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Explicit expression of stationary response probability density for nonlinear stochastic systems

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Identifying the exactly or approximately explicit expression of the stationary response probability density for general nonlinear stochastic dynamical systems is of great significance in the fields of stochastic dynamics and control. Almost all the existing methods are devoted to determine the exact or approximate solution for specific values of system and excitation parameters. Herein, aimed at stochastic systems with polynomial nonlinearity and excited by Gaussian white noises, a novel method is proposed to identify the stationary response probability density which explicitly includes system and excitation parameters. The stationary probability density is first written as an exponential function according to the maximum entropy principle, the power of the exponential function is then expressed as a linear combination of prescribed nondimensional parameter clusters constituted by system and excitation parameters, and state variables, with the coefficients to be determined. The undetermined coefficients are derived by minimizing the residual of the associated Fokker-Planck- Kolmogorov equation. The application and efficacy of the proposed method are illustrated by a typical numerical example.

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Data Availability

All data and materials that support the findings of this study are available from the corresponding author upon reasonable request.

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Codes are available from the corresponding author upon reasonable request.

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Funding

This study was supported by the National Natural Science Foundation of China under Grant Nos. 11532011, 11972317, 11872328, and 11672262.

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Authors and Affiliations

  1. The State Key Laboratory of Fluid Power & Mechatronic Systems, Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

    Xiaoling Jin, Yong Wang & Zhilong Huang

  2. College of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou, 310018, China

    Yanping Tian

Authors
  1. Xiaoling Jin
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  2. Yanping Tian
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  3. Yong Wang
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  4. Zhilong Huang
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Correspondence to Yong Wang.

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Jin, X., Tian, Y., Wang, Y. et al. Explicit expression of stationary response probability density for nonlinear stochastic systems. Acta Mech 232, 2101–2114 (2021). https://doi.org/10.1007/s00707-021-02939-0

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  • DOI: https://doi.org/10.1007/s00707-021-02939-0

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