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

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Moscow Mathematical Journal, 2001, Volume 1, Number 4, Pages 583–599
DOI: https://doi.org/10.17323/1609-4514-2001-1-4-583-599 (Mi mmj38)

This article is cited in 20 scientific papers (total in 20 papers)

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

Yu. S. Ilyashenko^abcd, A. Panov^ab

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

Full-text PDF Citations (20)

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

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, Poincarй map, Liйnard equation.

Received: October 30, 2001; in revised form December 19, 2001

Bibliographic databases:

Document Type: Article

MSC: 34Cxx, 34Mxx

Language: English

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

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\issue 4

\pages 583--599

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Linking options:

https://www.mathnet.ru/eng/mmj38

https://www.mathnet.ru/eng/mmj/v1/i4/p583

This publication is cited in the following 20 articles:

Albert C. J. Luo, “Bifurcations for Homoclinic Networks in Two-Dimensional Polynomial Systems”, Int. J. Bifurcation Chaos, 34:03 (2024)
Albert C. J. Luo, “Limit cycles and homoclinic networks in two-dimensional polynomial systems”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 34:2 (2024)
Diego M. Benardete, “Complex Methods for Bounds on the Number of Periodic Solutions with an Application to a Neural Model”, The American Mathematical Monthly, 129:2 (2022), 133
M.J. Álvarez, B. Coll, P. De Maesschalck, R. Prohens, “Asymptotic lower bounds on Hilbert numbers using canard cycles”, Journal of Differential Equations, 268:7 (2020), 3370
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
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
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
Ioakim X., “Generalized Van der Pol Equation and Hilbert'S 16Th Problem”, Electron. J. Differ. Equ., 2014, 120
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
Maoan Han, Pei Yu, Applied Mathematical Sciences, 181, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, 2012, 81
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
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
Yu. Ilyashenko, Jaume Llibre, “A restricted version of Hilbert's 16th problem for quadratic vector fields”, Mosc. Math. J., 10:2 (2010), 317–335
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
Ilyashenko Yu., “Some Open Problems in Real and Complex Dynamical Systems”, Nonlinearity, 21:7 (2008), T101–T107
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
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
Yu. S. Il'yashenko, Mathematical Events of the Twentieth Century, 2006, 101
M. Briskin, Y. Yomdin, “Tangential version of Hilbert 16th problem for the Abel equation”, Mosc. Math. J., 5:1 (2005), 23–53
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

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