Limit cycles of a perturbed cubic polynomial differential center

In this paper we study the limit cycles of the system over(x, ̇) = - y (x + a) (y + b) + ε P (x, y), over(y, ̇) = x (x + a) (y + b) + ε Q (x, y) for ε sufficiently small, where a, b ∈ R {minus 45 degree rule} {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. © 2005 Elsevier Ltd. All rights reserved.