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

• Published in: Chinese physics B 2016-02, Vol.25 (2), p.13-21
• Main Author: 杨勇歌 徐伟 孙亚辉 谷旭东
• Format: Article
• Language: English
• Subjects: Derivatives; Equivalence; Gaussian; Mathematical analysis; Mathematical models; Oscillators; Pol振子; Stochasticity; van; White noise; 分数阶导数; 噪声激励; 振荡器; 蒙特卡罗数值模拟; 随机响应; 高斯白噪声
• ISSN: 1674-1056; 2058-3834; 1741-4199
• DOI: 10.1088/1674-1056/25/2/020201

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.