Abstract
In this paper we investigate the perturbations from a kind of quartic Hamiltonians under general cubic polynomials. It is proved that the number of isolated zeros of the related abelian integrals around only one center is not more than 12 except the case of global center. It is also proved that there exists a cubic polynomial such that the disturbed vector field has at least 3 limit cycles while the corresponding vector field without perturbations belongs to the saddle loop case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dumortier F, Li C. Perturbations from an elliptic Hamiltonian of degree four: I. Saddle loop and two saddle cycle. J Differential Equations, 176: 114–157 (2001)
Dumortier F, Li C. Perturbations from an elliptic hamiltonian of degree four: II. Cuspidal loop. J Differential Equations, 175: 209–243 (2001)
Dumortier F, Li C. Perturbation from an elliptic Hamiltonian of degree four: III global center. J Differential Equations, 188: 473–511 (2003)
Dumortier F, Li C. Perturbation from an elliptic Hamiltonian of degree four: IV figure eight-loop. J Differential Equations, 188: 512–554 (2003)
Zhao Y, Zhang Z. Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians. J Differential Equations, 155: 73–88 (1999)
Liu C. Estimate of the number of zeros of Abelian integrals for an elliptic Hamiltonian with figure-of-eight loop. Nonlinearity, 16: 1151–1163 (2003)
Chow S N, Li C, Wang D. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge: Cambridge University Press, 1994
Dumortier F, Morsalani M EL, Rousseau C. Hilbert’s 16th problem for quadratic systems and cyclicity of elementary graphics. Nonlinearity, 9: 1209–1261 (1996)
Morsalani M E, Mourtada A. Degenerate and non-trivial hyperbolic 2-polycycles: appearance of two independant Ecalle-Roussarie compensators and Khovanskiis theory. Nonlinearity, 7: 1593–1604 (1994)
Zhu H, Rousseau C. Finite cyclicity of graghics with a nilpotent singularity of saddle or elliptic type. J Differential Equations, 178: 325–436 (2002)
Gavrilov L. Remark on the number of critical points of the period. J Differential Equations, 101: 58–65 (1993)
Roussarie R. On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol Soc Bras Mat, 17: 67–101 (1986)
Joyal P, Rousseau C. Saddle quantities and applications. J Differential Equations, 78: 374–399 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by National Natural Science Foundation of China (Grant No. 10671020)
Rights and permissions
About this article
Cite this article
Zhao, L., Wang, Q. Perturbations from a kind of quartic Hamiltonians under general cubic polynomials. Sci. China Ser. A-Math. 52, 427–442 (2009). https://doi.org/10.1007/s11425-009-0009-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-009-0009-7