有界噪声激励下带有时滞反馈的随机Mathieu-Duffing系统的响应

研究了参数激励下带有时滞反馈的随机Mathieu-Duffing方程的主参数共振响应问题.运用多尺度方法分离了系统的快慢变量.分析了系统的分岔性质，发现调谐参数、时滞、时滞项的系数以及非线性项的强度等都可以影响系统的分岔行为，适当选择这些参数可以改变系统的分岔响应.同时，还讨论了非零解的稳定性，得到了非零解稳定的充要条件，而且发现在随机激励的带宽较小时，系统的多解现象仍然存在，分岔和跳跃现象仍会发生，数值模拟验证了理论推导的有效性.关键词：随机Mathieu-Duffing系统多尺度稳定性分岔

Response to bounded noise excitation of stochastic Mathieu-Duffing system with time delay state feedback

We investigate the principal parametric resonance of Mathieu-Duffing Equation under a narrow-band random excitation with time delay feedback. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The bifurcation of the system is discussed. We find that the bifurcation can be influenced by the detuning parameter, time delay, and the intensity of the non-linear term, and an appropriate choice of these parameters can change the response of bifurcation. In addition the stability of nontrivial solution is studied. The nontrivial solution of necessary and sufficient condition for stability is obtained. Moreover, we find that when the bandwidth of the random excitation is smaller, the multi-solution phenomenon still exists, and bifurcation and jumping phenomenon will occur. Theoretical analysis is verified by numerical results.