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 Mosc. Math. J., 2001, Volume 1, Number 4, Pages 583–599 (Mi mmj38)

Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations

Yu. S. Ilyashenkoabcd, A. Panovab

a M. V. Lomonosov Moscow State University
b Independent University of Moscow
c Steklov Mathematical Institute, Russian Academy of Sciences
d Cornell University

Abstract: We estimate the number of limit cycles of planar vector fields through the size of the domain of the Poincaré map, the increment of this map, and the width of the complex domain to which the Poincaré map may be analytically extended. The estimate is based on the relationship between the growth and zeros of holomorphic functions [IYa], [I]. This estimate is then applied to getting the upper bound of the number of limit cycles of the Liénard equation $\dot x=y-F(x)$, $\dot y=-x$ through the (odd) power of the monic polynomial $F$ and magnitudes of its coefficients.

Key words and phrases: Limit cycles, Poincaré map, Liénard equation.

DOI: https://doi.org/10.17323/1609-4514-2001-1-4-583-599

Full text: http://www.ams.org/.../abst1-4-2001.html
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MSC: 34Cxx, 34Mxx
Received: October 30, 2001; in revised form December 19, 2001
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Citation: Yu. S. Ilyashenko, A. Panov, “Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations”, Mosc. Math. J., 1:4 (2001), 583–599

Citation in format AMSBIB
\Bibitem{IlyPan01} \by Yu.~S.~Ilyashenko, A.~Panov \paper Some upper estimates of the number of limit cycles of planar vector fields with applications to Li\'enard equations \jour Mosc. Math.~J. \yr 2001 \vol 1 \issue 4 \pages 583--599 \mathnet{http://mi.mathnet.ru/mmj38} \crossref{https://doi.org/10.17323/1609-4514-2001-1-4-583-599} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1901077} \zmath{https://zbmath.org/?q=an:1008.34027} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208587600007} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Ilyashenko Y., “Selected topics in differential equations with real and complex time”, Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Science Series, Series II: Mathematics, Physics and Chemistry, 137, 2004, 317–354
2. M. Briskin, Y. Yomdin, “Tangential version of Hilbert 16th problem for the Abel equation”, Mosc. Math. J., 5:1 (2005), 23–53
3. K. P. Khorev, “On the Number of Limit Cycles of a Monodromic Polynomial Vector Field on the Plane”, Proc. Steklov Inst. Math., 254 (2006), 231–237
4. Chen X., Llibre J., Zhang Zh., “Sufficient conditions for the existence of at least n or exactly n limit cycles for the Lienard differential systems”, Journal of Differential Equations, 242:1 (2007), 11–23
5. Ilyashenko Yu., “Some Open Problems in Real and Complex Dynamical Systems”, Nonlinearity, 21:7 (2008), T101–T107
6. A. Yu. Fishkin, “On the Number of Zeros of an Analytic Perturbation of the Identically Zero Function on a Compact Set”, Math. Notes, 85:1 (2009), 101–108
7. Kolyutsky G.A., “Upper bounds on the number of limit cycles in generalized Li,nard equations of odd type”, Doklady Mathematics, 81:2 (2010), 176–179
8. Yu. Ilyashenko, Jaume Llibre, “A restricted version of Hilbert's 16th problem for quadratic vector fields”, Mosc. Math. J., 10:2 (2010), 317–335
9. Kolutsky G., “An Upper Estimate for the Number of Limit Cycles of Even-Degree Lienard Equations in the Focus Case”, J Dynam Control Systems, 17:2 (2011), 231–241
10. Afsharnezhad Z., Amaleh M.K., “Extension of Chicone's Method for Perturbation Systems of Three Parameters with Application to the Lienard System”, Int. J. Bifurcation Chaos, 22:3 (2012), 1250065
11. Ioakim X., “Generalized Van der Pol Equation and Hilbert'S 16Th Problem”, Electron. J. Differ. Equ., 2014, 120
12. Llibre J., Teixeira M.A., “Limit Cycles For M-Piecewise Discontinuous Polynomial Li,Nard Differential Equations”, Z. Angew. Math. Phys., 66:1 (2015), 51–66
13. Llibre J., Zhang X., “Limit Cycles of the Classical Lienard Differential Systems: a Survey on the Lins Neto, de Melo and Pugh'S Conjecture”, Expo. Math., 35:3 (2017), 286–299
14. Jiang F., Ji Zh., Wang Ya., “On the Number of Limit Cycles of Discontinuous Lienard Polynomial Differential Systems”, Int. J. Bifurcation Chaos, 28:14 (2018), 1850175