联合噪声激励下分数阶三稳van der Pol振子的随机P分岔  

Stochastic P bifurcation in a tri-stable van der Pol oscillator with fractional derivative excited by combined Gaussian white noises

作　　者：李亚杰 吴志强[2,3] 兰奇逊 郝颖[4] 张祥云 LI Yajie;WU Zhiqiang;LAN Qixun;HAO Ying;ZHANG Xiangyun(School of Mathematics and Physics,Henan University of Urban Construction,Pingdingshan 467036,China;School of Mechanical Engineering,Tianjin University,Tianjin 300072,China;Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control,Tianjin 300072,China;School of Architectural Engineering and Mechanics,Yanshan University,Qinhuangdao 066004,China;Basic Course Department,Tianjin Sino-German University of Applied Sciences,Tianjin300350,China)

机构地区：[1]河南城建学院数理学院,河南平顶山467036 [2]天津大学机械学院,天津300072 [3]天津市非线性动力学与混沌控制重点实验室,天津300072 [4]燕山大学建筑工程与力学学院,河北秦皇岛066004 [5]天津中德应用技术大学基础课部,天津300350

出　　处：《振动与冲击》2021年第16期275-280,293,共7页Journal of Vibration and Shock

基　　金：国家973计划(2014CB046805);国家自然科学基金(12002120,11672349,61503122,11902287);河南省科技攻关项目(202102210142);天津市自然科学基金(18JCQNJC01500)。

摘　　要：研究了联合高斯白噪声激励下含分数阶导数项的三稳态van der Pol系统的随机P分岔问题。利用均方误差最小原则,将分数阶导数项等效为阻尼力与回复力的线性组合,从而将原系统转化为等价的整数阶系统。运用随机平均法得到了系统幅值的稳态概率密度函数(PDF),利用奇异性理论,得到了系统发生随机P分岔的临界参数条件。在转迁集曲线围成的各区域内分别选取相应参数定性分析了系统幅值稳态概率密度曲线的类型,并通过Monte Carlo模拟的方法将所得数值结果与解析结果进行了比较,从数值仿真与解析结果的符合程度来看,该研究的推导过程及系统转迁集的计算是准确的。该方法对于设计用于调整系统响应的分数阶控制器有直接的指导作用。The stochastic P bifurcation behavior of tri-stability in a generalized van der Pol system with fractional derivative under additive and multiplicative Gaussian white noise excitations was investigated.Firstly,based on the minimal mean square error principle,the fractional derivative was found to be equivalent to a linear combination of damping and restoring forces,and the original system was simplified into an equivalent integer order system.Secondly,the stationary probability density function(PDF)of system amplitude was obtained by stochastic averaging,and according to the singularity theory,the critical parameters for stochastic P bifurcation of the system were found.Finally,the nature of stationary PDF curves of the system amplitude were qualitatively analyzed by choosing corresponding parameters in each region divided by the transition set curves.The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

关 键 词：高斯白噪声 分数阶导数 随机P分岔 转迁集 Monte Carlo模拟 

分 类 号：O324[理学—一般力学与力学基础]