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) + epsilon P(x, y) y = x(x + a) (y + b) + epsilon Q(x, y) for epsilon sufficiently small, where a, b is an element of R \ {0}, and P, Q are polynomials of degree n. We obtain that 3[(n - 1)/2] + 4 if a not equal b and, respectively, 2[(n - 1)/2] + 2 if a = b, up to first order in epsilon, are upper bounds for the number of the limit cycles that bifurcate from the period annulus of the cubic center given by epsilon = 0. Moreover, there are systems with at least 3[(n - 1)/2] + 2 limit cycles if a not equal b and, respectively, 2[(n - 1)/2] + 1 if a = b. (c) 2005 Elsevier Ltd. All rights reserved.