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Bifurcation of Limit Cycles and Center in 3D Cubic Systems with Z 3 -Equivariant Symmetry

Author

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  • Ting Huang

    (School of Computer Engineering, Guangzhou City University of Technology, Guangzhou 510800, China
    College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China)

  • Jieping Gu

    (School of Education, Guangxi Vocational Normal University, Nanning 530007, China)

  • Yuting Ouyang

    (College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China)

  • Wentao Huang

    (College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China)

Abstract

This paper focuses on investigating the bifurcation of limit cycles and centers within a specific class of three-dimensional cubic systems possessing Z 3 -equivariant symmetry. By calculating the singular point values of the systems, we obtain a necessary condition for a singular point to be a center. Subsequently, the Darboux integral method is employed to demonstrate that this condition is also sufficient. Additionally, we demonstrate that the system can bifurcate 15 small amplitude limit cycles with a distribution pattern of 5 − 5 − 5 originating from the singular points after proper perturbation. This finding represents a novel contribution to the understanding of the number of limit cycles present in three-dimensional cubic systems with Z 3 -equivariant symmetry.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2563-:d:1163145
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    References listed on IDEAS

    as
    1. 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.
    2. Guo, Laigang & Yu, Pei & Chen, Yufu, 2019. "Bifurcation analysis on a class of three-dimensional quadratic systems with twelve limit cycles," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    3. 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.
    4. 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.
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