "Ordinary differential equations" , Wiley (1969)
[a2]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)