On the number of limit cycles of a perturbed cubic polynomial differential center

For εε sufficiently small, we consider the planar polynomial differential system ẋ=−y(y2+Ax2+Bx+C)+εP(x,y),ẏ=x(y2+Ax2+Bx+C)+εQ(x,y), where P(x,y)P(x,y) and Q(x,y)Q(x,y) are polynomials of degree nn. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system.