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The objective of this paper is to investigate a Regular Perturbation dynamic system of the type, non- autonomous type of an ordinary differential equations of the form; x′ = f(x) + εg(x,t,ε), and to establish the sufficient and necessary conditions for the differences between the dynamics for which ε = 0 and ε > 0, and also establish x* as a stable equilibrium point of the unperturbed autonomous system of x′ = f(x) The paper further establish, that the above given equation, admit a stable T- Periodic orbit in a neighborhood of x*. Some properties of the Hamiltonian system, which form important source of the differential equations given above provided the motivation for the study. Proposition 1.1, Example 2.1, 2.2, 4.1, Theorem3.1 corollary3.1and Theorem 3.2 gives the results that established the objective for the study. My approach in this study has advantage over (3) and the results obtained in this study generalize the results in (3) in the case where four arguments were proved.