Floquet Theory

Abstract

The solutions of periodic linear systems of differential equations are not always periodic, but their global qualitative behaviour can be analysed by studying the first period interval only. After a brief summary of the necessary concepts and results from the theory of ordinary differential equations, mainly to introduce the terminology, we establish the existence of Floquet solutions of periodic systems in section 1.3. These are solutions of a particularly simple structure which reveal whether the system is globally stable or unstable. In the case of Hill’s equation and periodic Dirac systems, which include a spectral parameter, this classification is determined by a single function called Hill’s discriminant, which will be a fundamental tool throughout the book. Its essential properties are studied in sections 1.4 and 1.6. The discriminant function also provides full information about the eigenvalues of periodic and related boundary-value problems on the period interval and its multiples. This connection is explored in section 1.8.