Small-Amplitude Limit Cycles in Liénard Systems via Multiplicity

Journal of Differential Equations

Volumn 159, Issue 1, 1999, Pages 186-211

  Gasull, Armengol ^a   Torregrosa, Joan ^a  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

Author keywords

Li nard equation; limit cycles; multiplicity

Indexed keywords

EID: 0033589604     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1999.3649     Document Type: Article

References (23)

1. Aizenberg I. A., Yuzhakov A. P. Integral Representations and Residues in Multidimensional Complex Analysis. Transl. Math. Monogr. 58:1983;Amer. Math. Soc. Providence.
2. Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G. Theory of Bifurcations of Dynamic Systems on a Plane. 1973;Halsted, New York/Toronto.
3. Arnol'd V. I., Varchenko A., Goussein-Zadé S. Singularités des applications différentiables. 1982;Mir, Moscow.
4. Arnol'd V. I. Normal forms of functions in the neighbourhood of degenerate critical points. Russian Math. Surveys. 29:1974;19-48.
5. Blows T. R., Lloyd N. G. The number of small-amplitude limit cycles of Liénard equations. Math. Proc. Cambridge Philos. Soc. 95:1984;359-366.
6. Cherkas L. A. Conditions for a Liénard equation to have a centre. Differential Equations. 12:1976;201-206.
7. Cherkas L. A. Estimation of the number of limit cycles of autonomous systems. Differential Equations. 13:1977;529-547.
8. Christopher C. J., Lloyd N. G., Pearson J. M. On Cherkas's method for centre conditions. Nonlinear World. 2:1995;459-469.
9. Christopher C. J., Lloyd N. G. Small-amplitude limit cycles in polynomial Liénard systems. NoDEA Nonlinear Differential Equations Appl. 3:1996;183-190.
10. Gasull A., Torregrosa J. Center problem for several differential equations via Cherkas method. J. Math. Anal. Appl. 228:1998;322-343.
11. A. Gasull, Differential equations that can be transformed into equations of Liénard type, in, 17° Colóquio Brasileiro de Matemática, 1989.
12. Lins A., de Melo W., Pugh C. C. On Liénard's equation. Geometry and Topology. Lecture Notes in Math. 597:1977;Springer-Verlag, New York/Berlin. p. 335-357.
13. Lloyd N. G., Lynch S. Small amplitude limit cycles of certain Liénard systems. Proc. Roy. Soc. London Ser. A. 418:1988;199-208.
14. Lyapunov A. M. Stability of Motion. Mathematics in Science and Engineering. 30:1966;Academic Press, New York/London.
15. Lynch S. Small-amplitude limit cycles of Liénard systems. Calcolo. 27:1990;1-32.
16. Lynch S. More results on the bifurcation of limit cycles for systems of Liénard type. J. Egyptian Math. Soc. 2:1994;75-87.
17. Lynch S. Recent results on liénard systems. Bainov D. 6th Int. Coll. on Differential Equations. 1996;151-157.
18. Lynch S. Liénard systems and the second part of Hilbert's sixteenth problem. Nonlinear Anal. 30:1997;1395-1403.
19. Lynch S. Generalized quadratic Liénard equations. Appl. Math. Lett. 11:1998;7-10.
20. S. Lynch, Generalized cubic liénard equations, preprint, 1997.
21. Ye Yan Qian, Cai Sui Lin, Chen Lan Sun, Huang Ke Cheng, Luo Ding Jun, Ma Zhi En, Wang Er Nian, Wang Ming Shu, Yang Xin An. Theory of Limit Cycles. Transl. Math. Monogr. 66:1986;Amer. Math. Soc. Providence.
22. Zhang Zhi-Fen, Ding Tong-Ren, Huang Wen-Zao, Dong Zhen-Xi. Qualitative Theory of Differential Equations. Transl. Math. Monogr. 101:1992;Amer. Math. Soc. Providence.
23. Zuppa C. Order of cyclicity of the singular point of Liénard polynomial vector fields. Bol. Soc. Brasil. Mat. 12:1981;105-111.