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Stabilization of unstable periodic orbits in a Duffing system using an optimal time-delayed feedback control and pseudospectral method

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  • Piccirillo, Vinícius
  • Balthazar, Jose Manoel
  • Lima, Jeferson José
  • Tusset, Angelo Marcelo

Abstract

Delayed feedback control is one of the most important methods for stabilizing unstable periodic orbits (UPOs) embedded within chaotic attractors. This paper compares two delay control techniques commonly used to stabilize UPOs: time-delay feedback control (TDF) and extended time-delay feedback control (ETDF). These chaos control strategies are investigated using the nonlinear Duffing system. The Chebyshev pseudospectral collocation method is employed to convert a delay differential equation (DDE) into a system of ordinary differential equations (ODEs). The chaos control problem is then transformed into an optimization problem, where the cost function minimizes a combination of the maximum magnitude of the Floquet multiplier and the maximum Lyapunov exponent of the target UPO. A comparison of the TDF and ETDF methods, along with the pseudospectral Chebyshev method, is made in terms of the stability of the controlled orbit and control effort.

Suggested Citation

  • Piccirillo, Vinícius & Balthazar, Jose Manoel & Lima, Jeferson José & Tusset, Angelo Marcelo, 2025. "Stabilization of unstable periodic orbits in a Duffing system using an optimal time-delayed feedback control and pseudospectral method," Chaos, Solitons & Fractals, Elsevier, vol. 194(C).
    • Handle: RePEc:eee:chsofr:v:194:y:2025:i:c:s0960077925002140
    DOI: 10.1016/j.chaos.2025.116201
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    References listed on IDEAS

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    1. Chagas, T.P. & Toledo, B.A. & Rempel, E.L. & Chian, A.C.-L. & Valdivia, J.A., 2012. "Optimal feedback control of the forced van der Pol system," Chaos, Solitons & Fractals, Elsevier, vol. 45(9), pages 1147-1156.
    2. 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).
    3. Costa, Dimitri & Pavlovskaia, Ekaterina & Wiercigroch, Marian, 2024. "Feedback control of chaos in impact oscillator with multiple time-delays," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    4. 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.
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