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Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method

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  • de Paula, Aline Souza
  • Savi, Marcelo Amorim

Abstract

Chaos control is employed for the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The extended time-delayed feedback control uses a continuous feedback loop incorporating information from previous states of the system in order to stabilize unstable orbits. This article deals with the chaos control of a nonlinear pendulum employing the extended time-delayed feedback control method. The control law leads to delay-differential equations (DDEs) that contain derivatives that depend on the solution of previous time instants. A fourth-order Runge–Kutta method with linear interpolation on the delayed variables is employed for numerical simulations of the DDEs and its initial function is estimated by a Taylor series expansion. During the learning stage, the UPOs are identified by the close-return method and control parameters are chosen for each desired UPO by defining situations where the largest Lyapunov exponent becomes negative. Analyses of a nonlinear pendulum are carried out by considering signals that are generated by numerical integration of the mathematical model using experimentally identified parameters. Results show the capability of the control procedure to stabilize UPOs of the dynamical system, highlighting some difficulties to achieve the stabilization of the desired orbit.

Suggested Citation

  • de Paula, Aline Souza & Savi, Marcelo Amorim, 2009. "Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2981-2988.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:5:p:2981-2988
    DOI: 10.1016/j.chaos.2009.04.039
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    References listed on IDEAS

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    1. de Paula, Aline Souza & Savi, Marcelo Amorim, 2009. "A multiparameter chaos control method based on OGY approach," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1376-1390.
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    Cited by:

    1. Zheng, Y.G. & Yu, J.L., 2022. "Stabilization of multi-rotation unstable periodic orbits through dynamic extended delayed feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    2. Nana, B. & Yamgoué, S.B. & Tchitnga, R. & Woafo, P., 2018. "On the modeling of the dynamics of electrical hair clippers," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 14-23.
    3. Nana, B. & Yamgoué, S.B. & Tchitnga, R. & Woafo, P., 2017. "Dynamics of a pendulum driven by a DC motor and magnetically controlled," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 18-27.
    4. Gois, Sandra R.F.S.M. & Savi, Marcelo A., 2009. "An analysis of heart rhythm dynamics using a three-coupled oscillator model," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2553-2565.
    5. Ferreira, Bianca Borem & de Paula, Aline Souza & Savi, Marcelo Amorim, 2011. "Chaos control applied to heart rhythm dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 44(8), pages 587-599.

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