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+ϵω=0. The first nonzero Melnikov function Mμ=Mμ(F,γ,ω) of the Poincaré map along a loop γ of dF=0 is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral Mμ by a geometric number k=k(F,γ) 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 γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations dF+ϵω with arbitrary high length first nonzero Melnikov function Mμ along γ. We construct deformations dF+ϵω=0 whose first nonzero Melnikov function Mμ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions Mμ.