IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v31y2007i3p617-630.html
   My bibliography  Save this article

On limit cycles of the Liénard equation with Z2 symmetry

Author

Listed:
  • Yu, P.
  • Han, M.

Abstract

This paper considers the limit cycles in the Liénard equation, described by x¨+f(x)x˙+g(x)=0, with Z2 symmetry (i.e., the vector filed is symmetric with the y-axis). Particular attention is given to the existence of small-amplitude (local) limit cycles around fine focus points when g(x) is a third-degree, odd polynomial function and f(x) is an even function. Such a system has three fixed points on the x-axis, with one saddle point at the origin and two linear centres which are symmetric with the origin. Based on normal form computation, it is shown that such a system can generate more limit cycles than the existing results for which only the origin is considered. In general, such a Liénard equation can have 2m small limit cycles, i.e., H(2m,3)⩾2m, where H denotes the Hilbert number of the system, 2m and 3 are the degrees of f and g, respectively.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:3:p:617-630
    DOI: 10.1016/j.chaos.2005.10.013
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077905009665
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2005.10.013?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

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

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Sabatini, M., 2010. "Existence and uniqueness of limit cycles in a class of second order ODE’s with inseparable mixed terms," Chaos, Solitons & Fractals, Elsevier, vol. 43(1), pages 25-30.
    2. Zhang, Chunrui & Zheng, Baodong, 2009. "Bifurcation in Z2-symmetry quadratic polynomial systems with delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3078-3086.

    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, 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Yu, P. & Han, M., 2006. "Limit cycles in generalized Liénard systems," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1048-1068.
    9. 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.
    10. 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.
    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. 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.
    13. 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.
    14. 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.

    More about this item

    Statistics

    Access and download statistics

    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:eee:chsofr:v:31:y:2007:i:3:p:617-630. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.