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Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11

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  • Wang, S.
  • Yu, P.

Abstract

In this article, a systematic procedure has been explored to studying general Zq-equivariant planar polynomial Hamiltonian vector fields for the maximal number of closed orbits and the maximal number of limit cycles after perturbation. Following the procedure by taking special consideration of Z12-equivariant vector fields of degree 11, the maximal of 99 closed orbits are obtained under a well-defined coefficient group. Consequently, perturbation parameter control in limit cycle computation leads to the existence of 121 limit cycles in the perturbed Hamiltonian vector field, which gives rise to the lower bound of Hilbert number of 11th-order systems as H(11)⩾112. Two conjectures are proposed regarding the maximal number of closed orbits for equivariant polynomial Hamiltonian vector fields and the maximal number of limit cycles bifurcated from the well defined Hamiltonian vector fields after perturbation.

Suggested Citation

  • Wang, S. & Yu, P., 2006. "Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 606-621.
    • Handle: RePEc:eee:chsofr:v:30:y:2006:i:3:p:606-621
    DOI: 10.1016/j.chaos.2005.12.016
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    References listed on IDEAS

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    1. Wang, S. & Yu, P., 2005. "Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1317-1335.
    2. Yu, P. & Han, M., 2005. "Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields," Chaos, Solitons & Fractals, Elsevier, vol. 24(1), pages 329-348.
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    Cited by:

    1. Ting Huang & Jieping Gu & Yuting Ouyang & Wentao Huang, 2023. "Bifurcation of Limit Cycles and Center in 3D Cubic Systems with Z 3 -Equivariant Symmetry," Mathematics, MDPI, vol. 11(11), pages 1-22, June.
    2. Han, Maoan & Xiong, Yanqin, 2014. "Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters," Chaos, Solitons & Fractals, Elsevier, vol. 68(C), pages 20-29.
    3. Li, J. & Tian, Y. & Zhang, W., 2009. "Investigation of relation between singular points and number of limit cycles for a rotor–AMBs system," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1627-1640.

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