Bifurcation of limit cycles from a cubic center

I. D. Iliev [Math. Proc. Cambridge Philos. Soc. 127, 317–322 (1999)] studied the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center ẋ=−y, ẏ=x when they are perturbed inside the class of all polynomial differential systems using the Melnikov function method. Here we consider the cubic center ẋ=−y(x2+y2−1), ẏ=−x(x2+y2−1), and we study the maximum number of limit cycles that bifurcate from the periodic orbits of this cubic center when they are perturbed inside the class of all polynomial differential systems of degree three using the averaging theory. It is known that for differential systems in the plane the Melnikov function method and the averaging theory produce the same number of limit cycles.