Liénard systems and the second part of Hilbert's sixteenth problem

Nonlinear Analysis, Theory, Methods and Applications

Volumn 30, Issue 3, 1997, Pages 1395-1403

  Lynch, S ^a  

^a MANCHESTER METROPOLITAN UNIVERSITY   (United Kingdom)

Author keywords

Global and local bifurcations; Liapunov quantities; Small amplitude limit cycles and Hilbert numbers

Indexed keywords

EID: 0001272623     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0362-546X(97)00200-9     Document Type: Article

References (28)

1. CHRISTOPHER C. J. & LLOYD N. G., Polynomial systems: a lower bound for the Hilbert numbers, Proc. R. Soc. Lond. A 450, 219-224 (1995).
2. SONGLING S., A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica A 23, 153-158 (1980).
3. LI JIBIN & LI CHUNFU, Global bifurcations of planar disturbed Hamiltonian systems and distributions of limit cycles of cubic systems. Acta. Math. Sinica 28, 509-521 (1985).
4. BLOWS T. R. & PERKO L. M., Bifurcation of limit cycles from centres and separatrix cycles of planar analytic systems, SIAM Review 36 (3), 341-376 (1994).
5. GIACOMINI H., LLIBRE J. & KIANO M., On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity 9 (2), 501-516 (1996).
6. BLOWS T. R. & ROUSSEAU C., Bifurcation at infinity in polynomial vector fields, J. Diff. Eqn., 104, 215-242 (1993).
7. GUINEZ V., SAEZ E. & SZANTO I., Small amplitude limit cycles for cubic systems, Canad. Math. Bull, 36 (1), 54-63 (1993).
8. LIÉNARD A., Étude des oscillations entrenues, Revue Génerale de l'Électricité, 23, 946-954 (1928).
9. 5, ẏ = -x is two, Differentsial'nye Uravneniya, 11, 380-391 (1973).
10. CHERKAS L. A., Conditions for a Liénard equation to have a centre, Differentsial'nye Uravneniya, 12, 201-206 (1976).
11. LINS A., de MELO W. & PUGH C., On Liénards equation with linear damping, Lecture Notes in Mathematics, 597, (Edited by J. Palis and M. do Carmo), 335-357, Springer-Verlag, Berlin (1977).
12. XIANWU Z., Remarks on the uniqueness of limit cycles, Kexue Tongbao, 28, 452-455 (1983).
13. DUMORTIER F. & ROUSSARIE R., Cubic Liénard equations with linear damping, Nonlinearity, 3, 1015-1039 (1990).
14. XIANWU Z., ZHI-FEN Z. & SUZHI G., On the uniqueness of the limit cycle of the generalized Liénard equation, Bull. Lond. Math. Soc., 26, 213-247 (1994).
15. ODANI K., The limit cycle of the Van der Pol equation is not algebraic, J. Diff. Eqn., 115, 146-152 (1995).
16. BAUTIN N., On the number of limit cycles which appear with the variation of the coefficients from an equilibrium point of focus or centre type, Amer. Math. Soc. Trans., 5, 396-414 (1962).
17. SIBIRSKII K. S., The number of limit cycles in the neighbourhood of a critical point, Differential Equations, 1, 36-47 (1965).
18. JAMES E. M. & LLOYD N. G., A cubic system with eight small-amplitude limit cycles, I.M.A. J. Applied Math., 47, 163-171 (1991).
19. ZOLADEK H., Eleven small limit cycles in a cubic vector field, Nonlinearity 8, 843-860 (1995).
20. BLOWS T. R. & LLOYD N. G., The number of small-amplitude limit cycles of Liénard equations, Math. Proc. Camb. Philos. Soc., 95, 359-366 (1984).
21. LLOYD N. G. & LYNCH S., Small-amplitude limit cycles of certain Liénard systems, Proc. Roy. Soc. Lond. Ser. A, 418, 199-208 (1988).
22. LYNCH S., Small-amplitude limit cycles of Liénard equations, Calcolo, 127 (1-2), (1990).
23. LYNCH S., More results on the bifurcation of limit cycles for systems of Liénard type, J. Egypt. Math. Soc., 2, 75-87 (1994).
24. LYNCH S., Small-amplitude limit cycles of the generalized mixed Rayleigh-Liénard oscillator, Journal of Sound and Vibration, 178 (5), 615-620 (1994).
25. LYNCH S., Limit cycles of Generalized Liénard equations, Applied Math. Lett., 8 (6), 15-17 (1995).
26. URBINA A. M., LEÓN de la BARRA G., LEÓN de la BARRA M. & CAN̄AS M., Limit cycles of Liénard equations with nonlinear damping, Canad. Math. Bull., 36 2, 251-256 (1993).
27. CHRISTOPHER C. J. & LLOYD N. G., Small-amplitude limit cycles in Liénard Systems, Nonlinear Differential Equations and Applications, (to appear).
28. LLOYD N. G., PEARSON J. M., SAEZ E. & SZANTO I., Limit cycles of a cubic Kolmogorov system, Applied Math. Lett. 9 (1), 15-18 (1996).