  Using the new periodicity concept based on shifts, we construct a unified
Floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on
domains having continuous, discrete or hybrid structure. New periodicity
concept based on shifts enables the construction of Floquet theory on hybrid
domains that are not necessarily additive periodic. This makes periodicity and
stability analysis of hybrid periodic systems possible on non-additive domains.
In particular, this new approach can be useful to know more about Floquet
theory for linear $q$-difference systems defined on
$\overline{q^{\mathbb{Z}}}:=\{q^{n}% :n\in\mathbb{Z}\} \cup \{0\}$ where $q>1$.
By constructing the solution of matrix exponential equation we establish a
canonical Floquet decomposition theorem. Determining the relation between
Floquet multipliers and Floquet exponents, we give a spectral mapping theorem
on closed subsets of reals that are periodic in shifts. Finally, we show how
the constructed theory can be utilized for the stability analysis of dynamic
systems on periodic time scales in shifts.
