Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-30T12:41:50.084Z Has data issue: false hasContentIssue false
  • English
  • Français

The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations

Part of: Qualitative theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  09 April 2009

Guowei Chen ,
Yongbin Wu  and
Xinan Yang
Xinan Yang
Affiliation:
Department of Mathematics, Fuzhou University, Fuzhou 350002, China e-mail: gnchen@fzu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

Keywords

quintic Hamiltonian systemlimit cyclebifurcation

MSC classification

Secondary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 37C10: Vector fields, flows, ordinary differential equations

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 1 , August 2002 , pp. 37 - 54
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Chen, L. and Wang, M., ‘The number and relation position of limit cycles for quadratic system’, Acta Math. Sinica 22 (1979), 751–758 (Chinese).Google Scholar
[2]Chow, S. N., Li, C. and Wang, D., Normal forms and bifurcation of planar vector fields (Cambridge University Press, 1994).CrossRefGoogle Scholar
[3]Dulac, H., ‘Sur les cycles limits’, Bull. Soc. Math. France 51 (1923), 45–188.CrossRefGoogle Scholar
[4]Hilbert, D., ‘Mathematische Problems’, Arch. Math. Phys. 3 (1901), 44–63, 213–237.Google Scholar
[5]Li, J. and Huang, Q., ‘Bifurcations of limit cycles forming compound eyes in the cubic system’, Chinese Ann. Math. 813 (1987), 391–403.Google Scholar
[6]Shi, S., ‘An example of a quadratic system having at least four limit cycles’, Scientia Sinica 11 (1979), 1051–1056 (Chinese).Google Scholar
[7]Ye, Y., Theory of limit cycles (Shanghai Scientific and Technology Press, 1984).Google Scholar
[8]Zhang, Z., Qualitative theory of differential equations (Science Press, Beijing, China, 1985).Google Scholar
[9]Zoladek, H., ‘Eleven small limit cycles in a cubic field’, Nonlinearity 8 (1995), 843–860.CrossRefGoogle Scholar