Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises

doi:10.1007/s10483-021-2804-8

Applied Mathematics and Mechanics (English Edition) ›› 2021, Vol. 42 ›› Issue (12): 1759-1770.doi: https://doi.org/10.1007/s10483-021-2804-8

Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises

Mengjiao HUA¹, Yu WU^1,2,3,4

1. Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China;
2. Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, China;
3. Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China;
4. State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, China

收稿日期:2021-08-17 修回日期:2021-09-30 发布日期:2021-11-23
通讯作者: Yu WU ,E-mail:ywu@zju.edu.cn
基金资助:
the National Natural Science Foundation of China (Nos. 11932017, 11402227, 11432012, and 11621062) and the Natural Science Foundation of Zhejiang Province of China (No. LR20A020001)

Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises

Mengjiao HUA¹, Yu WU^1,2,3,4

1. Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China;
2. Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, China;
3. Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China;
4. State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, China

Received:2021-08-17 Revised:2021-09-30 Published:2021-11-23
Contact: Yu WU ,E-mail:ywu@zju.edu.cn
Supported by:
the National Natural Science Foundation of China (Nos. 11932017, 11402227, 11432012, and 11621062) and the Natural Science Foundation of Zhejiang Province of China (No. LR20A020001)

摘要/Abstract

摘要： The phenomenon of stochastic bifurcation driven by the correlated non-Gaussian colored noise and the Gaussian white noise is investigated by the qualitative changes of steady states with the most probable phase portraits. To arrive at the Markovian approximation of the original non-Markovian stochastic process and derive the general approximate Fokker-Planck equation (FPE), we deal with the non-Gaussian colored noise and then adopt the unified colored noise approximation (UCNA). Subsequently, the theoretical equation concerning the most probable steady states is obtained by the maximum of the stationary probability density function (SPDF). The parameter of the uncorrelated additive noise intensity does enter the governing equation as a non-Markovian effect, which is in contrast to that of the uncorrelated Gaussian white noise case, where the parameter is absent from the governing equation, i.e., the most probable steady states are mainly controlled by the uncorrelated multiplicative noise. Additionally, in comparison with the deterministic counterpart, some peculiar bifurcation behaviors with regard to the most probable steady states induced by the correlation time of non-Gaussian colored noise, the noise intensity, and the non-Gaussian noise deviation parameter are discussed. Moreover, the symmetry of the stochastic bifurcation diagrams is destroyed when the correlation between noises is concerned. Furthermore, the feasibility and accuracy of the analytical predictions are verified compared with those of the Monte Carlo (MC) simulations of the original system.

关键词: stochastic bifurcation, non-Gaussian colored noise, most probable steady state, Fokker-Planck equation (FPE), Monte Carlo (MC) simulation

Abstract: The phenomenon of stochastic bifurcation driven by the correlated non-Gaussian colored noise and the Gaussian white noise is investigated by the qualitative changes of steady states with the most probable phase portraits. To arrive at the Markovian approximation of the original non-Markovian stochastic process and derive the general approximate Fokker-Planck equation (FPE), we deal with the non-Gaussian colored noise and then adopt the unified colored noise approximation (UCNA). Subsequently, the theoretical equation concerning the most probable steady states is obtained by the maximum of the stationary probability density function (SPDF). The parameter of the uncorrelated additive noise intensity does enter the governing equation as a non-Markovian effect, which is in contrast to that of the uncorrelated Gaussian white noise case, where the parameter is absent from the governing equation, i.e., the most probable steady states are mainly controlled by the uncorrelated multiplicative noise. Additionally, in comparison with the deterministic counterpart, some peculiar bifurcation behaviors with regard to the most probable steady states induced by the correlation time of non-Gaussian colored noise, the noise intensity, and the non-Gaussian noise deviation parameter are discussed. Moreover, the symmetry of the stochastic bifurcation diagrams is destroyed when the correlation between noises is concerned. Furthermore, the feasibility and accuracy of the analytical predictions are verified compared with those of the Monte Carlo (MC) simulations of the original system.

Key words: stochastic bifurcation, non-Gaussian colored noise, most probable steady state, Fokker-Planck equation (FPE), Monte Carlo (MC) simulation

中图分类号:

Mengjiao HUA, Yu WU. Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(12): 1759-1770.

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Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises

Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises

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