On the number of zeros of Melnikov functions

Sergey Benditkis[1]; Dmitry Novikov[1]

  • [1] Weizmann Institute of Science, Rehovot, Israel

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 3, page 465-491
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Benditkis, Sergey, and Novikov, Dmitry. "On the number of zeros of Melnikov functions." Annales de la faculté des sciences de Toulouse Mathématiques 20.3 (2011): 465-491. <http://eudml.org/doc/219745>.

@article{Benditkis2011,
abstract = {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.},
affiliation = {Weizmann Institute of Science, Rehovot, Israel; Weizmann Institute of Science, Rehovot, Israel},
author = {Benditkis, Sergey, Novikov, Dmitry},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Limit cycles; Hilbert 16th problem; Hamiltonian systems; Abelian integrals; Melnikov functions; Iterated integrals},
language = {eng},
month = {7},
number = {3},
pages = {465-491},
publisher = {Université Paul Sabatier, Toulouse},
title = {On the number of zeros of Melnikov functions},
url = {http://eudml.org/doc/219745},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Benditkis, Sergey
AU - Novikov, Dmitry
TI - On the number of zeros of Melnikov functions
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/7//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 3
SP - 465
EP - 491
AB - 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.
LA - eng
KW - Limit cycles; Hilbert 16th problem; Hamiltonian systems; Abelian integrals; Melnikov functions; Iterated integrals
UR - http://eudml.org/doc/219745
ER -

References

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  1. 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 Zbl0659.58002MR966191
  2. 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). Zbl1207.34039MR2657426
  3. Chen (K.-T.).— Collected papers of K.-T. Chen. Contemporary Mathematicians. Birkhauser Boston, Inc., Boston, MA (2001). Zbl0977.01042MR1847673
  4. Deligne (P.).— Équations différentielles à points singuliers réguliers Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin-New York (1970). Zbl0244.14004MR417174
  5. Ecalle (J.).— Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, Paris (1992). MR 97f:58104 MR1399559
  6. 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). Zbl0852.34008MR1375128
  7. Gavrilov (L.).— Petrov modules and zeros of Abelian integrals, Bull. Sci. Math. 122, no. 8, p. 571-584 (1998). MR 99m:32043 Zbl0964.32022MR1668534
  8. 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) Zbl1104.34024MR2188587
  9. 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. Zbl1205.34030MR2643055
  10. Harris (B.).— Iterated Integrals and Cycles on Algebraic Manifolds, Nankai Tracts in Mathematics, vol. 7, World Scientific (2004). Zbl1063.14010MR2063961
  11. Ilyashenko (Yu. S.).— Finiteness theorems for limit cycles, American Mathematical Society, Providence, RI (1991). MR 92k:58221 Zbl0743.34036
  12. Ilyashenko (Yu. S.).— The origin of limit cycles under perturbation of the equation d w d z = - R z R w , where R ( z , w ) is a polynomial USSR Math.Sb. 78, No. 3, p. 360-373 (1969). Zbl0194.40102MR243155
  13. 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 Zbl1004.34017MR1898209
  14. Ilyashenko (Yu) and Yakovenko (S.).— Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, (2008). Zbl1186.34001MR2363178
  15. Kashiwara (M.).— Quasi-unipotent constructible sheaves, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, p. 757-773 (1982). Zbl0508.32013MR656052
  16. Movasati (H.), Nakai (I.).— Commuting holonomies and rigidity of holomorphic foliations Bull. Lond. Math. Soc. 40, no. 3, p. 473-478 (2008). Zbl1157.57020MR2418803
  17. Novikov (D.).— Modules of Abelian integrals and Picard-Fuchs systems, Nonlinearity 15, no.5, p. 1435-1444 (2002). Zbl1073.14513MR1925422
  18. Yakovenko (S.).— A geometric proof of the Bautin theorem, Concerning the Hilbert 16th problem, Amer. Math. Soc., Providence, RI, p. 203-219 (1995). Zbl0828.34026MR1334344
  19. 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 Zbl1103.32015MR2216496

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