Weak centers and bifurcation of critical periods in reversible cubic systems

In this paper, we investigate nonhomogeneous cubic differential systems with reversibility, which guarantees that the systems have a center at the origin. We apply the Ritt-Wu method to process algebraic equations and inequalities using Maple V.3 on a computer. We give an inductive algorithm for computing the period coefficient polynomials, we find the structure of solutions of systems of algebraic equations corresponding to isochronous centers and weak centers of every finite order, and we derive conditions on the parameters under which the origin is an isochronous center or a weak center of finite order. We show that with appropriate perturbations local bifurcation of critical periods will occur from weak centers of finite order and isochronous centers.