Generalized Liénard equations, Cyclicity and Hopf-Takens bifurcations

Qualitative Theory of Dynamical Systems

Volumn 5, Issue 2, 2004, Pages 195-222

  Caubergh, Magdalena ^a   Françoise, Jean Pierre ^b  

^a HASSELT UNIVERSITY   (Belgium)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Centers; Cyclicity; Hopf bifurcations; Li nard equations; Limit cycles; Lyapunov quantities

Indexed keywords

EID: 84896693375     PISSN: 15755460     EISSN: 16623592     Source Type: Journal    
DOI: 10.1007/BF02972678     Document Type: Article

References (16)

1. M. Briskin, J.-P. Françoise and Y. Yomdin, The Bautin ideal of the Abel equation, Nonlinearity 11, (1998), no. 3, 431-443.
2. T.R. Blows and N.G. Lloyd, The number of small-amplitude limit cycles of Liénard equations, Math. Proc. Cambridge Philos. Soc. 95, (1984), no. 2, 359-366.
3. L. Cherkas, A condition for distinguishing between the center and the focus of Liénard's equation, Dokl. Akad. Nauk BSSR 22, (1978), no. 11, 969-1051 (Russian).
4. M. Caubergh and F. Dumortier, Hopf-Takens bifurcations and centers, J. of Differential Equations 202, (2004), no. 1, 1-31.
5. C.J. Christopher and N.G. Lloyd, Small-amplitude limit cycles in Liénard systems, Non-linear Differential Equations Appl. 3, (1996), no. 2, 183-190.
6. J.-P. Françoise, Théorie des singularités et systèmes dynamiques, Presses Universitaires de France, Paris (1995).
7. J.-P. Françoise, Analytic properties of the return mapping of Liénard equations, Mathematical Research Letters 9, (2002), 255-266.
8. J.-P. Françoise and C.C. Pugh, Keeping track of Limit Cycles, Journal of Differential Equations 65, (1986), 139-157.
9. J.-P. Françoise and Y. Yomdin, Bernstein Inequalities and applications to analytic geometry and differential equations, J. Funct. Anal. 146, (1997), no. 1, 185-205.
10. A. Gasull and J. Torregrosa, Small-amplitude limit cycles in Liénard systems via multiplicity, J. Differential Equations 159, (1999), no. 1, 186-211.
11. N.G. Lloyd, Liénard systems with several limit cycles, Math. Proc. Cambridge Philos. Soc. 102, (1987), no. 3, 565-572.
12. B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press, London (1966).
13. R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's sixteenth Problem, Birkhauser-Verlag, Basel (1998).
14. D. Schlomiuk, Algebraic and Geometric Aspects of the theory of polynomial vector fields, in Bifurcations and periodic orbits of Vector Fields(ed. D. Schlomiuk) NATO ASI Series, Math. and Physical Sciences 408, (1993), 429-463.
15. F. Takens, Unfoldings of Certain Singularities of Vector Fields: Generalised Hopf Bifurcations, J. Differential Equations 14, (1973), no. 3, 476-493.
16. C. Zuppa, Order of cyclicity of the singular point of Liénard polynomial vector fields, Bol. Soc. Brasil. Mat. 12, (1981), no. 3, 105-111.