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Singular perturbation methods for nonlinear dynamic systems with time delays

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  • Hu, H.Y.
  • Wang, Z.H.

Abstract

This review article surveys the recent advances in the dynamics and control of time-delay systems, with emphasis on the singular perturbation methods, such as the method of multiple scales, the method of averaging, and two newly developed methods, the energy analysis and the pseudo-oscillator analysis. Some examples are given to demonstrate the advantages of the methods. The comparisons with other methods show that these methods lead to easier computations and higher accurate prediction on the local dynamics of time-delay systems near a Hopf bifurcation.

Suggested Citation

  • Hu, H.Y. & Wang, Z.H., 2009. "Singular perturbation methods for nonlinear dynamic systems with time delays," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 13-27.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:1:p:13-27
    DOI: 10.1016/j.chaos.2007.07.048
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    References listed on IDEAS

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    1. Liao, Xiaofeng & Ran, Jiouhong, 2007. "Hopf bifurcation in love dynamical models with nonlinear couples and time delays," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 853-865.
    2. Wang, Zhanfeng & Chu, Tianguang, 2006. "Delay induced Hopf bifurcation in a simplified network congestion control model," Chaos, Solitons & Fractals, Elsevier, vol. 28(1), pages 161-172.
    3. Sun, Chengjun & Cao, Zhijie & Lin, Yiping, 2007. "Analysis of stability and Hopf bifurcation for a viral infectious model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 234-245.
    4. Yang, Hong-Yong & Tian, Yu-Ping, 2005. "Hopf bifurcation in REM algorithm with communication delay," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1093-1105.
    5. Wang, Hongbin & Liu, Jiaqi, 2005. "Stability and bifurcation analysis in a magnetic bearing system with time delays," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 813-825.
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    Cited by:

    1. Naik, Raghavendra D. & Singru, Pravin M., 2012. "Stability and Hopf bifurcation of a Nonlinear oscillator with multiple time-delays," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1387-1396.
    2. 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.
    3. Youhua Qian & Shuli Wang & Shuping Chen, 2023. "Multi-Frequency Homotopy Analysis Method for Coupled Van der Pol-Duffing System with Time Delay," Mathematics, MDPI, vol. 11(2), pages 1-17, January.

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