THE CYCLICITY OF THE PERIOD ANNULUS OF TWO CLASSES OF CUBIC ISOCHRONOUS SYSTEMS

Yi Shao; Kuilin Wu

doi:10.11948/2013020

2013 Volume 3 Issue 3

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Yi Shao, Kuilin Wu. THE CYCLICITY OF THE PERIOD ANNULUS OF TWO CLASSES OF CUBIC ISOCHRONOUS SYSTEMS[J]. Journal of Applied Analysis & Computation, 2013, 3(3): 279-290. doi: 10.11948/2013020

Citation:

Yi Shao, Kuilin Wu. THE CYCLICITY OF THE PERIOD ANNULUS OF TWO CLASSES OF CUBIC ISOCHRONOUS SYSTEMS[J]. Journal of Applied Analysis & Computation, 2013, 3(3): 279-290. doi: 10.11948/2013020

THE CYCLICITY OF THE PERIOD ANNULUS OF TWO CLASSES OF CUBIC ISOCHRONOUS SYSTEMS

Yi Shao¹,
Kuilin Wu²

1 Department of Mathematics, Zhaoqing University, Guangdong, 526061, China;
2 Department of Mathematics, Guizhou University, Guizhou, 550025, China

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Abstract

Abstract

In this paper, we investigate the cyclicity of the period annulus of two classes of cubic isochronous systems. By using the Chebyshev criterion, we prove that the two systems have respectively at most three and four limit cycles produced from the period annulus around the isochronous center under cubic perturbations.
- Isochronous center /
- limit cycle /
- cubic perturbations
MSC: 34C05;34C07;34C14

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References

[1]	V. Arnold, Some unsolved problems in the theory of diferential equations and mathematical physics, Russian Math. Surveys, 44(1989), 157-171. Google Scholar
[2]	S. Gautier, L. Gavrilov and I. Iliev, Perturbations of quadratic centers of genus one, Discrete Contin. Dyn. Syst., 25(2009), 511-535. Google Scholar
[3]	A. Gasull, W. Li, J. Llibre and Z. Zhang, Chebyshev property of complete elliptic integrals and its application to Abelian integrals, Pac. J. Math., 202(2002), 341-361. Google Scholar
[4]	M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363(2011), 109-129. Google Scholar
[5]	I. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122(1998), 107-161. Google Scholar
[6]	I. Iliev, C. Li and J. Yu, Bifurcations of limit cycles in a reversible quadratic system with a centre, a saddle and two nodes, Comm. Pure. Anal. Appl., 9(2010), 583-610. Google Scholar
[7]	S. Karlin and W. Studden, Tchebycheff systems:with applications in analysis and statistics, Pure and Applied Mathematics, Interscience Publishers, New York, 1966. Google Scholar
[8]	C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 1(2012), 111-128. Google Scholar
[9]	C. Li and J. Llibre, The cyclicity of period annulus of a quadratic reversible Lotka-Volterra system, Nonlinearity, 22(2009), 2971-2979. Google Scholar
[10]	C. Li, W. Li, J. Llibre and Z. Zhang, Linear estimation of the number of zeros of Abelian integrals for some cubic isochronous centers, J. Differential Equations, 180(2002), 307-333. Google Scholar
[11]	, C. Li, W. Li, Llibre, Jaume and Z. Zhang, New families of centers and limit cycles for polynomial differential systems with homogeneous nonlinearities, Ann. Differential Equations, 19(2003), 302-317. Google Scholar
[12]	C. Li and Z. Zhang, Remarks on 16th weak Hilbert problem for n=2, Nonlinearity, 15(2002), 1975-1992. Google Scholar
[13]	P. Mardešić, Chbyshev systems and the versal unfolding of the cusp of order n, Travaux en cours, vol. 57, Hermann, Paris, 1998. Google Scholar
[14]	F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differential Equations, 6(2011), 1656-1669. Google Scholar
[15]	L. Pontryagin, On dynamic systems closed to Hamiltonian systems, Zh. Eksper. Teoret. Fiz., 4(1934), 234-238(in Russian). Google Scholar
[16]	I. Pleshkan, A new method of investigating the isochronicity of a system of two differential equations, J. Differential Equations, 5(1969), 796-802. Google Scholar
[17]	Y. Shao and Y. Zhao, The cyclicity and period function of a class of quadratic reversible Lotka-Volterra system of genus one, J. Math. Anal. Appl., 377(2011), 817-827. Google Scholar
[18]	Y. Shao and Y. Zhao, The cyclicity of the period annulus of a class of quadratic reversible system, Commun. Pure Appl. Anal., 3(2012), 1269-1283. Google Scholar
[19]	K. Wu and Y. Zhao, The cyclicity of the period annulus of the cubic isochronous center, Inter. J. Bifur. Chaos, 22(2012), 1250016(9 pages). Google Scholar
[20]	H. Żoladek, Quadratic systems with centers and their perturbations, J. Differential Equations, 109(1994), 223-273. Google Scholar
[21]	Y. Zhao, On the number of limit cycles in quadratic perturbations of quadratic codimension-four centres, Nonlinearity, 24(2011), 2505-2522. Google Scholar