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Bifurcations of the limit cycles in a z3-equivariant quartic planar vector field

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  • Wu, Yuhai
  • Gao, Yongxi
  • Han, Maoan

Abstract

In this paper, a Z3-equivariant quartic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the polycycle with three hyperbolic saddle points. It is found that this special quartic planar polynomial system has at least three large limit cycles which surround 13 singular points. By applying the double homoclinic loops bifurcation method and Poincaré–Bendixson theorem, we conclude that 16 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful to the study of the second part of 16th Hilbert Problem.

Suggested Citation

  • Wu, Yuhai & Gao, Yongxi & Han, Maoan, 2008. "Bifurcations of the limit cycles in a z3-equivariant quartic planar vector field," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1177-1186.
    • Handle: RePEc:eee:chsofr:v:38:y:2008:i:4:p:1177-1186
    DOI: 10.1016/j.chaos.2007.02.019
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    Cited by:

    1. Liang, Feng & Han, Maoan, 2012. "Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 454-464.
    2. Yanli Tang & Dongmei Zhang & Feng Li, 2022. "Limit Cycles and Integrability of a Class of Quintic System," Mathematics, MDPI, vol. 10(16), pages 1-11, August.

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