Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces

Nonlinearity

Volumn 12, Issue 4, 1999, Pages 1099-1112

  Christopher, Colin ^a   Lynch, Stephen ^b  

^a UNIVERSITY OF PLYMOUTH   (United Kingdom)

^b MANCHESTER METROPOLITAN UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033163981     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/12/4/321     Document Type: Article

References (25)

1. Blows T R and Lloyd N G 1984 The number of small-amplitude limit cycles of Liénard equations Math. Proc. Camb. Phil. Soc. 95 359-66
2. Christopher C J 1999 An algebraic approach to the classification of centres in Liénard systems J. Math. Anal. Appl. 229 319-29
3. Christopher C J and Lloyd N G 1995 Polynomial systems: a lower bound for the Hilbert numbers Proc. R. Soc. 450 219-24
4. Christopher C J and Lloyd N G 1996 Small-amplitude limit cycles in polynomial Liénard systems Nonlinear Diff. Eqns Appl. 3 183-90
5. Dumortier F and Chengzhi L 1996 On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations Nonlinearity 9 1489-500
6. Dumortier F and Chengzhi L 1997 Quadratic Liénard equations with quadratic damping J. Diff. Eqns 139 41-59
7. Dumortier F and Rousseau C 1990 Cubic Liénard equations with linear damping Nonlinearity 3 1015-39
8. Gasull A and Torregrosa J Small-amplitude limit cycles in Liénard systems via multiplicity Preprint (Universitat Autònoma de Barcelona)
9. Giacomini H and Neukirch S 1997 Number of limit cycles of the Liénard equation Phys. Rev. E 56 3809-13
10. Giacomini H and Neukirch S 1998 Improving a method for the study of limit cycles of the Liénard equation Phys. Rev. E 57 6573-6
11. Han M 1998 On the maximal number of limit cycles in global bifurcations Preprint (Georgia Institute of Technology)
12. Kooij R E and Zegeling A 1997 Qualitative properties of two-dimensional predator-prey systems Nonlinear Anal. Theory Methods Appl. 29 693-715
13. Lins A, de Melo W and Pugh CC 1977 On Liénard's equation (Lecture Notes in Math. vol 597) (Berlin: Springer) pp 335-57
14. Lloyd N G and Lynch S 1988 Small-amplitude limit cycles of certain Liénard systems Proc. R. Soc. A 418 199-208
15. Lynch S 1990 Small amplitude limit cycles of Liénard systems Calcolo 27 1-32
16. Lynch S 1997 Liénard systems and the second part of Hilbert's 16th problem Nonlinear Anal. Theory Methods Appl. 30 1395-403
17. Lynch S 1998 Generalized quadratic Liénard equations Appl. Math. Lett. 11 7-10
18. Lynch S 1999 Generalized cubic Liénard equations Appl. Math. Lett. 12 1-6
19. Odani 1995 The limit cycle of the van der Pol equation is not algebraic J. Diff. Eqns 115 146-52
20. Petrov G S 1984 Number of zeros of complete elliptic integrals Funct. Anal. Appl. 18 148-9
21. Sanjuán 1998 Liénard systems, limit cycles and Melnikov theory Phys. Rev. E 57 340-4
22. Seigel C K and Moser J K 1971 Lectures on Celestial Mechanics (New York: Springer)
23. Zegeling A and Kooij R E 1994 Uniqueness of limit cycles in polynomial systems with algebraic invariants Bull. Aust. Math. Soc. 49 7-20
24. Zeng X, Zhang Z and Gao S 1994 On the uniqueness of the limit cycle of the generalised Liénard equation Bull. Lond. Math. Soc. 26 213-47
25. Żoła̧dek H 1998 Algebraic invariant curves for the Liénard equation Trans. Am. Math. Soc. 350 1681-701