On third-order nilpotent critical points: integral factor method. (English) Zbl 1248.34048

Liu, Yirong; Li, Jibin

Int. J. Bifurcation Chaos Appl. Sci. Eng. 21, No. 5, 1293-1309 (2011).

Summary: For third-order nilpotent critical points of a planar dynamical system, the problem of characterizing its center and focus is completely solved in this article by using the integral factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic systems under small perturbations, it is proved that there exist eight small-amplitude limit cycles created from a nilpotent critical point.

MSC:

34C23	Bifurcation theory for ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
34C07	Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34A05	Explicit solutions, first integrals of ordinary differential equations
34C45	Invariant manifolds for ordinary differential equations

Keywords:

nilpotent critical point; center-focus problem; bifurcation of limit cycle; integral factor; quasi-Lyapunov constant

References:

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