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On the study of limit cycles of a cubic polynomials system under Z4-equivariant quintic perturbation

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  • Wu, Yuhai
  • Han, Maoan
  • Liu, Xuanliang

Abstract

This paper is concerned with the number and distribution of limit cycles of a perturbed cubic Hamiltonian system which has 5 centers and 4 saddle points. The singular point and singular close orbits’ stability theory and perturbation skills of differential equations are applied to study the Hopf, homoclinic loop and heteroclinic loop bifurcation of such system under Z4-equivariant quintic perturbation. It is found that the perturbed system has at least 16 limit cycles bifurcated from the focus. Further, at least 14 limit cycles with three different distributions appear in the heteroclinic loops bifurcation.

Suggested Citation

  • Wu, Yuhai & Han, Maoan & Liu, Xuanliang, 2005. "On the study of limit cycles of a cubic polynomials system under Z4-equivariant quintic perturbation," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 999-1012.
    • Handle: RePEc:eee:chsofr:v:24:y:2005:i:4:p:999-1012
    DOI: 10.1016/j.chaos.2004.09.079
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    Cited by:

    1. Singh, Vimal, 2007. "A new frequency-domain criterion for elimination of limit cycles in fixed-point state-space digital filters using saturation arithmetic," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 813-816.
    2. Singh, Vimal, 2008. "Suppression of limit cycles in second-order companion form digital filters with saturation arithmetic," Chaos, Solitons & Fractals, Elsevier, vol. 36(3), pages 677-681.
    3. Wu, Yuhai & Tian, Lixin & Hu, Yingjing, 2007. "On the limit cycles of a Hamiltonian under Z4-equivariant quintic perturbation," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 298-307.
    4. Han, Maoan & Zang, Hong & Zhang, Tonghua, 2007. "A new proof to Bautin’s theorem," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 218-223.
    5. Giné, Jaume, 2007. "On some open problems in planar differential systems and Hilbert’s 16th problem," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1118-1134.
    6. Singh, Vimal, 2008. "Novel frequency-domain criterion for elimination of limit cycles in a class of digital filters with single saturation nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 178-183.

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