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
![]() | (1) |
the matrix is periodic in
with period
and is summable on every compact interval in
.
1) Every fundamental matrix of the system (1) has a representation
![]() | (2) |
called the Floquet representation (see [1]), where is some
-periodic matrix and
is some constant matrix. There is a basis
of the space of solutions of (1) such that
has Jordan form in this basis; this basis can be represented in the form
![]() |
where are polynomials in
with
-periodic coefficients, and the
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)
. 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
are simply
-periodic functions. The matrices
and
in the representation (2) are, generally speaking, complex valued. If one restricts oneself just to the real case, then
does not have to be
-periodic, but must be
-periodic.
2) The system (1) can be reduced to a differential equation with a constant matrix, , by means of the Lyapunov transformation
![]() | (3) |
where and
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 be the spectrum of the matrix
. For every
such that
,
, in view of (2) the space
splits into the direct sum of two subspaces
and
![]() |
such that
![]() |
![]() |
here is the fundamental matrix of (1) normalized at zero. This implies exponential dichotomy of (1) if
for any
.
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) |
Comments
References
[a1] | J.K. Hale, "Ordinary differential equations" , Wiley (1969) |
[a2] | P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) |
Floquet theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Floquet_theory&oldid=46944