Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop

In this paper we consider Lienard equations of the form x = y, y = -(x - 2x(3) + x(5)) - epsilon(alpha + beta x(2) + gamma x(4))y, where 0 vertical bar epsilon vertical bar << 1, (alpha, beta, gamma) is an element of Lambda subset of R-3 and Lambda is bounded. We prove that the least upper bound for the number of zeros of the related Abelian integrals 1(h) = closed integral(Gamma h)(alpha + beta(2)(x) + gamma x(4))y dx is 2 (taking into account their multiplicities) for h is an element of (0, 1/6) and this upper bound is a sharp one. This implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for epsilon = 0 inside an eye-figure loop is less than or equal to 2. (C) 2007 Elsevier Ltd. All rights reserved.