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

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
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References:

[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] DOI: 10.1142/S0218127409025110 · Zbl 1182.34044
[5] DOI: 10.1142/S0218127409024669 · Zbl 1179.34030
[6] Y. X. Qin, On Integral Surfaces Defined by Ordinary Differential Equations (Northwest University Press, 1985) pp. 52–57.
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