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Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise

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  • Yang, Yongge
  • Xu, Wei
  • Gu, Xudong
  • Sun, Yahui

Abstract

The stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise is considered. Firstly, the generalized harmonic function technique is applied to the fractional self-excited systems. Based on this approach, the original fractional self-excited systems are reduced to equivalent stochastic systems without fractional derivative. Then, the analytical solutions of the equivalent stochastic systems are obtained by using the stochastic averaging method. Finally, in order to verify the theoretical results, the two most typical self-excited systems with fractional derivative, namely the fractional van der Pol oscillator and fractional Rayleigh oscillator, are discussed in detail. Comparing the analytical and numerical results, a very satisfactory agreement can be found. Meanwhile, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the self-excited fractional systems are also discussed in detail.

Suggested Citation

  • Yang, Yongge & Xu, Wei & Gu, Xudong & Sun, Yahui, 2015. "Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 190-204.
    • Handle: RePEc:eee:chsofr:v:77:y:2015:i:c:p:190-204
    DOI: 10.1016/j.chaos.2015.05.029
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    References listed on IDEAS

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    1. Shen, Yongjun & Yang, Shaopu & Sui, Chuanyi, 2014. "Analysis on limit cycle of fractional-order van der Pol oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 94-102.
    2. Ahmad, Wajdi M. & El-Khazali, Reyad, 2007. "Fractional-order dynamical models of love," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1367-1375.
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    Cited by:

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    3. Yan, Zhi & Wang, Wei & Liu, Xianbin, 2018. "Analysis of a quintic system with fractional damping in the presence of vibrational resonance," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 780-793.
    4. Li Jiang & Tao Wang & Qing-Xue Huang, 2023. "Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations," Mathematics, MDPI, vol. 11(7), pages 1-15, March.
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    7. Ning, Xin & Ma, Yanyan & Li, Shuai & Zhang, Jingmin & Li, Yifei, 2018. "Response of non-linear oscillator driven by fractional derivative term under Gaussian white noise," Chaos, Solitons & Fractals, Elsevier, vol. 113(C), pages 102-107.
    8. Guo, Feng & Wang, Xue-yuan & Qin, Ming-wei & Luo, Xiang-dong & Wang, Jian-wei, 2021. "Resonance phenomenon for a nonlinear system with fractional derivative subject to multiplicative and additive noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 562(C).

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