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

Salomón Rebollo Perdomo

• 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 Poincaré–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 Poincaré–Pontryagin– Melnikov function associated with some perturbed Hamiltonian equations.