Geodesic
Infinite orbit depth and length of Melnikov functions
Mardešić, Pavao1 ; Novikov, Dmitry2 ; Ortiz-Bobadilla, Laura3 ; Pontigo-Herrera, Jessie2
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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In this paper we study polynomial Hamiltonian systems dF=0 in the plane and their small perturbations: dF+ϵω=0. The first nonzero Melnikov function Mμ=Mμ(F,γ,ω) of the Poincaré map along a loop γ of dF=0 is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral Mμ 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 dF+ϵω with arbitrary high length first nonzero Melnikov function Mμ along γ. We construct deformations dF+ϵω=0 whose first nonzero Melnikov function Mμ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions Mμ.

DOI : 10.1016/j.anihpc.2019.07.003
Classification : 34C07, 34C05, 34C08
Keywords: Iterated integrals, Center problem

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 
@article{AIHPC_2019__36_7_1941_0,
     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://geodesic.mathdoc.fr/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

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