IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v341y2019icp256-276.html
   My bibliography  Save this article

Double Hopf bifurcation of differential equation with linearly state-dependent delays via MMS

Author

Listed:
  • Pei, Lijun
  • Wang, Shuo

Abstract

In this paper the dynamics of differential equation with two linearly state-dependent delays is considered, with particular attention focused on non-resonant double Hopf bifurcation. Firstly, we identify the sufficient and necessary conditions of the double Hopf bifurcation by formal linearization, linear stability analysis and Hopf bifurcation theorem. Secondly, in the first time the method of multiple scales (MMS) is employed to classify the dynamics in the neighborhoods of two kinds of non-resonant double Hopf bifurcation points, i.e. Cases III and Ib, in the bifurcation parameters plane. Finally, numerical simulation is executed qualitatively to verify the previously analytical results and demonstrate the rich phenomena, including the stable equilibrium, stable periodic solutions, bistability and stable quasi-periodic solution and so on. It also implies that MMS is simple, effective and correct in higher co-dimensional bifurcation analysis of the state-dependent DDEs. Besides, the other complicated dynamics, such as the switch between torus and phase-locked solution, period-doubling and the route of the break of torus to chaos are also found in this paper.

Suggested Citation

  • Pei, Lijun & Wang, Shuo, 2019. "Double Hopf bifurcation of differential equation with linearly state-dependent delays via MMS," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 256-276.
    • Handle: RePEc:eee:apmaco:v:341:y:2019:i:c:p:256-276
    DOI: 10.1016/j.amc.2018.08.040
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300318307549
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2018.08.040?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.

    Citations

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


    Cited by:

    1. Pei, Lijun & Wang, Shuo, 2019. "Dynamics and the periodic solutions of the delayed non-smooth Internet TCP-RED congestion control system via HB–AFT," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 689-702.

    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:apmaco:v:341:y:2019:i:c:p:256-276. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.