IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v40y2009i1p28-49.html
   My bibliography  Save this article

Dynamics for a class of nonlinear systems with time delay

Author

Listed:
  • Xu, Jian
  • Chung, Kwok-Wai

Abstract

Starting from the delay Liénard and van der Pol equations with time delay, this paper represents an attempt to give an introductory presentation of dynamics for a class of nonlinear systems with time delay and emphasizes the recent development on this field in china. It is clearly seen that the time delay occurred in systems can lead to the rich dynamical behaviors such as death island, Hopf bifurcation, double Hopf bifurcation, period-doubling bifurcation, phase locked (periodic) and phase shifting solutions, co-existing motions, quasi-periodic motion and even chaos. In the quantitative and qualitative treatment of the analytical method, a new method, called perturbation-incremental scheme (PIS), is introduced by two steps, namely perturbation and increment for continuation to the critical value at a simple Hopf bifurcation. This paper shows that time delay may be used as a simple but efficient “switch” to control motions of a system: either from order to complex motions or from complex to order motions for different applications. As well as the PIS can bypass and avoid the tedious calculation of the center manifold reduction (CMR) and normal form.

Suggested Citation

  • Xu, Jian & Chung, Kwok-Wai, 2009. "Dynamics for a class of nonlinear systems with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 28-49.
  • Handle: RePEc:eee:chsofr:v:40:y:2009:i:1:p:28-49
    DOI: 10.1016/j.chaos.2007.07.032
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077907005310
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2007.07.032?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. Arik, Sabri, 2005. "Global robust stability analysis of neural networks with discrete time delays," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1407-1414.
    2. Li, Chuandong & Liao, Xiaofeng & Zhang, Rong & Prasad, Ashutosh, 2005. "Global robust exponential stability analysis for interval neural networks with time-varying delays," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 751-757.
    3. Zhang, Hongbin & Li, Chunguang & Liao, Xiaofeng, 2005. "A note on the robust stability of neural networks with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 25(2), pages 357-360.
    4. Shahverdiev, E.M. & Nuriev, R.A. & Hashimov, R.H. & Shore, K.A., 2005. "Parameter mismatches, variable delay times and synchronization in time-delayed systems," Chaos, Solitons & Fractals, Elsevier, vol. 25(2), pages 325-331.
    5. Cammarata, L. & Fichera, A. & Pagano, A., 2002. "Neural prediction of combustion instability," Applied Energy, Elsevier, vol. 72(2), pages 513-528, June.
    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. Jiang, Heping & Song, Yongli, 2015. "Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 1102-1126.

    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. Gau, R.S. & Lien, C.H. & Hsieh, J.G., 2007. "Global exponential stability for uncertain cellular neural networks with multiple time-varying delays via LMI approach," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1258-1267.
    2. Ou, Ou, 2007. "Global robust exponential stability of delayed neural networks: An LMI approach," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1742-1748.
    3. Cui, Shihua & Zhao, Tao & Guo, Jie, 2009. "Global robust exponential stability for interval neural networks with delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1567-1576.
    4. Li, Chuandong & Chen, Jinyu & Huang, Tingwen, 2007. "A new criterion for global robust stability of interval neural networks with discrete time delays," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 561-570.
    5. Sun, Yeong-Jeu, 2007. "Duality between observation and output feedback for linear systems with multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 879-884.
    6. Xiong, Wenjun & Ma, Deyi & Liang, Jinling, 2009. "Robust convergence of Cohen–Grossberg neural networks with time-varying delays," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1176-1184.
    7. Sun, Yeong-Jeu, 2007. "Stability criterion for a class of descriptor systems with discrete and distributed time delays," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 986-993.
    8. Wang, Jiafu & Huang, Lihong, 2012. "Almost periodicity for a class of delayed Cohen–Grossberg neural networks with discontinuous activations," Chaos, Solitons & Fractals, Elsevier, vol. 45(9), pages 1157-1170.
    9. He, Yong & Wang, Qing-Guo & Zheng, Wei-Xing, 2005. "Global robust stability for delayed neural networks with polytopic type uncertainties," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1349-1354.
    10. Zhao, Hongyong & Ding, Nan & Chen, Ling, 2009. "Almost sure exponential stability of stochastic fuzzy cellular neural networks with delays," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1653-1659.
    11. Qiu, Jiqing & Yang, Hongjiu & Zhang, Jinhui & Gao, Zhifeng, 2009. "New robust stability criteria for uncertain neural networks with interval time-varying delays," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 579-585.
    12. Ding, Ke & Huang, Nan-Jing, 2008. "A new class of interval projection neural networks for solving interval quadratic program," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 718-725.
    13. Yucel, Eylem & Arik, Sabri, 2009. "Novel results for global robust stability of delayed neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1604-1614.
    14. Sun, Yeong-Jeu & Gau, Ruey-Shyan & Hsieh, Jer-Guang, 2009. "Simple criteria for sector root clustering of uncertain systems with multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 65-71.
    15. Zhao, Dan & Li, Shihuai & Yang, Wenming & Zhang, Zhiguo, 2015. "Numerical investigation of the effect of distributed heat sources on heat-to-sound conversion in a T-shaped thermoacoustic system," Applied Energy, Elsevier, vol. 144(C), pages 204-213.
    16. Huang, Zai-Tang & Luo, Xiao-Shu & Yang, Qi-Gui, 2007. "Global asymptotic stability analysis of bidirectional associative memory neural networks with distributed delays and impulse," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 878-885.
    17. Hoang, Thang Manh, 2011. "Complex synchronization manifold in coupled time-delayed systems," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 48-57.
    18. Sun, Yuze & Rao, Zhuming & Zhao, Dan & Wang, Bing & Sun, Dakun & Sun, Xiaofeng, 2020. "Characterizing nonlinear dynamic features of self-sustained thermoacoustic oscillations in a premixed swirling combustor," Applied Energy, Elsevier, vol. 264(C).
    19. Li, Xinyan & Huang, Yong & Zhao, Dan & Yang, Wenming & Yang, Xinglin & Wen, Huabing, 2017. "Stability study of a nonlinear thermoacoustic combustor: Effects of time delay, acoustic loss and combustion-flow interaction index," Applied Energy, Elsevier, vol. 199(C), pages 217-224.
    20. Park, Ju H. & Lee, S.M. & Kwon, O.M., 2009. "On exponential stability of bidirectional associative memory neural networks with time-varying delays," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1083-1091.

    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:40:y:2009:i:1:p:28-49. 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.