A STUDY ON ZOLADEK'S EXAMPLE

Pei Yu; Maoan Han

doi:10.11948/2011009

2011 Volume 1 Issue 1

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Pei Yu, Maoan Han. A STUDY ON ZOLADEK'S EXAMPLE[J]. Journal of Applied Analysis & Computation, 2011, 1(1): 143-153. doi: 10.11948/2011009

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Pei Yu, Maoan Han. A STUDY ON ZOLADEK'S EXAMPLE[J]. Journal of Applied Analysis & Computation, 2011, 1(1): 143-153. doi: 10.11948/2011009

A STUDY ON ZOLADEK'S EXAMPLE

Pei Yu¹,
Maoan Han²

1 Department of Applied Mathematics, The University of Western Ontario London, Ontario, Canada N6A 5B7;
2 Department of Mathematics, Shanghai Normal University, Shanghai, 200234 China

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Abstract

Abstract

In this paper, we consider an example of third-order polynomial planar system, proposed by Zoladek who claimed that this example had eleven small-amplitude limit cycles around a center. We use focus value computation to show that for this example there may exist maximal nine small-amplitude limit cycles around the center due to Hopf bifurcation.
- Hilbert's 16th problem /
- integrable system /
- limit cycle /
- focus value /
- Hopf bifurcation
MSC: 34C07;34C23

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References

[1]	D. Hilbert, Mathematical problems(M. Newton, Transl.), Bull. Amer. Math., 8(1902), 437-479. Google Scholar
[2]	N. N. Bautin, On the number of limit cycles which appear with the variation of coe-cients from an equilibrium position of focus or center type, Mat. Sbornik (N.S.), 30(1952), 181-196. Google Scholar
[3]	J. Li and J. Bai, The cyclicity of multiple Hopf bifurcation in planar difierential cubic systems:M(3) ≥ 7, Preprint, Hunming Institute Technology, 1989. Google Scholar
[4]	C. J. Christopher and N. G. Lloyd, On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems, Bull. London Math. Soc., 22(1990), 5-12. Google Scholar
[5]	N. G. Lloyd, T. R. Blows, M. C. Kalenge, Some cubic systems with several limit cycles, Nonlinearity, 1(1988), 653-669. Google Scholar
[6]	A. P. Sadovskii, Cubic systems of nonlinear oscillations with seven limit cycles, Difierential Equations, 39(2003), 505-516. (Translated from Difierentsial'nye Uravneniy, 39(4)(2003),472-481. Google Scholar
[7]	E. M. James, N. G. Lloyd, A cubic system with eight small-amplitude limit cycles, IMA J. Appl. Math., 47(1991), 163-171. Google Scholar
[8]	P. Yu and R. Corless, Symbolic computation of limit cycles associated with Hilbert's 16th problem, Communications in Nonlinear Science and Numerical Simulation, 14(2009), 4041-4056. Google Scholar
[9]	M. Han, Y. Lin and P. Yu, A study on the existence of limit cycles of a planar system with 3rd-degree polynomials, Int. J. Bifurcation and Chaos, 14(2004), 41-60. Google Scholar
[10]	J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector flelds, Int. J. Bifurcations and Chaos, 13(2003), 47-106. Google Scholar
[11]	T. Zhang, H. Zang and M. Han, Bifurcation of limit cycles in a cubic system, Chaos, Solitons and Fractals, 20(2004), 629-638. Google Scholar
[12]	P. Yu and M. Han, Twelve limit cycles in a 3rd-order planar system with Z₂ symmetry, Communication on Pure and Applied Analysis, 3(2004), 515-526. Google Scholar
[13]	P. Yu and M. Han, Twelve limit cycles in a cubic case of the 16th Hilbert problem, Int. J. Bifurcation and Chaos, 15(2005), 2191-2205. Google Scholar
[14]	P. Yu and M. Han, Small limit cycles bifurcating from flne focus points in cubic order Z₂-equivariant vector flelds, Chaos, Solitons and Fractals, 24(2005), 329-348. Google Scholar
[15]	C. Li, L. Liu and J. Yang, A cubic system with thirteen limit cycles, J. Difi. Eqns 246(2009), 3609-3619. Google Scholar
[16]	Y. Liu and J. Li, New results on the study of Zq-equivariant planar polynomial vector flelds, Qual. Theory Dyn. Syst., 9(2010), 167-219. Google Scholar
[17]	J. Yang, M. Han, J. Li and P. Yu, Existence conditions of thirteen limit cycles in a cubic system, Int. J. Bifurcation and Chaos, 20(2010), 2569-2577. Google Scholar
[18]	H. Żolądek, _Quadratic systems with center and their perturbations, J. Difi. Eqns., 109(1994), 223-273. Google Scholar
[19]	P. Yu, Computation of normal forms via a perturbation technique, J. Sound and Vib., 211(1998), 19-38. Google Scholar