  In this study, we focus on the existence of a periodic solution for the
neutral nonlinear dynamic systems with delay% \[
x^{\Delta}(t)=A(t)x(t)+Q^{\Delta}\left(t,x\left(\delta_{-}(s,t)\right) \right)
+G\left(t,x(t),x\left(\delta_{-}(s,t)\right) \right) . \] We utilize the new
periodicity concept in terms of shifts operators, which allows us to extend the
concept of periodicity to time scales where the additivity requirement $t\pm
T\in\mathbb{T}$ for all $t\in\mathbb{T}$ and for a fixed $T>0,$ may not hold.
More, importantly, the new concept will easily handle time scales that are not
periodic in the conventional way such as; $\overline{q^{\mathbb{Z}}}$ and
$\cup_{k=1}^{\infty}\left[ 3^{\pm k},2.3^{\pm k}\right] \cup\left\{0\right\} .$
Hence, we develop a tool that enables the investigation of periodic solutions
of $q$-difference systems. Since we are dealing with systems, in order to
convert our equation to an integral systems, we resort to the transition matrix
of the homogeneous Floquet system $y^{\Delta}(t)=A(t)y(t)$ and then make use of
Krasnoselskii's fixed point theorem to obtain a fixed point.
