Journal of Dynamical and Control Systems

, Volume 4, Issue 2, pp 149–189 | Cite as

Cyclicity of Graphics with Semi-Hyperbolic Points Inside Quadratic Systems

  • C. Rousseau
  • G. Świrszcz
  • H. Żoładek

Abstract

This paper is a part of the proof of the existential part of Hilbert's 16th problem for quadratic vector fields initiated in [2]. Its principal aim is the proof of the finite cyclicity of four elementary graphics among quadratic systems with two semi-hyperbolic points and surrounding a center, namely, the graphics (I52, (I91), (H52) and (H111) (using the names introduced in [2]). The technique used is a refinement of the technique of Khovanskii as adapted to the finite cyclicity of graphics by Il'yashenko and Yakovenko, together with an equivalent of the Bautin trick to treat the center case. We show that the cyclicity of each of the first three graphics is equal to 2 and that the cyclicity of the fourth one is equal to three. We improve the known results about finite cyclicity of the graphics (H81), (H101), (I42) by showing that their cyclicities are equal to 2.

Polynomial vector fields finite number of limit cycles finite cyclicity of graphics semi-hyperbolic points Khovanskii procedure 

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References

  1. 1.
    N. N. Bautin, On the number of limit cycles appearing with variations of coeficients from an equilibrium state of the type of a focus or a center. (Russian) Mat. Sb. (N.S.) 30 (1952), 181–196; English translation: Am. Math. Soc. Transl. (1954). Reprinted in: Stability and Dynamical Systems. Am. Math. Soc. Transl. Ser. 1, 5 (1962), 396–413.Google Scholar
  2. 2.
    F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert's 16th problem for quadratic vector fields. J. Differ. Equ. 110 (1994), 86–133.CrossRefGoogle Scholar
  3. 3.
    _____, Elementary graphics of cyclicity one and two. Nonlinearity 7 (1994), 1001–1043.CrossRefGoogle Scholar
  4. 4.
    F. Dumortier, M. El Morsalani, and C. Rousseau, Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics. Nonlinearity 9 (1996), 1209–1261.CrossRefGoogle Scholar
  5. 5.
    M. El Morsalani, Bifurcations de polycycles infinis pour les champs de vecteurs polynomiaux. Ann. Faculté Sci. Toulouse 3 (1994), 387–410.Google Scholar
  6. 6.
    _____, Perturbations of grahics with semi-hyperbolic singularities. Bull. Sci. Math. 120 (1996), 337–366.Google Scholar
  7. 7.
    A. Guzmán and C. Rousseau, Genericity conditions for finite cyclicity of elementary graphics. Preprint, Univ. Montréal. Sumbitted to J. Differ. Equ. (1997).Google Scholar
  8. 8.
    D. Hilbert, Mathematische Probleme (lecture). The second Int. Cong. Math., Paris, 1990, Nachr. Ges. Wiss. Gottingen Math.-Phys. Kl. (1990), 253–297; Mathematical developments arising from Hilbert's problems. In: Proc. Symp. Pure Math., F. Brower, Ed., Am. Math. Soc. 28 (1976), 50–51.Google Scholar
  9. 9.
    Y. Il'yashenko and S. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Russ. Math. Surv. 46 (1991), 1–43.Google Scholar
  10. 10.
    _____, Finite cyclicity of elementary polycycles in generic families. Am. Math. Soc. Transl. 165 (1995), 21–95.Google Scholar
  11. 11.
    A. G. Khovanskii, Fewnomials. Am. Math. Soc. Transl. Math. Monographs 88 (1991).Google Scholar
  12. 12.
    A. Mourtada, Cyclicité finie des polycycles hyperboliques des champs de vecteurs du plan: mise sous forme normale. Springer Lect. Notes Math. 1455 (1990), 272–314.Google Scholar
  13. 13.
    R. Moussu and C. Roche, Théorie de Khovanskii et problème de Dulac. Inv. Math. 105 (1991), 431–441.Google Scholar
  14. 14.
    M. El Morsalani and A. Mourtada, Degenerate and nontrivial hyperbolic 2-polycycles: appearance of two independant Ecalle—Roussarie compensators and Khovanskii's theory. Nonlinearity 7 (1994), 68–83.Google Scholar
  15. 15.
    R. Roussarie, A note on finite cyclicity and Hilbert's 16th problem. Springer Lect. Notes Math. 1331 (1988), 161–168.Google Scholar
  16. 16.
    _____, Cyclicité finie des lacets et des points cuspidaux. Nonlinearity 2 (1989), 73–117.Google Scholar
  17. 17.
    H. Żoł{ie189-1}dek, Asymptotic properties of abelian integrals arising in quadratic systems. In: Proc. Conf. “Bifurcations in Differentiable Dynamics” Diepenbeek, 1992 (to appear).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • C. Rousseau
    • 1
  • G. Świrszcz
    • 2
  • H. Żoładek
    • 2
  1. 1.Département de Mathématiques et de Statistique and CRMUniversité de Montréal, Montréal, QcCanada
  2. 2.Institute of Mathematics, University of Warsaw, Polish Academy of SciencesWarsawPoland

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