We provide an effective uniform upper bound for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order of the Melnikov function. The generic case was considered by Binyamini, Novikov and Yakovenko [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.
Nous donnons une borne supérieure effective et uniforme pour le nombre de zéros de la première fonction de Melnikov d’une perturbation polynomiale d’un champ de vecteurs hamiltonien polynomial sur le plan. La borne dépend des degrés du champ et de la perturbation, et de l’ordre de la fonction de Melnikov. Le cas générique a été considéré par Binyamini, Novikov et Yakovenko [BNY10]. La borne est obtenue à l’aide d’une construction effective de la connection de Gauss-Manin pour les intégrales itérées.
@article{AFST_2011_6_20_3_465_0,
author = {Sergey Benditkis and Dmitry Novikov},
title = {On the number of zeros of {Melnikov} functions},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {465--491},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 20},
number = {3},
year = {2011},
doi = {10.5802/afst.1314},
mrnumber = {2894835},
zbl = {1243.34042},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1314/}
}
TY - JOUR AU - Sergey Benditkis AU - Dmitry Novikov TI - On the number of zeros of Melnikov functions JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2011 SP - 465 EP - 491 VL - 20 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1314/ DO - 10.5802/afst.1314 LA - en ID - AFST_2011_6_20_3_465_0 ER -
%0 Journal Article %A Sergey Benditkis %A Dmitry Novikov %T On the number of zeros of Melnikov functions %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 465-491 %V 20 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1314/ %R 10.5802/afst.1314 %G en %F AFST_2011_6_20_3_465_0
Sergey Benditkis; Dmitry Novikov. On the number of zeros of Melnikov functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 3, pp. 465-491. doi : 10.5802/afst.1314. https://afst.centre-mersenne.org/articles/10.5802/afst.1314/
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