The third order melnikov function of a cubic integrable system under quadratic perturbations

Abstract

• This paper consider a class of cubic integrable systems perturbed by general quadratic polynomials. • This paper focus to study the limit cycles of a perturbed polynomial system. • The most important part of the paper is computing the higher-order Melnikov functions of any order. • The subject of the paper is closely related to the Hilbert 16th problem. In this work, we consider a cubic integrable system under quadratic perturbations. We then study the limit cycles of the perturbed system by using Melnikov functions up to order three. We prove that the sharp upper bound of the number of limit cycles lies between six and seven. Also, we give an example that shows six limit cycles.