Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T07:33:00.466Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Liénard system which has no periodic solutions

Published online by Cambridge University Press:  24 October 2008

Jitsuro Sugie  and
Toshiaki Yoneyama
Jitsuro Sugie
Affiliation:
Department of Mathematics, Okayama University, Okayama 700, Japan
Toshiaki Yoneyama
Affiliation:
Department of Applied Sciences, Miyazaki University, Miyazaki 889-21, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

The problem of periodicity of solutions of the generalized Liénard equation

has attracted much attention. Many efforts have been made to give sufficient conditions to guarantee the existence and the uniqueness of periodic solutions (limit cycles) of (1·1). There are also some papers on the number of limit cycles of (1·1) (see, for example, [3, 5, 6, 13]). However, there are only a few results on non-existence of periodic solutions of (1·1).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 2 , March 1993 , pp. 413 - 422
Copyright
Copyright © Cambridge Philosophical Society 1993

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]Hara, T. and Yoneyama, T.. On the global center of generalized Liénard equation and its application to stability problems. Funkcial. Ekvac. 28 (1985), 171–192.Google Scholar
[2]Lins, A., Melo, W. De and Pugh, C. C.. On Liénard's equation. In Geometry and Topology, Lecture Notes in Math. vol. 597 (Springer-Verlag, 1977), pp. 335–357.CrossRefGoogle Scholar
[3]Lloyd, N. G.. Liénard systems with several limit cycles. Math. Proc. Cambridge Philos. Soc. 102 (1987), 565–572.CrossRefGoogle Scholar
[4]Sansone, G.. Sopra una classe di equazioni di Liénard prive di integrali periodici. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 6 (1949), 156–160.Google Scholar
[5]Sansone, G. and Conti, R.. Non-Linear Differential Equations (Pergamon Press, 1964).Google Scholar
[6]Staude, U.. Uniqueness of periodic solutions of the Liénard equation. In Recent Advances in Differential Equations (Academic Press, 1981), pp. 421–429.CrossRefGoogle Scholar
[7]Sugie, J.. The global centre for the Liénard system. Nonlinear Anal. 17 (1991), 333–345.CrossRefGoogle Scholar
[8]Sugie, J. and Hara, T.. Non-existence of periodic solutions of the Liénard system. J. Math. Anal. Appl. 159 (1991), 224–236.CrossRefGoogle Scholar
[9]Villari, G.. A new system for Liénard equation. Boll. Un. Mat. Ital. A (7) 1 (1987), 375–381.Google Scholar
[10]Villari, G.. On the qualitative behaviour of solutions of Liénard equation. J. Differential Equations 67 (1987), 269–277.CrossRefGoogle Scholar
[11]Villari, G. and Zanolin, F.. On a dynamical system in the Liénard plane. Necessary and sufficient conditions for the intersection with the vertical isocline and applications. Funkcial. Ekvac. 33 (1990), 19–38.Google Scholar
[12]Ke, Wang. On the existence of limit cycles of the equation x' = h(y)−F(x), y' = −g(x). Kodai Math. J. 12 (1989), 98–108.Google Scholar
[13]Zhifen, Zhang. What influences on the number of limit cycles of Liénard equation. In Nonlinear Analysis and Applications, Lecture Notes in Pure and Appl. Math. vol. 109 (Dekker, 1987), pp. 623–628.Google Scholar