IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i11p2563-d1163145.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/11/2563/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/11/2563/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    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.
    3. 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.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Singh, Vimal, 2007. "Modified LMI condition for the realization of limit cycle-free digital filters using saturation arithmetic," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1448-1453.
    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. 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.
    4. 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.
    5. 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.
    6. Yu, P. & Han, M., 2007. "On limit cycles of the Liénard equation with Z2 symmetry," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 617-630.
    7. 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.
    8. 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.
    9. Cui, Yan & Liu, Suhua & Tang, Jiashi & Meng, Yimin, 2009. "Amplitude control of limit cycles in Langford system," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 335-340.
    10. Yu, P. & Han, M., 2006. "Limit cycles in generalized Liénard systems," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1048-1068.
    11. 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.
    12. Li, Yongjun & Romanovski, Valery G., 2021. "Isochronous solutions of a 3-dim symmetric quadratic system," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    13. 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.
    14. 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.
    15. Yang, Junmin & Yu, Pei, 2017. "Nine limit cycles around a singular point by perturbing a cubic Hamiltonian system with a nilpotent center," Applied Mathematics and Computation, Elsevier, vol. 298(C), pages 141-152.
    16. Ye, Zhiyong & Han, Maoan, 2006. "Singular limit cycle bifurcations to closed orbits and invariant tori," Chaos, Solitons & Fractals, Elsevier, vol. 27(3), pages 758-767.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2563-:d:1163145. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.