Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields

Abstract

In this paper, bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields are studied. First, the system is simplified by using some proper transformations and the first five Lyapunov constants at a nilpotent singular point are calculated by applying the inverse integrating factor method. Then, sufficient and necessary conditions are obtained for two nilpotent singular points of the system being centers. A new perturbation scheme is present to prove the existence of 12 small-amplitude limit cycles in cubic Z2-equivariant vector fields, which bifurcate from two nilpotent singular points. This is a new lower bound of the number of limit cycles bifurcating in such systems.