JournalsaihpcVol. 36, No. 7pp. 1941–1957

Infinite orbit depth and length of Melnikov functions

  • Pavao Mardešić

    Université de Bourgogne, Institute de Mathématiques de Bourgogne - UMR 5584 CNRS, Université de Bourgogne-Franche-Comté, 9 avenue Alain Savary, BP 47870, 21078 Dijon, France
  • Dmitry Novikov

    Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel
  • Laura Ortiz-Bobadilla

    Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, 04510, Ciudad de México, Mexico
  • Jessie Pontigo-Herrera

    Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel
Infinite orbit depth and length of Melnikov functions cover
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Abstract

In this paper we study polynomial Hamiltonian systems dF=0dF = 0 in the plane and their small perturbations: dF+ϵω=0dF + \epsilon \omega = 0. The first nonzero Melnikov function Mμ=Mμ(F,γ,ω)M_{\mu } = M_{\mu }(F,\gamma ,\omega ) of the Poincaré map along a loop γ of dF=0dF = 0 is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral MμM_{\mu } by a geometric number k=k(F,γ)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+ϵωdF + \epsilon \omega with arbitrary high length first nonzero Melnikov function MμM_{\mu } along γ. We construct deformations dF+ϵω=0dF + \epsilon \omega = 0 whose first nonzero Melnikov function MμM_{\mu } is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions MμM_{\mu }.

Cite this article

Pavao Mardešić, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Pontigo-Herrera, Infinite orbit depth and length of Melnikov functions. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 1941–1957

DOI 10.1016/J.ANIHPC.2019.07.003