Cyclicity of elementary polycycles with fixed number of singular points in generic $k$-parameter families
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P. I. Kaleda and I. V. Shchurov
Translated by: N. Yu. Netsvetaev - St. Petersburg Math. J. 22 (2011), 557-571
- DOI: https://doi.org/10.1090/S1061-0022-2011-01158-6
- Published electronically: May 2, 2011
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Abstract:
An estimate is found for the number of limit cycles arising from polycycles in generic finite-parameter families of differential equations on the two-sphere. It is proved that if the polycycles have a fixed number of singular points and all the singular points are elementary, then an estimate of cyclicity holds true, which is polynomial in the number of parameters of the family.References
- V. I. Arnol′d, V. S. Afrajmovich, Yu. S. Il′yashenko, and L. P. Shil′nikov, Bifurcation theory, Current problems in mathematics. Fundamental directions, Vol. 5 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986, pp. 5–218, i (Russian). MR 895653
- Yu. S. Il′yashenko, Normal forms for local families and nonlocal bifurcations, Astérisque 222 (1994), 5, 233–258. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). MR 1285390
- Yu. S. Il′yashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields, Uspekhi Mat. Nauk 46 (1991), no. 1(277), 3–39, 240 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 1, 1–43. MR 1109035, DOI 10.1070/RM1991v046n01ABEH002733
- Yu. Il′yashenko and S. Yakovenko, Finite cyclicity of elementary polycycles in generic families, Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, vol. 165, Amer. Math. Soc., Providence, RI, 1995, pp. 21–95. MR 1334340, DOI 10.1090/trans2/165/02
- V. Kaloshin, The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles, Invent. Math. 151 (2003), no. 3, 451–512. MR 1961336, DOI 10.1007/s00222-002-0244-9
- A. G. Khovanskiĭ, Malochleny, Biblioteka Matematika [Mathematics Library], vol. 2, Izdatel′stvo FAZIS, Moscow, 1997 (Russian, with Russian summary). Appendix A by Yu. S. Il′yashenko; Appendix B by Lu Van den Driz [Lou van den Dries]. MR 1619432
- R. Roussarie, A note on finite cyclicity property and Hilbert’s 16th problem, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 161–168. MR 961099, DOI 10.1007/BFb0083072
Bibliographic Information
- P. I. Kaleda
- Affiliation: OJSC “N. A. Dollezhal Research and Development Insitute of Power Engineering”, M. Krasnoselskaya 2/8, Moscow 107140, Russia
- Email: pkaleda@yandex.ru
- I. V. Shchurov
- Affiliation: National Research University Higher School of Economics, Kochnovsky 3, Moscow, Russia
- Email: ilya.schurov@noo.ru
- Received by editor(s): July 5, 2009
- Published electronically: May 2, 2011
- Additional Notes: Partially supported by RFBR (grant no. 7-01-00017-a), and RFBR/CNRS (grant no. 05-01-02801-CNRSa)
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 557-571
- MSC (2010): Primary 34C07
- DOI: https://doi.org/10.1090/S1061-0022-2011-01158-6
- MathSciNet review: 2768961