Liu, Yirong; Li, Jibin On third-order nilpotent critical points: integral factor method. (English) Zbl 1248.34048 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. Cited in 19 Documents 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 PDF BibTeX XML Cite \textit{Y. Liu} and \textit{J. Li}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 21, No. 5, 1293--1309 (2011; Zbl 1248.34048) Full Text: DOI OpenURL 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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.