A KIND OF BIFURCATION OF LIMIT CYCLES FROM A NILPOTENT CRITICAL POINT

Tao Liu; Yirong Liu; Feng Li

doi:10.11948/2018.10

2018 Volume 8 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(1): 10-18

Next Article Previous Article

Tao Liu, Yirong Liu, Feng Li. A KIND OF BIFURCATION OF LIMIT CYCLES FROM A NILPOTENT CRITICAL POINT[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 10-18. doi: 10.11948/2018.10

Citation:

Tao Liu, Yirong Liu, Feng Li. A KIND OF BIFURCATION OF LIMIT CYCLES FROM A NILPOTENT CRITICAL POINT[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 10-18. doi: 10.11948/2018.10

A KIND OF BIFURCATION OF LIMIT CYCLES FROM A NILPOTENT CRITICAL POINT

1 School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, China;
2 Hunan Academy of Social Sciences, Changsha, Hunan, 410003, P. R. China.;
3 School of Mathematics and Statistics, Linyi University, Linyi, Shandong, 276005, China

Fund Project:

Abstract

Abstract

In this paper, an interesting and new bifurcation phenomenon that limit cycles could be bifurcated from nilpotent node (focus) by changing its stability is investigated. It is different from lowing its multiplicity in order to get limit cycles. We prove that n² + n -1 limit cycles could be bifurcated by this way for 2n + 1 degree systems. Moreover, this upper bound could be reached. At last, we give two examples to show that N(3)=1 and N(5)=5 respectively. Here, N(n) denotes the number of small-amplitude limit cycles around a nilpotent node (focus) with n being the degree of polynomials in the vector field.
- Nilpotent critical point /
- limit cycle /
- bifurcation
MSC: 34C05;37G15

FullText(HTML)

References

[1]	A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcation of Dynamic Systems on a Plane, John Wily and Sons, New York, 1973. Google Scholar
[2]	A. Andreev, Investigation of the behavior of the integrability curves of a system of two differential equations in the neighborhood of a singularity point, Transl. Amer. Math. Soc., 1958, 2, 183-207. Google Scholar
[3]	V. V. Amel'kin, N. A. Lukashevich, A. P. Sadovskii, Nonlinear Oscillations in Second Order Systems, (Russian), Minsk:Belarusian State University, 1982. Google Scholar
[4]	M. J. Álvarez and A. Gasull, Monodromy and stablility for nipotent critical points, Int. Int. J. Bifur. Chaos., 2005, 15(4), 1253-1265. Google Scholar
[5]	M. J. Álvarez and A. Gasull, Cenerating limits cycles from a nipotent critical point via normal forms, J.Math. Anal. Appl., 2006, 318(2), 271-287. Google Scholar
[6]	M. Han and P. Yu, Normal forms, melnikov functions and bifurcations of limit cycles, Springer, 2012. Google Scholar
[7]	M. Han, J. Jiang and H. Zhu, Limit cycle bifurcations in near-Hamiltonian systems by perturbing a nilpotent center, Int. J. Bifur. Chaos., 2008, 18(10), 3013-3027. Google Scholar
[8]	M. Han and V. G. Romanovski, Limit Cycle Bifurcations from a Nilpotent Focus or Center of Planar Systems, Abstract and Applied Analysis, vol. 2012, Article ID 720830, 28 pages. doi:10.1155/2012/720830. Google Scholar
[9]	A. M. Lyapunov, Stability of Motion, Mathematics in Science and Engineering, Academic Press,NY-London, 1966. Google Scholar
[10]	Y. Liu, Multiplicity of higher order singular point of differential autonomous system, J. Cent. South Univ. Techonol., 1999, 30(3), 622-623. Google Scholar
[11]	Y. Liu, J. Li and W. Huang, Singular Point Values, Center Problem and Bifurcations of Limit Cycles of Two Dimensions Differential Autonomous Systems, Science Press, Beijing, 2008. Google Scholar
[12]	Y. Liu and J. Li, New study on the center problem and bifurcations of limit cycles for The Lyapunov system (I), Int. J. Bifur. Chaos. 2009, 19(11), 3791-3801. Google Scholar
[13]	Y. Liu and J. Li, New study on the center problem and bifurcations of limit cycles for The Lyapunov system (Ⅱ), Int. J. Bifur. Chaos. 2009, 19(9), 3087-3099. Google Scholar
[14]	Y. Liu and J. Li, Bifurcations of limit cycles created by a multiple nilponent critical point of planar dynamical systems, Int. J. Bifur. Chaos. 2011, 21(2), 497-504. Google Scholar
[15]	Y. Liu and J. Li, On the study of three-ordernilpotent critical points:Integral factor method, Int. J. Bifur. Chaos. 2011, 21(5), 1293-1309. Google Scholar
[16]	P. Maesschalck and F. Dumortier, SlowCfast BogdanovCTakens bifurcations, J. Diff. Eqs., 2011, 250(2), 1000-1025. Google Scholar
[17]	R. Moussu, Symétrie et forme normale des centres et foyers dégénérés, Ergod. Th. Dynam Sys., 1982, 2, 241-251. Google Scholar
[18]	E. Strozyna and H. Zoladek, The analytic normal for the nilpotent singularty, J.Diff. Eqs., 2002, 179(2), 479-537. Google Scholar
[19]	Y. Tang and W. Zhang, Bogdanov-Takens bifurcation of a polynomialdifferential system in biochemical reaction, Comp. and Math. with App., 2004, 48(5), 869-883. Google Scholar
[20]	F. Takens, Singularities of vector fields, Inst.Hautes Études Sci. Publ.Math., 1974, 43(1), 47-100. Google Scholar
[21]	D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Bio, 2007, 208(2), 419-429. Google Scholar
[22]	D. Xiao, Bifurcations on a five-parameter family of planar vector field, J. Dyn. and Diff. Equa, 2008, 20(4), 961-980. Google Scholar
[23]	Z. Zhang, Qualitative Theory of Ordinary Differential Equations, Science Press, Beijing, 1985. Google Scholar