LettersNo Access

LIMIT CYCLES FOR A CLASS OF QUINTIC NEAR-HAMILTONIAN SYSTEMS NEAR A NILPOTENT CENTER

JIAO JIANG

Department of Mathematics, Shanghai Maritime University, Shanghai 200135, P. R. China

Author for correspondence.

Search for more papers by this author

JIZHOU ZHANG

Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China

Search for more papers by this author

, and

MAOAN HAN

Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0218127409023949Cited by:9 (Source: Crossref)

Abstract

In this paper we deal with a centrally symmetric quintic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have ten limit cycles by using a homoclinic bifurcation method based on stability-change.

Supported by Program for New Century Excellent Talents in University (NCET-04-0388) and Shanghai Shuguang Genzong Project (04SGG05).

Keywords:

References

M. Alvarez and A. Gasull, Int. J. Bifurcation and Chaos 15, 1253 (2005). Link, Web of Science, Google Scholar
M. Alvarez and A. Gasull, J. Math. Anal. Appl. 318, 271 (2006), DOI: 10.1016/j.jmaa.2005.05.064. Crossref, Web of Science, Google Scholar
A. Andreev, Am. Math. Soc. 8, 183 (1958). Google Scholar
A. A. Andronov et al. , Qualitative Theory of Second Order Dynamic Systems ( Wiley , NY , 1973 ) . Google Scholar
V. I. Arnold, Funct. Anal. Appl. 11, 1 (1977). Web of Science, Google Scholar
J. Chavarrigaet al., Ergod. Th. Dyn. Syst. 23, 417 (2003). Web of Science, Google Scholar
F. Dumortier, R. Roussarie and C. Rousseau, J. Diff. Eqns. 110, 86 (1994), DOI: 10.1006/jdeq.1994.1061. Crossref, Web of Science, Google Scholar
F. Dumortier, R. Roussarie and C. Rousseau, Nonlinearity 7, 1001 (1994), DOI: 10.1088/0951-7715/7/3/013. Crossref, Web of Science, Google Scholar
F. Dumortier, R. Roussarie and S. Sotomayor, Nonlinearity 10, 1369 (1997), DOI: 10.1088/0951-7715/10/6/001. Crossref, Web of Science, Google Scholar
A. Gasull and A. Guillamon, J. Math. Anal. Appl. 211, 190 (1997), DOI: 10.1006/jmaa.1997.5455. Crossref, Web of Science, Google Scholar
M. Han, Sci. China Ser. A 40, 1247 (1997), DOI: 10.1007/BF02876370. Crossref, Web of Science, Google Scholar
M. Han and J. Chen, Sci. China Ser. A 43, 914 (2000), DOI: 10.1007/BF02879797. Crossref, Web of Science, Google Scholar
M. Han , Periodical Solutions and Bifurcation Theory of Dynamic Systems ( Science Publishing House , Beijing , 2002 ) . Google Scholar
M. Han, S. Hu and X. Liu, Nonlin. Anal. 53, 701 (2003), DOI: 10.1016/S0362-546X(02)00301-2. Crossref, Web of Science, Google Scholar
M. Han and C. Fan, Chin. Ann. Math. A 26, 825 (2005). Google Scholar
M. Han , Bifurcation Theory of Limit Cycles of Planar Systems , Handbook of Differential Equations, Ordinary Differential Equations 3 , eds. A. Canada , P. Drabek and A. Fonda ( Elsevier , 2006 ) . Google Scholar
M. Han and T. Zhang, Math. Biosci. Engin. 3, 67 (2006). Crossref, Web of Science, Google Scholar
Yu. S. Ilyashenko, Russ. Math. Surv. 40, 143 (1990). Web of Science, Google Scholar
J. Llibre and X. Zhang, Dyn. Contin. Discr. Impuls. Syst. Series A: Math. Anal. 8, 161 (2001). Web of Science, Google Scholar
V. Manosa, Int. J. Bifurcation and Chaos 12, 687 (2002). Link, Web of Science, Google Scholar
M. Morsalani and A. Mourtada, Nonlinearity 7, 1593 (1994), DOI: 10.1088/0951-7715/7/6/004. Crossref, Web of Science, Google Scholar
R. Moussu, Erg. Th. Dyn. Syst. 2, 241 (1982). Crossref, Google Scholar
R. Roussarie, Bol. Soc. Bras. Mat. 17, 67 (1986), DOI: 10.1007/BF02584827. Crossref, Google Scholar
R. Roussarie, Nonlinearity 2, 73 (1989), DOI: 10.1088/0951-7715/2/1/006. Crossref, Web of Science, Google Scholar
H. Zhu and C. Rousseau, J. Diff. Eqns. 178, 325 (2002), DOI: 10.1006/jdeq.2001.4017. Crossref, Web of Science, Google Scholar