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

Liu, Yirong; Li, Jibin

Int. J. Bifurcation Chaos Appl. Sci. Eng. 20, No. 8, 2579-2584 (2010).

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.

MSC:

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

Keywords:

cubic nilpotent system; Lyapunov constant

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