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 + epsilon omega = 0. The first nonzero Melnikov function M-mu = M-mu(F, gamma, omega) of the Poincare map along a loop gamma 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 gamma having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + 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. (C) 2019 Elsevier Masson SAS. All rights reserved.