Infinite orbit depth and length of Melnikov functions

Mardešić, Pavao; Novikov, Dmitry; Ortiz-Bobadilla, Laura; Pontigo-Herrera, Jessie

doi:10.1016/j.anihpc.2019.07.003

Mardešić, Pavao ¹ ; Novikov, Dmitry ² ; Ortiz-Bobadilla, Laura ³ ; Pontigo-Herrera, Jessie ²

¹ 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
² Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel
³ Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, 04510, Ciudad de México, Mexico

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 1941-1957.

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Résumé

In this paper we study polynomial Hamiltonian systems $d F = 0$ in the plane and their small perturbations: $d F + ϵ ω = 0$ . The first nonzero Melnikov function $M_{μ} = M_{μ} (F, γ, ω)$ of the Poincaré map along a loop γ of $d F = 0$ is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral $M_{μ}$ by 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 F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations $d F + ϵ ω$ with arbitrary high length first nonzero Melnikov function $M_{μ}$ along γ. We construct deformations $d F + ϵ ω = 0$ whose first nonzero Melnikov function $M_{μ}$ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions $M_{μ}$ .

MR Zbl

DOI : 10.1016/j.anihpc.2019.07.003

Classification : 34C07, 34C05, 34C08
Mots-clés : Iterated integrals, Center problem

Affiliations des auteurs :

Mardešić, Pavao ¹ ; Novikov, Dmitry ² ; Ortiz-Bobadilla, Laura ³ ; Pontigo-Herrera, Jessie ²

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     author = {Marde\v{s}i\'c, Pavao and Novikov, Dmitry and Ortiz-Bobadilla, Laura and Pontigo-Herrera, Jessie},
     title = {Infinite orbit depth and length of {Melnikov} functions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1941--1957},
     publisher = {Elsevier},
     volume = {36},
     number = {7},
     year = {2019},
     doi = {10.1016/j.anihpc.2019.07.003},
     mrnumber = {4020529},
     zbl = {1435.37086},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.07.003/}
}

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Mardešić, Pavao; Novikov, Dmitry; Ortiz-Bobadilla, Laura; Pontigo-Herrera, Jessie. Infinite orbit depth and length of Melnikov functions. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 1941-1957. doi : 10.1016/j.anihpc.2019.07.003. http://www.numdam.org/articles/10.1016/j.anihpc.2019.07.003/

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