# 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) |

**How to Cite This Entry:**

Floquet theory.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Floquet_theory&oldid=18199