基于Chebyshev多项式逼近的随机van der Pol系统的倍周期分岔分析

Period-doubling bifurcation analysis of stochastic van der Pol system via Chebyshev polynomial approximation

应用Chebyshev多项式逼近法研究了谐和激励作用下具有随机参数的随机vanderPol系统的倍周期分岔现象.随机系统首先被转化成等价的确定性系统,然后通过数值方法求得响应,借此探索了随机vanderPol系统丰富的随机倍周期分岔现象.数值模拟显示随机vanderPol系统存在与确定性系统极为相似的倍周期分岔行为,但受随机因素的影响,又有与之不同之处.数值结果表明,Chebyshev多项式逼近是研究非线性系统动力学问题的一种新的有效方法.

Chebyshev polynomial approximation is applied to the period-doubling bifurcation problem of a stochastic van der Pol system with bounded random parameters and subjected to harmonic excitations. Firstly, the stochastic system is reduced to its equivalent deterministic one, through which the response of the stochastic system can be obtained by numerical methods. Nonlinear dynamical behavior related to various forms of stochastic period-doubling bifurcation in the stochastic system is explored. Numerical simulations show that similar to their counterpart in deterministic nonlinear system, various forms of period-doubling bifurcation may occur in the stochastic van der Pol system, but with some modified features. Numerical results also show that Chebyshev polynomial approximation can provide an effective approach to dynamical problems in stochastic nonlinear systems.