Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise

doi:10.1088/1674-1056/26/9/090501

中国物理B ›› 2017, Vol. 26 ›› Issue (9): 90501-090501.doi: 10.1088/1674-1056/26/9/090501

Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise

Wei Li(李伟), Mei-Ting Zhang(张美婷), Jun-Feng Zhao(赵俊锋)

1 School of Mathematics and Statistics, Xidian University, Xi'an 710071, China;
2 Applied Mathematics Department, School of Science, Northwestern Polytechnical University, Xi'an 710072, China

收稿日期:2017-03-01 修回日期:2017-05-08 出版日期:2017-09-05 发布日期:2017-09-05
通讯作者: Wei Li E-mail:liweilw@mail.xidian.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 11302157), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JM1028), the Fundamental Research Funds for the Central Universities, China (Grant No. JB160706), and Chinese-Serbian Science and Technology Cooperation for the Years 2015-2016 (Grant No. 3-19).

Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise

Wei Li(李伟)¹, Mei-Ting Zhang(张美婷)¹, Jun-Feng Zhao(赵俊锋)²

1 School of Mathematics and Statistics, Xidian University, Xi'an 710071, China;
2 Applied Mathematics Department, School of Science, Northwestern Polytechnical University, Xi'an 710072, China

Received:2017-03-01 Revised:2017-05-08 Online:2017-09-05 Published:2017-09-05
Contact: Wei Li E-mail:liweilw@mail.xidian.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 11302157), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JM1028), the Fundamental Research Funds for the Central Universities, China (Grant No. JB160706), and Chinese-Serbian Science and Technology Cooperation for the Years 2015-2016 (Grant No. 3-19).

摘要/Abstract

摘要： The stochastic bifurcation of a generalized Duffing-van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.

关键词: stochastic bifurcation, fractional derivative, color noise, stochastic averaging method

Abstract: The stochastic bifurcation of a generalized Duffing-van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.

Key words: stochastic bifurcation, fractional derivative, color noise, stochastic averaging method

中图分类号: (Nonlinear dynamics and chaos)

05.45.-a

05.10.Gg (Stochastic analysis methods) 05.10.Ln (Monte Carlo methods) 05.40.Ca (Noise)

李伟, 张美婷, 赵俊锋. Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise[J]. 中国物理B, 2017, 26(9): 90501-090501.

Wei Li(李伟), Mei-Ting Zhang(张美婷), Jun-Feng Zhao(赵俊锋). Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise[J]. Chin. Phys. B, 2017, 26(9): 90501-090501.

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Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise

Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise

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