BIFURCATION OF LIMIT CYCLES FROM THE GLOBAL CENTER OF A CLASS OF INTEGRABLE NON-HAMILTON SYSTEMS

Shiyou Sui; Baoyi Li

doi:10.11948/2018.1441

2018 Volume 8 Issue 5

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Shiyou Sui, Baoyi Li. BIFURCATION OF LIMIT CYCLES FROM THE GLOBAL CENTER OF A CLASS OF INTEGRABLE NON-HAMILTON SYSTEMS[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1441-1451. doi: 10.11948/2018.1441

Citation:

Shiyou Sui, Baoyi Li. BIFURCATION OF LIMIT CYCLES FROM THE GLOBAL CENTER OF A CLASS OF INTEGRABLE NON-HAMILTON SYSTEMS[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1441-1451. doi: 10.11948/2018.1441

BIFURCATION OF LIMIT CYCLES FROM THE GLOBAL CENTER OF A CLASS OF INTEGRABLE NON-HAMILTON SYSTEMS

Shiyou Sui¹,
Baoyi Li²

1 School of Mathematical Sciences, Beijing Normal University, Xinjikouwai Str. 19, Beijing 100875, China;
2 School of Mathematical Sciences, Tianjin Normal University, Extention of Binshui West Road, Tianjin 300387, China

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Abstract

Abstract

In this paper, we consider the bifurcation of limit cycles for system ẋ=-y(x² + a²)^m,ẏ=x(x² + a²)^m under perturbations of polynomials with degree n, where a ≠0, m ∈ N. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from periodic orbits of the center of the unperturbed system. Particularly, if m=2,n=5, the sharp bound is 5.
- Limit cycle /
- averaging function /
- bifurcation
MSC: 37G15;34C05

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References

[1]	A. Atabaigi, N. Nyamoradi and H. R. Z. Zangeneh, The number of limit cycles of a quintic polynomial system, Comput. Math. Appl., 2009, 57, 677-684. Google Scholar
[2]	A. Buiča and J. Llibre, Limit cycles of a perturbed cubic polynomial differential center, Chaos Solitons Fractals, 2007, 32, 1059-1069. Google Scholar
[3]	T. R. Blows and L. M. Perko, Bifurcation of limit cycles from centers and separatrix cycles of analytic systems, SIAM Rev., 1994, 36, 341-376. Google Scholar
[4]	G. Chang and M. Han, Bifurcation of limit cycles by perturbing a periodic annulus with multiple critical points, Int. J. Bifurcation and Chaos, 2013, 23, 1350143(14 pages). Google Scholar
[5]	S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, 1994. Google Scholar
[6]	B. Coll, A. Gasull and R. Prohens, Bifurcation of limit cycles from two families of centers, Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal., 2005, 12, 275-287. Google Scholar
[7]	B. Coll, J. Llibre and R. Prohens, Limit cycles bifurcating from a perturbed quartic center, Chaos Solitons Fractals, 2011, 44, 317-334. Google Scholar
[8]	A. Gasull, J. T. Lázaro and J. Torregrosa, Upper bounds for the number of zeroes for some Abelian integrals, Nonlinear Anal., 2012, 75, 5169-5179. Google Scholar
[9]	A. Gasull, C. Li and J. Torregrosa, Limit cycles appearing from the perturbation of a system with a multiple line of critical points, Nonlinear Anal., 2012, 75, 278-285. Google Scholar
[10]	J. Giné,and J. Llibre, Limit cycles of cubic polynomial vector feilds via the averaging theory, Nonlinear Anal., 2007, 66, 1707-1721. Google Scholar
[11]	M. Han, On the maximum number of periodic solutions of piecewise smooth periodic equations by average method, Journal of Applied Analysis and Computation, 2017, 7(2), 788-794. Google Scholar
[12]	S. Li, Y. Zhao and J. Li, On the number of limit cycles of a perurbed cubic polynomial differential center, J. Math. Anal. Appl., 2013, 404, 212-220. Google Scholar
[13]	J. Llibre and J. S. Pérez del Río and J. A. Rodríguez, Averaging analysis of a perturbed quadratic center, Nonlinear Anal., 2001, 46, 45-51. Google Scholar
[14]	J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamic System, in:Applied Mathematical Science, Springer-Verlag, New York, 1985. Google Scholar
[15]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, in:Universitext, Springer-Verlag, Berlin, 1996. Google Scholar
[16]	G. Xiang and M. Han, Global bifurcation of limit cycles in a family of polynomial, J. Math. Anal. Appl., 2004, 295, 633-644. Google Scholar
[17]	Y. Xiong, The number of limit cycles in perturbations of polynomial systems with multiple circles of critical points, J. Math. Anal. Appl., 2016, 440, 220-239. Google Scholar