Melnikov analysis of a perturbed integrable non-Hamiltonian system

Abstract

This paper provides a completed Melnikov analysis of any order for a class of perturbed polynomial differential systems, where the unperturbed system is a cubic center with a straight line of singular points. The emphasis on any order Melnikov function is crucial because of two major reasons: one is that higher order Melnikov functions are not in general expressible explicitly, the other one is the essential difficulty of computing elliptic integrals due to the complexity of many iterations. The exact upper bounds of the number of limit cycles bifurcated from the period annulus under different polynomial perturbations are obtained by analyzing the algebraic structures from explicit expressions of any order Melnikov functions.