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Liénard systems with several limit cycles

Published online by Cambridge University Press:  24 October 2008

N. G. Lloyd
N. G. Lloyd
Affiliation:
Department of Mathematics, The University College of Wales, Aberystwyth, Dyfed
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Extract

There is an extensive literature on Liénard's equation

and numerous criteria for the existence of limit cycles have been developed: see the survey of Staude[7], for example. Broadly speaking, such results are proved in one of two ways: a bounded solution is shown to exist and the Poincaré–Bendixson theorem used, or an ‘a priori’ bound for periodic solutions is obtained and the methods of degree theory utilized.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 3 , November 1987 , pp. 565 - 572
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Blows, T. R. and Lloyd, N. G.. The number of small-amplitude limit cycles of Liénard equations. Math. Proc. Cambridge Philos. Soc. 95 (1984), 359–366.CrossRefGoogle Scholar
[2]Cartwright, M. L. and Swinnerton-Dyer., H. P. F.Boundedness theorems for some second-order differential equations, I. Ann. Polon. Math. 29 (1974), 233–256.CrossRefGoogle Scholar
[3]J. P.de Figueiredo, Rui. On the existence of N periodic solutions of Liénard's equation. Nonlinear Analysis 7 (1983), 483–499.CrossRefGoogle Scholar
[4]Ding, Sunhong. Theorem of existence of n limit cycles for Liénard's equation. Sci. Sinica Ser. A 26 (1983), 449–459.Google Scholar
[5]Lins, A., de Melo, W. and Pugh, C. C.. On Liénard's equation. Geometry and Topology (Rio de Janeiro, 1976), Lecture Notes in Math. Vol. 597 (Springer-Verlag, 1977), 335–357.Google Scholar
[6]Rychkov, G. S.. The maximum number of limit cycles of the system is two. Differentsial'nye Uravneniya 11 (1973), 390–391.Google Scholar
[7]Staude, U.. Uniqueness of periodic solutions of the Liénard equation. In Recent Advances in Differential Equations (Academic Press, 1981).Google Scholar
[8]Zeng, Xianwu. Remarks on the uniqueness of limit cycles. Kexue Tongbao 28 (1983), 452–455.Google Scholar