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.

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 k of the Melnikov function. The generic case k=1 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 k de la fonction de Melnikov. Le cas générique k=1 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.

DOI: 10.5802/afst.1314
Benditkis, Sergey 1; Novikov, Dmitry 1

1 Weizmann Institute of Science, Rehovot, Israel
     author = {Benditkis, Sergey and Novikov, Dmitry},
     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},
     zbl = {1243.34042},
     mrnumber = {2894835},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1314/}
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Benditkis, Sergey; Novikov, Dmitry. 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. http://www.numdam.org/articles/10.5802/afst.1314/

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