Center Problem for Several Differential Equations via Cherkas' Method

Journal of Mathematical Analysis and Applications

Volumn 228, Issue 2, 1998, Pages 322-343

  Gasull, Armengol ^a   Torregrosa, Joan ^a  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001494750     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1998.6112     Document Type: Article

References (29)

1. Andreev A. F. Investigation of the behavior of the integral curves of a system of two differential equations in the neighborhood of a singular point. Transl. Amer. Math. Soc. 8:1958;183-207.
2. Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G. Qualitative Theory of Second-Order Dynamic Systems. 1973;Wiley, New York.
3. Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G. Theory of Bifurcations of Dynamic Systems on a Plane. 1973;Wiley, New York.
4. Bautin N. N. On the number of limit cycles which appear with variation of coefficients from an equilibrium position of focus or center type. Amer. Math. Soc. Transl. 100:1954;397-413.
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. Broer H. W., Dumortier F., van Strien S. J., Takens F. de Jager E. M. Structures in Dynamics. Studies in Mathematical Physics. 2:1991;North-Holland, Amsterdam.
7. Chavarriga J., García I., Giné J. Integrability of centers perturbed by quasi-homogeneous polynomials. J. Math. Anal. Appl. 210:1997;268-278.
8. Chavarriga J., Giné J. Integrability of cubic with degenerate infinity. Proceedings of XIV CEDYA, Vic. 1995.
9. Chavarriga J., Giné J. Integrability of a linear center perturbed by fourth degree homogeneous polynomial. Publ. Mat. 40:1996;21-39.
10. Chavarriga J., Giné J. Integrability of a linear center perturbed by fifth degree homogeneous polynomial. Publ. Mat. 41:1997;335-356.
11. Cherkas L. A. Conditions for a Liénard equation to have a centre. Differential Equations. 12:1976;201-206.
12. Christopher C. J. Invariant algebraic curves and conditions for centre. Proc. Roy. Soc. Edinburgh Sect. A. 124:1994;1209-1229.
13. Christopher C. J., Lloyd N. G., Pearson J. M. On a Cherkas's method for centre conditions. Nonlinear World. 2:1995;459-469.
14. Christopher C. J., Lloyd N. G. Small-amplitude limit cycles in polynomial Liénard systems. NoDEA Nonlinear Differential Equations Appl. 3:1996;183-190.
15. Cima A., Gasull A., Mañosa F. Cyclicity of a family of vector fields. J. Math. Anal. Appl. 196:1995;921-937.
16. Coll B., Gasull A., Prohens R. Differential equations defined by the sum of two quasi-homogeneous vector fields. Canad. J. Math. 49:1997;212-231.
17. Conti R., Sansone G. Nonlinear Differential Equations. 1964;Pergamon, London.
18. A. Gasull, J. Torregrosa, Small-amplitude limit cycles on Liénard systems via multiplicity, 1997.
19. Lins A., De Melo W., Pugh C. C. On Liénard's equation. Lecture Notes in Math. 596:1977;335-357.
20. N. G. Lloyd, C. J. Christopher, J. Devlin, J. M. Pearson, N. Yasmin, Quadratic-like cubic systems, 1996.
21. Lloyd N. G., Pearson J. M. Five limit cycles for a simple cubic system. Publ. Mat. 41:1997;199-208.
22. Lyapunov A. M. Stability of Motion. Mathematics in Science and Engineering. 30:1966;Academic Press, New York/London.
23. Lynch S. Small-amplitude limit cycles of Liénard systems. Calcolo. 27:1990;1-32.
24. Pearson J. M., Lloyd N. G., Christopher C. J. Algorithmic derivation of centre conditions. SIAM Rev. 38:1996;619-636.
25. Sibirskii K. S. On the number of limit cycles on the neighborhood of a singular point. Differential Equations. 1:1965;36-47.
26. Staude U. Uniqueness of periodic solutions of the Liénard equation. Recent Advances in Differential Equations. 1981;Academic Press, New York. p. 421-429.
27. Yanqian Ye. Theory of Limit Cycles. Translations of Mathematical Monographs. 66:1986;Amer. Math. Soc. Providence.
28. Shu-Xiang Yu, Ji-Zhou Zhang. On the center of the Liénard equation. J. Differential Equations. 102:1993;53-61.
29. Zuppa C. Order of cyclicity of the singular point of Liénard polynomial vector fields. Bol. Soc. Brasil. Mat. 12:1981;105-111.