Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

doi:10.1088/1674-1056/25/2/020201

中国物理B ›› 2016, Vol. 25 ›› Issue (2): 20201-020201.doi: 10.1088/1674-1056/25/2/020201

Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

Yong-Ge Yang(杨勇歌), Wei Xu(徐伟), Ya-Hui Sun(孙亚辉), Xu-Dong Gu(谷旭东)

1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China;
2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, China;
3. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi'an 710129, China

收稿日期:2015-08-30 修回日期:2015-10-15 出版日期:2016-02-05 发布日期:2016-02-05
通讯作者: Wei Xu E-mail:weixu@nwpu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11472212, 11532011, and 11502201).

Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

Yong-Ge Yang(杨勇歌)¹, Wei Xu(徐伟)¹, Ya-Hui Sun(孙亚辉)², Xu-Dong Gu(谷旭东)³

1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China;
2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, China;
3. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi'an 710129, China

Received:2015-08-30 Revised:2015-10-15 Online:2016-02-05 Published:2016-02-05
Contact: Wei Xu E-mail:weixu@nwpu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11472212, 11532011, and 11502201).

摘要/Abstract

摘要：

This paper aims to investigate the stochastic response of the van der Pol (VDP) oscillator with two kinds of fractional derivatives under Gaussian white noise excitation. First, the fractional VDP oscillator is replaced by an equivalent VDP oscillator without fractional derivative terms by using the generalized harmonic balance technique. Then, the stochastic averaging method is applied to the equivalent VDP oscillator to obtain the analytical solution. Finally, the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional VDP oscillator. The numerical results not only demonstrate the accuracy of the proposed approach but also show that the fractional order, the fractional coefficient and the intensity of Gaussian white noise play important roles in the responses of the fractional VDP oscillator. An interesting phenomenon we found is that the effects of the fractional order of two kinds of fractional derivative items on the fractional stochastic systems are totally contrary.

关键词: stochastic averaging method, fractional derivative, van der Pol, equivalent stochastic system

Abstract:

Key words: stochastic averaging method, fractional derivative, van der Pol, equivalent stochastic system

中图分类号: (Probability theory, stochastic processes, and statistics)

02.50.-r

05.40.-a (Fluctuation phenomena, random processes, noise, and Brownian motion) 05.40.Ca (Noise)

杨勇歌, 徐伟, 孙亚辉, 谷旭东. Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation[J]. 中国物理B, 2016, 25(2): 20201-020201.

Yong-Ge Yang(杨勇歌), Wei Xu(徐伟), Ya-Hui Sun(孙亚辉), Xu-Dong Gu(谷旭东). Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation[J]. Chin. Phys. B, 2016, 25(2): 20201-020201.

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Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

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