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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+ω=0. The first nonzero Melnikov function Mμ=Mμ(F, γ, ω) of the Poincaré map along a loop γof dF=0 is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral Mμby 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 Fand its orbit γhaving infinite orbit depth. If our conjecture is true, for this example there should exist deformations dF+ω with arbitrary high length first nonzero Melnikov function Mμalong γ. We construct deformations dF+ω=0 whose first nonzero Melnikov function Mμis of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions Mμ.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-02288935
Contributeur : Imb - Université de Bourgogne <>
Soumis le : lundi 16 septembre 2019 - 11:22:01
Dernière modification le : jeudi 15 octobre 2020 - 14:42:04

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Pavao Mardešić, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Pontigo-Herrera. Infinite orbit depth and length of Melnikov functions. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2019, 36 (7), pp.1941-1957. ⟨10.1016/j.anihpc.2019.07.003⟩. ⟨hal-02288935⟩

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