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.