The number of limit cycles of a quintic polynomial system

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