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On the number of zeros of Melnikov functions
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 3, pp. 465-491.

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.

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.

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

1 Weizmann Institute of Science, Rehovot, Israel
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     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},
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     volume = {Ser. 6, 20},
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Sergey Benditkis; Dmitry Novikov. On the number of zeros of Melnikov functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 3, pp. 465-491. doi : 10.5802/afst.1314. https://afst.centre-mersenne.org/articles/10.5802/afst.1314/

[AGV] Arnold (V. I.), Gusein-Zade (S. M.), and Varchenko (A. N.).— Singularities of differentiable maps, vol. II, Monodromy and asymptotics of integrals, Birkhäuser Boston Inc., Boston, MA (1988). MR 89g:58024 | MR | Zbl

[BNY10] Binyamini (G.), Novikov (D.), Yakovenko (S.).— On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem Inventiones Mathematicae, 181 No. 2, p. 227-289 (2010). | MR | Zbl

[Ch] Chen (K.-T.).— Collected papers of K.-T. Chen. Contemporary Mathematicians. Birkhauser Boston, Inc., Boston, MA (2001). | MR | Zbl

[D] Deligne (P.).— Équations différentielles à points singuliers réguliers Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin-New York (1970). | MR | Zbl

[E92] Ecalle (J.).— Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris (1992). MR 97f:58104 | MR | Zbl

[F96] Françoise (J. P.).— Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. Systems 16, no. 1, p. 87-96 (1996). | MR | Zbl

[G98] Gavrilov (L.).— Petrov modules and zeros of Abelian integrals, Bull. Sci. Math. 122, no. 8, p. 571-584 (1998). MR 99m:32043 | MR | Zbl

[G05] Gavrilov (L.).— Higher order Poincaré-Pontryagin functions and iterated path integrals, Ann. Fac. Sci. Toulouse Math. (6) 14, no. 4, p. 663-682 (2005). MR2188587 (2006i:34074) | Numdam | MR | Zbl

[GN10] Gavrilov (L.) and Novikov (D.).— On the finite cyclicity of open period annuli, Duke Mathematical Journal. 152, No.  1, p. 1-26. (2010) 0807.0512. | MR | Zbl

[H] Harris (B.).— Iterated Integrals and Cycles on Algebraic Manifolds, Nankai Tracts in Mathematics, vol. 7, World Scientific (2004). | MR | Zbl

[I] Ilyashenko (Yu. S.).— Finiteness theorems for limit cycles, American Mathematical Society, Providence, RI (1991). MR 92k:58221 | Zbl

[I69] Ilyashenko (Yu. S.).— The origin of limit cycles under perturbation of the equation dw dz=-R z R w , where R(z,w) is a polynomial USSR Math.Sb. 78, No. 3, p. 360-373 (1969). | MR | Zbl

[I02] Ilyashenko (Yu. S.).— Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. (N.S.) 39, no. 3, p. 301-354 (electronic) (2002). MR 1 898 209 | MR | Zbl

[IY] Ilyashenko (Yu) and Yakovenko (S.).— Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, (2008). | MR | Zbl

[K81] Kashiwara (M.).— Quasi-unipotent constructible sheaves, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, p. 757-773 (1982). | MR | Zbl

[MN08] Movasati (H.), Nakai (I.).— Commuting holonomies and rigidity of holomorphic foliations Bull. Lond. Math. Soc. 40, no. 3, p. 473-478 (2008). | MR | Zbl

[N02] Novikov (D.).— Modules of Abelian integrals and Picard-Fuchs systems, Nonlinearity 15, no.5, p. 1435-1444 (2002). | MR | Zbl

[Y95] Yakovenko (S.).— A geometric proof of the Bautin theorem, Concerning the Hilbert 16th problem, Amer. Math. Soc., Providence, RI, p. 203-219 (1995). | MR | Zbl

[Z] Żołądek (H.).— The monodromy group, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 67, Birkhäuser Verlag, Basel, (2006). MR2216496 | MR | Zbl

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