The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Journal of Differential Equations

Volumn 205, Issue 2, 2004, Pages 488-520

  Li, Chengzhi ^a   Roussarie, Robert ^b  

^a PEKING UNIVERSITY   (China)

^b UNIVERSITÉ DE BOURGOGNE   (France)

Author keywords

Cyclicity of elliptic segment loops; Reversible quadratic Hamiltonian systems

Indexed keywords

EID: 4344718736     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2004.04.003     Document Type: Article

References (22)

1. F. Chen, C. Li, J. Llibre, Z. Zhang, A uniform proof on the weak Hilbert's 16th problem for n=2, (2003) preprint.
2. Chow S.-N. Li C. Yi Y. The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loops Ergodic Theory Dynamical System 22 2002 349-374
3. F. Dumortier, R. Roussarie, Abelian integrals and limit cycles, preprint.
4. Gavrilov L. The infinitesimal 16th Hilbert problem in the quadratic case Invent. Math. 143 2001 449-497
5. Gavrilov L. Horozov E. Limit cycles of perturbations of quadratic Hamiltonian vector fields J. Math. Pures Appl. 72 1993 213-238
6. Gavrilov L. Iliev I.D. Second order analysis in polynomially perturbed reversible quadratic Hamiltonian systems Ergodic Theory Dynamical System 20 2000 1671-1686
7. Horozov E. Iliev I.D. On the number of limit cycles in perturbations of quadratic Hamiltonian systems Proc. London Math. Soc. 69 1994 198-224
8. E. Horozov, I.D. Iliev, Hilbert-Arnold problem for cubic Hamiltonian and limit cycles, Proceedings of the Fourth International Collection of Differential Equations, VSP International Publication, Utrecht, 1994, pp. 115-124.
9. Iliev I.D. High-order Melnikov functions for degenerate cubic Hamiltonian Adv. Differential Equations 1 1996 689-708
10. Iliev I.D. The cyclicity of the period annulus of the quadratic Hamiltonian triangle J. Differential Equations 128 1996 309-326
11. Iliev I.D. Perturbations of quadratic centers Bull. Sci. Math. 122 1998 107-161
12. C. Li, J. Llibre, A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems, J. Dynamics and Differential Equations, to appear.
13. Li C. Zhang Ze. Remarks on 16th weak Hilbert problem for n=2 Nonlinearity 15 2002 1975-1992
14. P. Mardesic, Chebyshev systems and the versal unfolding of the cusps of order n, Travaux en Cours, Vol. 57, Hermann, Paris, 1998, 153 pp.
15. Roussarie R. Cyclicité finie des lacets et des points cuspidaux Nonlinearity 2 1989 73-117
16. Roussarie R. On the number of limit cycles which appear by perturbation of separate loop of planar vector fields Bol. Soc. Brasil Mat. 17 1996 67-101
17. R. Roussarie, Bifurcations of planar vector fields and Hilbert's sixteenth problem, Progress in Mathematics, Vol. 164, Birkhäuser, Basel, 1998, 200 pp.
18. Zegeling A. Kooij R.E. The distribution of limit cycles in quadratic systems with four finite singularities J. Differential Equations 151 1999 375-385
19. Zhang Z. Li C. On the number of limit cycles of a class of quadratic Hamiltonian systems under quadratic perturbations Adv. Math. 26 1997 445-460
20. Zhao Y. Liang Z. Lu G. The cyclicity of period annulus of the quadratic Hamiltonian systems with non-Morsean point J. Differential Equations 162 2000 199-223
21. Zhao Y. Zhu S. Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment Bull. Sci. Math. 125 2001 109-138
22. Zła̧dek H. The cyclicity of triangle and segments in quadratic systems J. Differential Equations 122 1995 137-159