# Infinite Orbit depth and length of Melnikov functions

## TL;DR

This paper investigates the complexity of Melnikov functions in polynomial Hamiltonian systems, providing examples of infinite orbit depth and discussing the challenges in constructing deformations with high-length Melnikov functions.

## Contribution

It introduces an example of a Hamiltonian system with infinite orbit depth and discusses the conjecture that the bound on Melnikov function length is optimal.

## Key findings

- Provided an example with infinite orbit depth.
- Constructed deformations with Melnikov functions of length three.
- Discussed difficulties in creating deformations with higher-length Melnikov functions.

## Abstract

In this paper we study polynomial Hamiltonian systems $dF=0$ in the plane and their small perturbations: $dF+\epsilon\omega=0$. The first nonzero Melnikov function $M_{\mu}=M_{\mu}(F,\gamma,\omega)$ of the Poincar\'e map along a loop $\gamma$ of $dF=0$ is given by an iterated integral. In a previous work (see arXiv 1703.03837), we bounded the length of the iterated integral $M_\mu$ by a geometric number $k=k(F,\gamma)$ which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system $F$ and its orbit $\gamma$ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations $dF+\epsilon\omega$ with arbitrary high length first nonzero Melnikov function $M_\mu$ along $\gamma$. We construct deformations $dF+\epsilon\omega=0$ whose first nonzero Melnikov function $M_\mu$ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions $M_\mu$.