Limit cycle bifurcations and analytic center conditions for a class of generalized nilpotent systems

Volume 17, Issue 2, pp 278-287

Publication Date: 2017-06-15


Yusen Wu - School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023 Henan, P. R. China.
Cui Zhang - School of Mathematical Science, Luoyang Normal University, Luoyang, 410022 Henan, P. R. China.
Sumin Yang - School of Humanities and Science, Guangxi Technological College of Machinery and Electricity, Nanning, 530007 Guangxi, P. R. China.


Bifurcation of limit cycles and analytic center conditions for a class of systems in which the origin is a generalized nilpotent singular point are discussed. An interesting phenomenon is that the exponent parameter \(n\) controls the singular point type of the studied system (1.1).


Limit cycle bifurcation, analytic center conditions, generalized nilpotent systems.


[1] V. V. Amelkin, N. A. Lukashevich, A. P. Sadovski˘ı, Neline˘ınye kolebaniya v sistemakh vtorogo poryadka, (Russian) [[Nonlinear oscillations in second-order systems]] Beloruss. Gos. Univ., Minsk, (1982), 208 pages.
[2] A. F. Andreev, Investigation of the behaviour of the integral curves of a system of two differential equations in the neighbourhood of a singular point, Amer. Math. Soc. Transl., 8 (1958), 183–207.
[3] M. Berthier, R. Moussu, R´eversibilit´e et classification des centres nilpotents, (French) [[Reversibility and classification of nilpotent centers]] Ann. Inst. Fourier (Grenoble), 44 (1994), 465–494.
[4] J. ´ Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, (French) [[Introduction to analyzable functions and constructive proof of the Dulac conjecture]] Actualit´es Math´ematiques, [Current Mathematical Topics] Hermann, Paris, (1992).
[5] I. A. Garc´ıa, J. Gin´e, Analytic nilpotent centers with analytic first integral, Nonlinear Anal., 72 (2010), 3732–3738.
[6] H. Giacomini, J. Gin´e, J. Llibre, The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. Differential Equations, 227 (2006), 406-426.
[7] H. Giacomini, J. Gin´e, J. Llibre, Corrigendum to: “The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems”, J. Differential Equations, 232 (2007), 702.
[8] J. Gin´e, Analytic integrability and characterization of centers for generalized nilpotent singular points, Appl. Math. Comput., 148 (2004), 849–868.
[9] M.-A. Han, V. G. Romanovski, Limit cycle bifurcations from a nilpotent focus or center of planar systems, Abstr. Appl. Anal., 2012 (2012), 28 pages.
[10] F. Li, Y.-Y. Liu, Limit cycles in a class of switching system with a degenerate singular point, Chaos Solitons Fractals, 92 (2016), 86–90.
[11] F. Li, P. Yu, Y.-R. Liu, Analytic integrability of two lopsided systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 14 pages.
[12] F. Li, P. Yu, Y. Tian, Y.-R. Liu, Center and isochronous center conditions for switching systems associated with elementary singular points, Commun. Nonlinear Sci. Numer. Simul., 28 (2015), 81–97.
[13] Y.-R. Liu, J.-B. Li, On third-order nilpotent critical points: integral factor method, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 1293–1309.
[14] T. Liu, L.-G. Wu, F. Li, Analytic center of nilpotent critical points, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 8 pages.
[15] R. Moussu, Sym´etrie et forme normale des centres et foyers d´eg´en´er´es, (French) [[Symmetry and normal form in degenerate centers and foci]] Ergodic Theory Dynamical Systems, 2 (1982), 241–251.
[16] A.P. Sadovskii, Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, translated from Differ. Uravn., 12 (1976), 1238–1246.
[17] F. Takens, Singularities of vector fields, Inst. Hautes ´ Etudes Sci. Publ. Math., 43 (1974), 47–100.
[18] M. A. Teixeira, J.-Z. Yang, The center-focus problem and reversibility, J. Differential Equations, 174 (2001), 237–251.


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