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Primary resonant optimal control for homoclinic bifurcations in single-degree-of-freedom nonlinear oscillators

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  • Cao, Hongjun

Abstract

A primary resonant optimal control method (PROCM) is presented based on the adjustable role played by the phase shift in a general single-degree-of-freedom nonlinear oscillator. By Melnikov’s method and rigorous mathematical deductions, the optimization solutions for the amplitude coefficients to be used as the control parameters can be obtained, and the force term as the controller can be designed. The main novelty of this PROCM is able to enlarge to the largest possible degree the control region where homoclinic transversal intersections embedding in systems dynamics do not occur, and this is accomplished at lowest cost. This method is confirmed by the hardening Helmholtz–Duffing oscillator, in which the various homoclinic bifurcations can be efficiently controlled locally and globally.

Suggested Citation

  • Cao, Hongjun, 2005. "Primary resonant optimal control for homoclinic bifurcations in single-degree-of-freedom nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1387-1398.
    • Handle: RePEc:eee:chsofr:v:24:y:2005:i:5:p:1387-1398
    DOI: 10.1016/j.chaos.2004.09.084
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    Cited by:

    1. El-Bassiouny, A.F., 2006. "Single-mode control and chaos of cantilever beam under primary and principal parametric excitations," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1098-1121.
    2. Amer, Y.A., 2007. "Vibration control of ultrasonic cutting via dynamic absorber," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1703-1710.
    3. Litak, Grzegorz & Borowiec, Marek & Syta, Arkadiusz & Szabelski, Kazimierz, 2009. "Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2414-2429.
    4. Tang, Yun & Yang, Fenghong & Chen, Guanrong & Zhou, Tianshou, 2006. "Classification of homoclinic tangencies for periodically perturbed systems," Chaos, Solitons & Fractals, Elsevier, vol. 28(1), pages 76-89.
    5. El-Bassiouny, A.F., 2009. "On methods for continuous systems with quadratic, cubic and quantic nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1308-1316.
    6. Litak, Grzegorz & Syta, Arkadiusz & Borowiec, Marek, 2007. "Suppression of chaos by weak resonant excitations in a non-linear oscillator with a non-symmetric potential," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 694-701.

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