Response of nonlinear system to random narrow-band excitation with time delay state feedback

The principal resonance response of a Duffing 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.