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Nonlinear response of a forced van der Pol–Duffing oscillator at non-resonant bifurcations of codimension two

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  • Ji, J.C.
  • Zhang, N.

Abstract

Non-resonant bifurcations of codimension two may appear in the controlled van der Pol–Duffing oscillator when two critical time delays corresponding to a double Hopf bifurcation have the same value. With the aid of centre manifold theorem and the method of multiple scales, the non-resonant response and two types of primary resonances of the forced van der Pol–Duffing oscillator at non-resonant bifurcations of codimension two are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. It is shown that the non-resonant response of the forced oscillator may exhibit quasi-periodic motions on a two- or three-dimensional (2D or 3D) torus. The primary resonant responses admit single and mixed solutions and may exhibit periodic motions or quasi-periodic motions on a 2D torus. Illustrative examples are presented to interpret the dynamics of the controlled system in terms of two dummy unfolding parameters and exemplify the periodic and quasi-periodic motions. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation.

Suggested Citation

  • Ji, J.C. & Zhang, N., 2009. "Nonlinear response of a forced van der Pol–Duffing oscillator at non-resonant bifurcations of codimension two," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1467-1475.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:3:p:1467-1475
    DOI: 10.1016/j.chaos.2008.06.008
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    References listed on IDEAS

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    1. Ji, J.C. & Hansen, C.H., 2006. "Stability and dynamics of a controlled van der Pol–Duffing oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 555-570.
    2. Czołczynski, K. & Kapitaniak, T. & Perlikowski, P. & Stefański, A., 2007. "Periodization of Duffing oscillators suspended on elastic structure: Mechanical explanation," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 920-926.
    3. Czołczynski, K. & Perlikowski, P. & Stefański, A. & Kapitaniak, T., 2007. "Synchronization of self-excited oscillators suspended on elastic structure," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 937-943.
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