Some computable results on the existence of periodic solutions for singular non-autonomous third order systems

Here we are concerned with the existence of periodic solution for nonlinear non-autonomous third order system of ordinary differential equations with singular terms. Our method here is based on the topological method in the sense that we conclude some computable results for the focal system from a homotopic nonsingular system. The aim is to obtain sufficient conditions for which the system has periodic solution whenever the value of deformation with respect to the first variation of the nonsingular subsystem is sufficiently small. The method presented here is constructive in the sense that the existence of periodic orbits can be verified numerically as well as computed if any. For this, we will show how classical methods like the Newton one for solving algebraic equations can be applied to the equations obtained analytically in this paper. Finally we treat a definite singular system numerically in order to verify the obtained results.