Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center


In this paper we deal with a cubic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have 5 limit cycles by using bifurcation theory.

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J. Jiang, "Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center," Applied Mathematics, Vol. 3 No. 7, 2012, pp. 772-777. doi: 10.4236/am.2012.37115.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. Andronov, E. Leontovich, I. Gordon and A. Maier, “Theory of Bifurcations of Dynamical Systems on a Plane,” Israel Program for Scientific Translations, Jerusalem, 1971.
[2] M. Han, “On Hopf Cyclicity of Planar Systems,” Journal of Mathematical Analysis and Applications, Vol. 245, No. 2, 2000, pp. 404-422. doi:10.1006/jmaa.2000.6758
[3] Y. A. Kuznetsov, “Elements of Applied Bifurcation Theory,” Springer-Verlag, New York, 1995.
[4] T. Carmon, R. Uzdin, C. Pigier, Z. Musslimani, M. Segev and A. Nepomnyashchy, “Rotating Propeller Solitons,” Physical Review Letters, Vol. 87, No. 14, 2001, p. 143901. doi:10.1103/PhysRevLett.87.143901
[5] J. Guckenheimer and P. Holmes, “Non-Linear Oscillations, Dynamical Systems and Bifurcation of Vector Fields,” Springer-Verlag, New York, 1983.
[6] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, “Theory and Applications of Hopf Bifurcation,” Cambridge University Press, Cambridge, 1981.
[7] S. Wiggins, “Global Bifurcations and Chaos: Analytical Methods,” Springer-Verlag, New York, 1988.
[8] M. Han J. Jiang and H. Zhu, “Limit Cycle Bifurcations in Near-Hamiltonian Systems by Perturbing a Nilpotent Center,” International Journal of Bifurcation and Chaos, Vol. 18, No. 10, 2008, pp. 3013-3027. doi:10.1142/S0218127408022226

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