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.