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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: [email protected]
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Abstract

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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
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