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Stability and dynamics of a controlled van der Pol–Duffing oscillator

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  • Ji, J.C.
  • Hansen, C.H.

Abstract

The trivial equilibrium of a van der Pol–Duffing oscillator under a linear-plus-nonlinear feedback control may change its stability either via a single or via a double Hopf bifurcation if the time delay involved in the feedback reaches certain values. It is found that the trivial equilibrium may lose its stability via a subcritical or supercritical Hopf bifurcation and regain its stability via a reverse subcritical or supercritical Hopf bifurcation as the time delay increases. A stable limit cycle appears after a supercritical Hopf bifurcation occurs and disappears through a reverse supercritical Hopf bifurcation. The interaction of the weakly periodic excitation and the stable bifurcating solution is investigated for the forced system under primary resonance conditions. It is shown that the forced periodic response may lose its stability via a Neimark–Sacker bifurcation. Analytical results are validated by a comparison with those of direct numerical integration.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:28:y:2006:i:2:p:555-570
    DOI: 10.1016/j.chaos.2005.08.021
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    References listed on IDEAS

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    1. Wang, Hongbin & Liu, Jiaqi, 2005. "Stability and bifurcation analysis in a magnetic bearing system with time delays," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 813-825.
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    Cited by:

    1. Wang, Mei-Qi & Ma, Wen-Li & Li, Yuan & Chen, En-Li & Liu, Peng-Fei & Zhang, Ming-Zhi, 2022. "Dynamic analysis of piecewise nonlinear systems with fractional differential delay feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    2. Li, Jiaorui & Feng, C.S., 2010. "First-passage failure of a business cycle model under time-delayed feedback control and wide-band random excitation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(24), pages 5557-5562.
    3. 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.
    4. Zhou, Jin & Cheng, Xuhua & Xiang, Lan & Zhang, Yecui, 2007. "Impulsive control and synchronization of chaotic systems consisting of Van der Pol oscillators coupled to linear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 607-616.
    5. Ji, J.C. & Zhang, N. & Gao, Wei, 2009. "Difference resonances in a controlled van der Pol-Duffing oscillator involving time delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 975-980.
    6. Niu, Ben & Wei, Junjie, 2008. "Stability and bifurcation analysis in an amplitude equation with delayed feedback," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1362-1371.
    7. Peng, Ya-Fu & Hsu, Chun-Fei, 2009. "Identification-based chaos control via backstepping design using self-organizing fuzzy neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1377-1389.
    8. Sah, Simohamed & Belhaq, Mohamed, 2008. "Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1489-1496.
    9. Iñarrea, Manuel, 2009. "Chaos and its control in the pitch motion of an asymmetric magnetic spacecraft in polar elliptic orbit," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1637-1652.

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