中文摘要：

基于正交多项式逼近理论,研究了在不同随机参数作用下参激双势阱Duffing系统的随机动力学行为.首先,借助Poincaré（庞加莱）截面分析系统的复杂动力学行为;其次,分别针对系统非线性项系数和阻尼项系数为随机参数的情况,运用正交多项式逼近法,将随机参数Duffing系统转化为与之等价的确定性扩阶系统,并证明其有效性;最后,运用等价确定性扩阶系统的集合平均响应,揭示随机系统的动力学特性,以及随机变量强度变化对系统产生的影响.数值结果表明,对于多吸引子共存情形,参激双势阱Duffing系统在随机非线性项系数影响下,其动力学行为较为稳定,共存吸引子与确定性情形保持一致;而当阻尼系数为随机参数时,随着随机变量强度的增加,部分共存吸引子将发生分岔现象.

英文摘要：

Based on the orthogonal polynomial approximation theory,the stochastic dynamical behaviors of double-well Duffing systems under random parametric excitations were investigated. Firstly,the complex dynamical behaviors of deterministic Duffing systems were studied by means of the Poincaré sections.Then,the Duffing systems with random stiffnesses and damping parameters were reduced to equivalent deterministic expanded-order systems,and the effectiveness of this approximation method was proved. Thus,the ensemble-mean responses of the equivalent systems were applied to reveal the stochastic dynamical properties and the effects of the random variable intensity on the double-well Duffing systems. The numerical simulation results indicate that,in the case of coexistent attractors,the double-well Duffing system with random stiffness parameters has the similar stable dynamical behaviors to those in deterministic cases. However,for the Duffing system with random damping parameters,during the increase of the random variable intensity,the bifurcation phenomena occur to some coexistent attractors.