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In this paper, we study periodic linear systems on periodic time scales whichinclude not only discrete and continuous dynamical systems but also systemswith a mixture of discrete and continuous parts (e.g. hybrid dynamicalsystems). We develop a comprehensive Floquet theory including Lyapunovtransformations and their various stability preserving properties, a unifiedFloquet theorem which establishes a canonical Floquet decomposition on timescales in terms of the generalized exponential function, and use these resultsto study homogeneous as well as nonhomogeneous periodic problems. Furthermore,we explore the connection between Floquet multipliers and Floquet exponents viamonodromy operators in this general setting and establish a spectral mappingtheorem on time scales. Finally, we show this unified Floquet theory has thedesirable property that stability characteristics of the original system can bedetermined via placement of an associated (but time varying) system's poles inthe complex plane. We include several examples to show the utility of thistheory.

A Unified Floquet Theory for Discrete, Continuous, and Hybrid Periodic Linear Systems