Abstract: In this paper the periodicity of a perturbed non homogeneous Markov system (P-NHMS) is studied. More specifically, the concept of a periodic P-NHMS is introduced, when the sequence of the total transition matrices 
does not converge, but oscillates among several different matrices, which are rather close to a periodic matrix Q, with period equal to d. It is proved that under this more realistic assumption, the sequence of the relative population structures 
splits into d subsequences that converge in norm, as t → ∞. Moreover, the asymptotic variability of the system is examined using the vector of means, variances and covariances of the state sizes, μ(t). More specifically, we prove that the vector μ(t) also splits into d subsequences that converge, as t → ∞, and we give the limits in an elegant closed analytic form.
