Bifurcations of limit cycles and center problem for a class of cubic nilpotent system. (English) Zbl 1202.34064

Summary: For a class of cubic nilpotent system, the formulae of the first eight quasi-Lyapunov constants are obtained. We show that the origin of this system is a center if and only if the first eight Lyapunov constants are zeros. Under a small perturbation, eight limit cycles can be created from the eight-order weakened focus.


34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] Amelikin B. B., Nonlinear Oscillations in Second Order Systems (1982)
[2] DOI: 10.1142/S0218127405012740 · Zbl 1088.34021
[3] DOI: 10.1016/j.jmaa.2005.05.064 · Zbl 1100.34030
[4] Lyapunov A. M., Stability of Motion, Mathematics in Science and Engineering 30 (1966) · Zbl 0161.06303
[5] Liu Y., J. Cent. South Univ. Technol. 30 pp 622–
[6] DOI: 10.1142/S0218127409025110 · Zbl 1182.34044
[7] DOI: 10.1142/S0218127409024669 · Zbl 1179.34030
[8] Liu Y. R., J. Diff. Eqs.
[9] Moussu R., Erg. Th. Dyn. Syst. 2 pp 241–
[10] DOI: 10.1006/jdeq.2001.4043 · Zbl 1005.34034
[11] DOI: 10.1007/BF02684366 · Zbl 0279.58009
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