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Bifurcation of limit cycles near equivariant compound cycles

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Abstract

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z 3 equivariant system can have 13 limit cycles.

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References

  1. Li J. Chaos and Melnikov Function. Chongqing: Chongqing University Press, 1989 (in Chinese)

    Google Scholar 

  2. Li J, Lu Z. Ordinary and Delay Differential Equations. Edinburg: TX, 1991, 116–128

    Google Scholar 

  3. Li J, Liu Z. Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system. Publ Mat, 35: 487–506 (1991)

    MATH  Google Scholar 

  4. Li J, Chan H S Y, Chung K W. Investigations of bifurcations of limit cycles in Z 2-equivariant planar vector fields of degree 5. Intern J Bifur Chaos Appl Sci Engrg, 12: 2137–2157 (2002)

    Article  MATH  Google Scholar 

  5. Li J, Chan H S Y, Chung K W. Bifurcations of limit cycles in a Z 6-equivariant planar vector field of degree 5. Sci China Ser A: Math, 45: 817–826 (2002)

    MATH  Google Scholar 

  6. Li J. Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Internat J Bifur Chaos Appl Sci Engrg, 13: 47–106 (2003)

    Article  MATH  Google Scholar 

  7. Shui S, Zhu D. Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips. Sci China Ser A: Math, 48(2): 248–260 (2005)

    Article  MATH  Google Scholar 

  8. Han M, Chen X. Existence and bifurcation of integral manifolds with applications. Sci China Ser A: Math, 48(7): 940–957 (2005)

    Article  MATH  Google Scholar 

  9. Ye Y. Theory of Limit Cycles (Chinese Translations of Mathematical Monographs Vol. 66). Providence: American Mathematical Society, 1986

    Google Scholar 

  10. Han M, Luo D, et al. Uniqueness of limit cycles bifurcating from a singular closed orbit, III. Acta Math Sinica, Chinese Series, 35: 673–684 (1992)

    MATH  Google Scholar 

  11. Han M, Hu S, et al. On the stability of double homoclinic and heteroclinic cycles. Nonlinear Analysis, 53: 701–713 (2003)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Mao-an Han, Tong-hua Zhang & Hong Zang

Authors
  1. Mao-an Han
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  2. Tong-hua Zhang
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  3. Hong Zang
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Corresponding author

Correspondence to Mao-an Han.

Additional information

The work was supported by the National Natural Science Foundation of China (Grant No. 10671127) and Program for New Century Excellent Talents in University (Grant No. NCET-04-0388)

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Han, Ma., Zhang, Th. & Zang, H. Bifurcation of limit cycles near equivariant compound cycles. SCI CHINA SER A 50, 503–514 (2007). https://doi.org/10.1007/s11425-007-2037-5

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  • DOI: https://doi.org/10.1007/s11425-007-2037-5

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