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

Abstract

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.

Keywords: stochastic averaging method fractional derivative van der Pol equivalent stochastic system

Received: 30 August 2015
Revised: 15 October 2015
Accepted manuscript online:

PACS:	02.50.-r	(Probability theory, stochastic processes, and statistics)
	05.40.-a	(Fluctuation phenomena, random processes, noise, and Brownian motion)
	05.40.Ca	(Noise)

Fund:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11472212, 11532011, and 11502201).

Corresponding Authors: Wei Xu
E-mail: weixu@nwpu.edu.cn

Cite this article:

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 2016 Chin. Phys. B 25 020201

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

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