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