Hostname: page-component-6bf8c574d5-w79xw Total loading time: 0 Render date: 2025-02-24T06:01:49.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit cycles of the generalized polynomial Liénard differential equations

Published online by Cambridge University Press:  12 November 2009

JAUME LLIBRE ,
ANA CRISTINA MEREU  and
MARCO ANTONIO TEIXEIRA
JAUME LLIBRE
Affiliation:
Departament de Matemtiques, Universitat Autnoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. e-mail: jllibre@mat.uab.cat
ANA CRISTINA MEREU
Affiliation:
Departamento de Matemtica, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P, Brazil. e-mail: anameren@ime.unicamp.br, teixeira@ime.unicamp.br
MARCO ANTONIO TEIXEIRA
Affiliation:
Departamento de Matemtica, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P, Brazil. e-mail: anameren@ime.unicamp.br, teixeira@ime.unicamp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 2 , March 2010 , pp. 363 - 383
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Blows, T. R. and Lloyd, N. G.The number of small-amplitude limit cycles of Liénard equations. Math. Proc. Camb. Phil. Soc. 95 (1984), 359–366.CrossRefGoogle Scholar
[2]Buică, A. and Llibre, J.Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128 (2004), 7–22.CrossRefGoogle Scholar
[3]Christopher, C. J. and Lynch, S.Small-amplitude limti cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces. Nonlinearity 12 (1999), 1099–1112.CrossRefGoogle Scholar
[4]Coppel, W. A. Some quadratic systems with at most one limit cycles. Dynamics Reported Vol. 2 Wiley, 1998, pp. 61–68.CrossRefGoogle Scholar
[5]Dumortier, F., Panazzolo, D. and Roussarie, R.More limit cycles than expected in Liénard systems. Proc. Amer. Math. Soc. 135 (2007), 1895–1904.CrossRefGoogle Scholar
[6]Dumortier, F. and Li, C.On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations. Nonlinearity 9 (1996), 1489–1500.CrossRefGoogle Scholar
[7]Dumortier, F. and Li, C.Quadratic Liénard equations with quadratic damping. J. Diff. Eqs. 139 (1997), 41–59.CrossRefGoogle Scholar
[8]Dumortier, F. and Rousseau, C.Cubic Liénard equations with linear dampimg. Nonlinearity 3 (1990), 1015–1039.CrossRefGoogle Scholar
[9]Gasull, A. and Torregrosa, J.Small-amplitude limit cycles in Liénard systems via multiplicity. J. Diff. Eqs. 159 (1998), 1015–1039.Google Scholar
[10]Ilyashenko, Y.Centennial history of Hilbert's 16th problem. Bull. Amer. Math. Soc. 39 (2002), 301–354.CrossRefGoogle Scholar
[11]Jibin, LiHilbert's 16th problem and bifurcations of planar polynomial vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 47–106.Google Scholar
[12]Liénard, A.Étude des oscillations entrenues. Revue Génerale de l' Électricité 23 (1928), 946–954.Google Scholar
[13]Lins, A., de Melo, W. and Pugh, C.C.On Liénard's equation. Lecture Notes in Math 597 Springer, (1977), pp. 335–357.CrossRefGoogle Scholar
[14]Lloyd, N. G. Limit cycles of polynomial systems-some recent developments. London Math. Soc. Lecture Note Ser. 127, Cambridge University Press, 1988, pp. 192–234.Google Scholar
[15]Lloyd, N. G. and Lynch, S.Small-amplitude limit cycles of certain Liénard systems. Proc. Royal Soc. London Ser. A 418 (1988), 199–208.Google Scholar
[16]Lloyd, N. and Pearson, J.Symmetric in planar dynamical systems. J. Symb. Comput. 33 (2002), 357–366.CrossRefGoogle Scholar
[17]Lynch, S.Limit cycles if generalized Liénard equations. Appl. Math. Lett. 8 (1995), 15–17,CrossRefGoogle Scholar
[18]Lynch, S.Generalized quadratic Liénard equations. Appl. Math. Lett. 11 (1998), 7–10,CrossRefGoogle Scholar
[19]Lynch, S.Generalized cubic Liénard equations. Appl. Math. Lett. 12 (1999), 1–6,CrossRefGoogle Scholar
[20]Lynch, S. and Christopher, C. J.Limit cycles in highly non-linear differential equations. J. Sound Vib. 224 (1999), 505–517CrossRefGoogle Scholar
[21]Rychkov, G. S.The maximum number of limit cycle of the system ẋ = y − a 1x 3 − a 2x 5, ẏ = −x is two. Differential'nye Uravneniya 11 (1975), 380–391.Google Scholar
[22]Smale, S.Mathematical problems for the next century. Math. Intelligencer 20 (1998), 7–15.CrossRefGoogle Scholar
[23]Yu, P. and Han, M.Limit cycles in generalized Liénard systems. Chaos Solitons Fractals 30 (2006), 1048–1068.CrossRefGoogle Scholar