Limit cycles of a perturbed cubic polynomial differential center

In this paper we study the limit cycles of the system x˙=-y(x+a)(y+b)+εP(x,y), y˙=x(x+a)(y+b)+εQ(x,y) for ε sufficiently small, where a,b∈R⧹{0}, and P, Q are polynomials of degree n. We obtain that 3[(n − 1)/2] + 4 if a ≠ b and, respectively, 2[(n − 1)/2] + 2 if a = b, up to first order in ε, are upper bounds for the number of the limit cycles that bifurcate from the period annulus of the cubic center given by ε = 0. Moreover, there are systems with at least 3[(n − 1)/2] + 2 limit cycles if a ≠ b and, respectively, 2[(n − 1)/2] + 1 if a = b.