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An improved path integration method for nonlinear systems under Poisson white noise excitation

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  • Ren, Zhicong
  • Xu, Wei

Abstract

In order to overcome some unsatisfactory trends and limitations of the traditional path integration (PI) method for Poisson white noise, a novel PI method is proposed in this paper, which includes two improved schemes. The first one is a new Transition Probability Density Function (TPDF) approximation which considers the randomness of the impulse happening time during each time intervals. The second one is a transformation of Chapman–Kolmogorov (CK) equation by a variable substitution instead of directly using it, whose numerical calculation is based on the back stepping Runge–Kutta scheme and the triangulation-based interpolation. Monte Carlo Simulations (MCS) are utilized to measure the accuracy of the improved algorithm with three illustrative nonlinear systems. The results show that compared with the traditional PI method, the improved PI method can give a more accurate description of the TPDF values, and provide more precise stationary Probability Density Function (PDF) results whenever the mean arrival rate is large or small. The improved algorithm has a wider range of choices in time interval values to maintain the accuracy of stationary PDF results. Besides, it is discovered that cubic interpolation deserves to be applied in the improved PI method more than linear and natural interpolations.

Suggested Citation

  • Ren, Zhicong & Xu, Wei, 2020. "An improved path integration method for nonlinear systems under Poisson white noise excitation," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    • Handle: RePEc:eee:apmaco:v:373:y:2020:i:c:s0096300320300059
    DOI: 10.1016/j.amc.2020.125036
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    References listed on IDEAS

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    1. Qiao, Yan & Xu, Wei & Jia, Wantao & Han, Qun, 2017. "Stochastic stationary response of a variable-mass system with mass disturbance described by Poisson white noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 122-134.
    2. Ren, Zhicong & Xu, Wei & Qiao, Yan, 2019. "Local averaged path integration method approach for nonlinear dynamic systems," Applied Mathematics and Computation, Elsevier, vol. 344, pages 68-77.
    3. 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.
    4. Jia, Wantao & Zhu, Weiqiu, 2014. "Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 398(C), pages 125-144.
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