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Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method

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  • Han, Qun
  • Xu, Wei
  • Sun, Jian-Qiao

Abstract

The stochastic response of nonlinear oscillators under periodic and Gaussian white noise excitations is studied with the generalized cell mapping based on short-time Gaussian approximation (GCM/STGA) method. The solutions of the transition probability density functions over a small fraction of the period are constructed by the STGA scheme in order to construct the GCM over one complete period. Both the transient and steady-state probability density functions (PDFs) of a smooth and discontinuous (SD) oscillator are computed to illustrate the application of the method. The accuracy of the results is verified by direct Monte Carlo simulations. The transient responses show the evolution of the PDFs from being Gaussian to non-Gaussian. The effect of a chaotic saddle on the stochastic response is also studied. The stochastic P-bifurcation in terms of the steady-state PDFs occurs with the decrease of the smoothness parameter, which corresponds to the deterministic pitchfork bifurcation.

Suggested Citation

  • Han, Qun & Xu, Wei & Sun, Jian-Qiao, 2016. "Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 115-125.
    • Handle: RePEc:eee:phsmap:v:458:y:2016:i:c:p:115-125
    DOI: 10.1016/j.physa.2016.04.006
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    References listed on IDEAS

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    1. He, Qun & Xu, Wei & Rong, Haiwu & Fang, Tong, 2004. "Stochastic bifurcation in Duffing–Van der Pol oscillators," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 338(3), pages 319-334.
    2. Yue, Xiaole & Xu, Wei & Jia, Wantao & Wang, Liang, 2013. "Stochastic response of a ϕ6 oscillator subjected to combined harmonic and Poisson white noise excitations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(14), pages 2988-2998.
    3. Hong, Ling & Sun, Jian-Qiao, 2006. "Bifurcations of forced oscillators with fuzzy uncertainties by the generalized cell mapping method," Chaos, Solitons & Fractals, Elsevier, vol. 27(4), pages 895-904.
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    Cited by:

    1. Yue, Xiaole & Yu, Bei & Li, Yongge & Xu, Yong, 2022. "Dynamic response and bifurcation for Rayleigh-Liénard oscillator under multiplicative colored noise," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    2. Han, Qun & Xu, Wei & Hu, Bing & Huang, Dongmei & Sun, Jian-Qiao, 2018. "Extinction time of a stochastic predator–prey model by the generalized cell mapping method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 351-366.

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