Limit cycles of cubic polynomial vector fields via the averaging theory

In this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method. More precisely, we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system ẋ=−yf(x,y), ẏ=xf(x,y), where f(x,y)=0f(x,y)=0 is a conic such that f(0,0)≠0f(0,0)≠0, by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method.