Infinite Orbit depth and length of 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 Poincaré 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 $M_{\mu }$.