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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

   

 

On the number of limit cycles of planar quadratic vector fields with a perturbed center


Author: A. Yu. Fishkin
Translated by: Alex Martsinkovsky
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 71 (2010).
Journal: Trans. Moscow Math. Soc. 2010, 105-139
MSC (2010): Primary 34C07; Secondary 37C10, 37C27
Published electronically: December 21, 2010
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Abstract: We investigate the number of limit cycles of a planar quadratic vector field with a perturbed center-like singular point. An upper bound is obtained on the number of $ \delta$-good limit cycles of such a vector field (Theorem 1). Here $ \delta$ is a parameter characterizing the limit cycles: it shows how far those cycles are from the singular points of the vector field and from the infinite points. The bound also includes another parameter, $ \kappa$, characterizing the vector field. More precisely, $ \kappa$ gives an estimate on the distance from the vector field to the set consisting of quadratic vector fields with a line of singular points. Earlier, Ilyashenko and Llibre found a bound on the number of $ \delta$-good limit cycles of those vector fields which are sufficiently far from the fields with a center-like singular point. Theorem 1 and that bound complement each other and yield a new bound on the number of $ \delta$-good limit cycles of a quadratic vector field, regardless of its distance to the vector fields with a center-like singular point (Theorem 2).


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Additional Information

A. Yu. Fishkin
Affiliation: M. V. Lomonosov Moscow State University
Email: fishkinalexey@gmail.com

DOI: http://dx.doi.org/10.1090/S0077-1554-2010-00181-1
Published electronically: December 21, 2010
Additional Notes: Supported by the RFFI Grants 7–01–00017-a and 05–01–02801–CNRS_a).
Article copyright: © Copyright 2010 American Mathematical Society