A new formal series method and its application to the center problem in Z2-equivariant nilpotent vector fields

In this paper, center problems and bifurcation of limit cycles are considered for $Z_2$-equivariant nilpotent vector fields. A new formal series method is developed for computing the focal values of the vector fields, which can be conveniently implemented using a computer algebraic system. As an application, the new method is applied to classify the centers for a class of quintic-order systems, which contains four conditions associated with a nilpotent singular point at the origin and two center conditions associated with an elementary center at infinity. Moreover, eight small-amplitude limit cycles in the neighborhood of the origin and nine large-amplitude limit cycles at infinity are obtained. This is the first time to investigate the synchronous bifurcation problem associated with a nilpotent singular point at the origin and a Hopf singular point at infinity.