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

This article is cited in 14 scientific papers (total in 14 papers)

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.


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MSC: 34Cxx, 34Mxx
Received: October 30, 2001; in revised form December 19, 2001

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
\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

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    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  crossref  mathscinet  isi
    2. M. Briskin, Y. Yomdin, “Tangential version of Hilbert 16th problem for the Abel equation”, Mosc. Math. J., 5:1 (2005), 23–53  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet
    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  crossref  mathscinet  zmath  isi
    5. Ilyashenko Yu., “Some Open Problems in Real and Complex Dynamical Systems”, Nonlinearity, 21:7 (2008), T101–T107  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet
    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    11. Ioakim X., “Generalized Van der Pol Equation and Hilbert'S 16Th Problem”, Electron. J. Differ. Equ., 2014, 120  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
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