Journal of Dynamical and Control Systems

, Volume 4, Issue 2, pp 149–189

Cyclicity of Graphics with Semi-Hyperbolic Points Inside Quadratic Systems

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

DOI: 10.1023/A:1022887001627

Cite this article as:
Rousseau, C., Świrszcz, G. & Żoładek, H. Journal of Dynamical and Control Systems (1998) 4: 149. doi:10.1023/A:1022887001627

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 fieldsfinite number of limit cyclesfinite cyclicity of graphicssemi-hyperbolic pointsKhovanskii procedure

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