Poincaré–Pontryagin–Melnikov Functions for a Type of Perturbed Degenerate Hamiltonian Equations

Abstract

In this paper we consider polynomial perturbations of a family of polynomial Hamiltonian equations whose associated Hamiltonian is not transversal to infinity, and its complexification is not a Morse polynomial. We look for an algorithm to compute the first non-vanishing Poincare–Pontryagin–Melnikov function of the displacement function associated with the perturbed equation. We show that the algorithm of the case when the Hamiltonian is transversal to infinity and its complexification is a Morse polynomial can be extended to our family of perturbed equations. We apply the result to study the maximum number of zeros of the first non-vanishing Poincare–Pontryagin–Melnikov function associated with some perturbed Hamiltonian equations.