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中文摘要：
研究了窄带激励下带有时滞反馈的Duffing振子的主共振响应。用多尺度法分离了系统的快慢变量。分析了稳态响应的稳定性和分叉，研究了时滞、调谐参数、噪声带宽和幅值对系统的影响。证明了由于时滞的存在，系统将表现出复杂的动力学行为：时滞会导致分叉、时滞会影响跳跃区域等;发现噪声幅值会导致系统多解或分叉现象的出现，且随着噪声带宽的增大系统非零稳态响应从-极限环变为-扩散的极限环。最后，给出了数值模拟。
英文摘要：
The principal resonance response of a Dulling oscillator subject to a random narrow-band excitation with delayed feedback is investigated in this paper. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The stability and bifurcation of the steady state response are studied by means of qualitative analyses. And the effects of delay, detuning, bandwidth and magnitude of random excitation on dynamics of the original system are investigated. The results show that the complex dynamics such as bifurcation, jump domain and so on are induced by time delay and the phenomena that multiple solutions or bifurcation are induced by noise. Moreover, as the bandwidth of noise increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. All the results are verified by numerical simulation and well agreements are obtained.
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