一类拟齐次多项式中心的极限环分支

Abstract

Abstract: Determining the condition such that a planar quasi-homogeneous polynomial differential system has a center is a difficult topic. In this paper, we first extend the quintic quasi-homogeneous system, provided by[12], to the quasi-homogeneous polynomial system of degree n (odd number), and then give the necessary and sufficient condition to ensure that it possess a global center. Using the first order Melnikov function, we obtain the least upper bound for the number of limit cycles bifurcating from the period annulus of the center of the system, under the perturbation of polynomial of degree n. Finally, we prove that this conclusion is also true for the limit cycles bifurcating from all the (m,1)-(or (1,m)-) planar quasi-homogeneous polynomial differential Hamiltonian system, under polynomial perturbation of degree 2m-1, where m is any positive integer.

Key words: Quasi-homogeneous polynomial, Center, Limit cycle, Melnikov function

CLC Number:

O175.12

Liang Haihua, Chen Yuming, Cen Xiuli. Limit Cycles Bifurcating from a Class of Quasi-Homogeneous Polynomial Center[J].Acta mathematica scientia,Series A, 2018, 38(1): 1-9.

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References

[1] Roussarie R. Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem. Basel:Birkhuser Verlag, 1998
[2] Yakovenko S. On functions and curves defined by ordinary differential equations. Fields Institute Communications, 1997, 24:497-525
[3] Khovansky A G. Real analytic manifolds with finiteness properties and complex Abelian integrals. Funct Anal Appl, 1984, 18:119-128
[4] Varchenko A N. Estimate of the number of zeros of an Abelian integral depending on a parameter and limit cycles. Funct Anal Appl, 1984, 18(2):98-108
[5] Binyamini G, Novikov D, Yakovenko S. On the number of zeros of Abelian integrals:a constructive solution of the infinitesimal Hilbert sixteenth problem. Invent Math, 2010, 181(2):227-289
[6] Li W, Llibre J, Yang J, Zhang Z. Limit cycles bifurcating from the period annulus of quasi-homogeneous centers. J Dyn Diff Equat, 2009, 21(1):133-152
[7] Zhao Y, Zhang Z. Abelian integrals and period function for quasi-homogeneous Hamiltonian vector fields. Appl Math Lett, 2005, 18(5):563-569
[8] Gavrilov L, Giné J, Grau M. On the cyclicity of weight-homogeneous centers. J Differential Equations, 2009, 246(8):3126-3135
[9] Geng F, Lian H. Bifurcation of limit cycles from a quasi-homogeneous degenerate center. Int J Bifurcation Chaos, 2015, 25(1):1-8
[10] Giné J, Grau M, Llibre J. Limit cycles bifurcating from planar polynomial quasi-homogeneous centers. J Differential Equations, 2015, 259(12):7135-7160
[11] Li C, Li W, Llibre J, Zhang Z. Polynomial systems:a lower bound for the weakened 16th Hilbert problem. Extracta Math, 2001, 16(3):441-447
[12] Tang Y, Zhang X. Center of planar quinic quasihomogeneous polynomial differential systems. Disc Contin Dyn Syst, 2015, 35(5):2177-2191
[13] Dumortier F, Llibre J, Artes J C. Qualitative Theory of Planar Differential Systems. Berlin:Springer-Verlag, 2006
[14] Xiong Y, Han M. Planar quasi-homogeneous polynomial systems with a given weight degree. Disc Contin Dyn Syst, 2016, 36(7):4015-4025

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Limit Cycles Bifurcating from a Class of Quasi-Homogeneous Polynomial Center

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