The number of limit cycles of a quintic polynomial system

In this paper we consider the bifurcation of limit cycles of the system (x) over dot = y(x(2) - a(2))(y(2) - b(2)) + epsilon P(x, y), (y) over dot = -x(x(2) - a(2))(y(2) - b(2)) + 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 up to first order in epsilon the upper bounds for tile number oflimir cycles that bifurcate from the period annulus of the quintic center given by epsilon = 0 are (3/2)(n + sin(2)(n pi/2)) + 1 if a not equal b and n - 1 if a = b. Moreover, there are systems with at least (3/2)(n + sin(2)(n pi/2)) + 1 if a not equal b and , n - 1 limit cycles if a = b. (C) 2008 Elsevier Ltd. All rights reserved.