Center problem and the bifurcation of limit cycles for a cubic polynomial system

In this study, we consider the center problem and the bifurcation of limit cycles for a cubic system that lies in a symmetrical vector field about the origin. By analyzing and calculating the focal values (or the Lyapunov constant), we obtain the conditions where two equilibrium points, (1,1) and (-1, -1), become a pair of simultaneous centers. Moreover, six limit cycles, including three stable limit cycles, can bifurcate from (1,1) under a specific condition. From the symmetric quality, (-1, -1) can also bifurcate into six limit cycles by simultaneous Hopf bifurcation, which is an interesting result. (C) 2015 Elsevier Inc. All rights reserved.