Third Order Melnikov Functions of a Cubic Center under Cubic Perturbations

Abstract

In this paper, cubic perturbations of the integral system (1+x)2dH where H=(x2+y2)/2 are considered. Some useful formulae are deduced that can be used to compute the first three Melnikov functions associated with the perturbed system. By employing the properties of the ETC system and the expressions of the Melnikov functions, the existence of exactly six limit cycles is given. Note that there are many cases for the existence of third-order Melnikov functions, and some existence conditions are very complicated—the corresponding Melnikov functions are not presented.