Sun Zhongkui, Fu Jin, Xiao Yuzhu, Xu Wei
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.
Department of Mathematics and Information Science, Chang'an University, Xi'an 710086, People's Republic of China.
Chaos. 2015 Aug;25(8):083102. doi: 10.1063/1.4927646.
In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses.
本文在受到白噪声作用的时滞杜芬 - 范德波尔振子中，从理论和数值两方面研究了噪声和时间延迟对随机分岔的影响。由于存在时间延迟，随机响应是非马尔可夫的。因此，采用了近似方法来获得福克 - 普朗克 - 柯尔莫哥洛夫方程以及响应幅度的稳态概率密度函数。基于随机分岔由稳态概率分布的定性性质所表征这一认识，发现时间延迟、反馈强度以及噪声强度都会引发随机P分岔的出现。此外，结果表明延迟位移反馈强度对随机分岔的影响伴随着对时间延迟的敏感依赖性。而且，数值模拟结果很好地证实了理论分析的有效性。
