Infinite orbit depth and length of Melnikov functions

In this paper we study polynomial Hamiltonian systems $dF=0$ in the plane and their small perturbations: $dF+?\omega =0$. The first nonzero Melnikov function $M_{\mu }=M_{\mu }(F,\gamma ,\omega )$ 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 $M_{\mu }$ 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 γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations $dF+?\omega$ with arbitrary high length first nonzero Melnikov function $M_{\mu }$ along γ. We construct deformations $dF+?\omega =0$ 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 $M_{\mu }$.