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
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Abstract

In this paper we study polynomial Hamiltonian systems in the plane and their small perturbations: . The first nonzero Melnikov function of the Poincaré map along a loop γ of is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral by a geometric number 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 with arbitrary high length first nonzero Melnikov function along γ. We construct deformations whose first nonzero Melnikov function is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions .

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