Poincaré--Pontryagin--Melnikov functions for a class of perturbed planar Hamiltonian equations

Abstract

In this paper we extend a well-known algorithm for studying higher order Poincaré--Pontryagin--Melnikov functions of polynomial perturbed Hamiltonian equations. We consider a family of unperturbed equations whose associated Hamiltonian is not transversal to infinity, and its complexification is no a Morse polynomial. We prove that the first non-vanishing Poincaré--Pontryagin--Melnikov function of the displacement function, associated with the perturbed equation, is an Abelian integral, and we provide the algorithm to compute it. Our result generalizes the algorithm for the case when the Hamiltonian is transversal to infinity, and its complexification is a Morse polynomial. We apply our result to study the maximum number of zeros of the first non-vanishing Poincaré--Pontryagin--Melnikov function associated with some particular perturbed degenerated Hamiltonian equations.