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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Cyclicity of elementary polycycles with fixed number of singular points in generic $ k$-parameter families

Authors: P. I. Kaleda and I. V. Shchurov
Translated by: N. Yu. Netsvetaev
Original publication: Algebra i Analiz, tom 22 (2010), nomer 4.
Journal: St. Petersburg Math. J. 22 (2011), 557-571
MSC (2010): Primary 34C07
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.

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Additional Information

P. I. Kaleda
Affiliation: OJSC “N. A. Dollezhal Research and Development Insitute of Power Engineering”, M. Krasnoselskaya 2/8, Moscow 107140, Russia

I. V. Shchurov
Affiliation: National Research University Higher School of Economics, Kochnovsky 3, Moscow, Russia

Keywords: Number of limit cycles, polycycle, Hilbert’s sixteenth problem, Hilbert–Arnol′d problem
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)
Article copyright: © Copyright 2011 American Mathematical Society