On the number of zeros of Melnikov functions

TL;DR

This paper establishes an explicit upper bound on the number of zeros of the first non-vanishing Melnikov function for polynomial perturbations of planar Hamiltonian systems, depending on degrees and order.

Contribution

It introduces an effective method to bound zeros of Melnikov functions using Gauss-Manin connections, extending previous results to higher orders.

Findings

Provides a uniform upper bound depending on degrees and order

Uses Gauss-Manin connection for iterated integrals

Extends results beyond the generic case k=1

Abstract

We provide an effective uniform upper bond for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order $k$ of the Melnikov function. The generic case $k=1$ was considered by Binyamini, Novikov and Yakovenko (\cite{BNY-Inf16}). The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.