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Stability switches and bifurcation in a two-degrees-of-freedom nonlinear quarter-car with small time-delayed feedback control

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  • Koumene Taffo, G.I.
  • Siewe Siewe, M.
  • Tchawoua, C.

Abstract

In this paper, we investigate a two-degrees-of-freedom nonlinear quarter-car model with time-delayed feedback control. It is well known that a time delay has destabilizing effects in mathematical models. However, delays are not necessarily destabilizing. In this work we explore a system where a time delay can be both stabilizing and destabilizing. Using the generalized Sturm criterion, the critical control gain for the delay-independent stability region and critical time delays for stability switches are derived. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo stability switches with the variation of the time delay. These stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Properties of Hopf bifurcation such as direction and stability of bifurcating periodic solutions are determined by using the normal form theory and centre manifold theorem. Numerical simulations are provided to support the theoretical analysis. The critical conditions can provide a theoretical guidance for the design of vehicles with significant reduction of vibration in order to increase passengers ride comfort.

Suggested Citation

  • Koumene Taffo, G.I. & Siewe Siewe, M. & Tchawoua, C., 2016. "Stability switches and bifurcation in a two-degrees-of-freedom nonlinear quarter-car with small time-delayed feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 226-239.
    • Handle: RePEc:eee:chsofr:v:87:y:2016:i:c:p:226-239
    DOI: 10.1016/j.chaos.2016.04.012
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    References listed on IDEAS

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    1. Xiaobing Zhou & Murong Jiang & Xiaomei Cai, 2011. "Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-16, May.
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    Cited by:

    1. Chen, Enli & Xing, Wuce & Wang, Meiqi & Ma, Wenli & Chang, Yujian, 2021. "Study on chaos of nonlinear suspension system with fractional-order derivative under random excitation," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Bauomy, H.S. & EL-Sayed, A.T., 2023. "Safety of a quarter-vehicle car through negative derivative feedback controller," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

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