Nonlinear resonance in Duffing oscillator with fixed and integrative time-delayed feedbacks

We study the nonlinear resonance, one of the fundamental phenomena in nonlinear oscillators, in a damped and periodically-driven Duffing oscillator with two types of time-delayed feedbacks, namely, fixed and integrative. Particularly, we analyse the effect of the time-delay parameter alpha and the strength gamma of the time-delayed feedback. Applying the perturbation theory we obtain a nonlinear equation for the amplitude of the periodic response of the system. For a range of values of gamma and alpha, the response amplitude is found to be higher than that of the system in the absence of delayed feedback. The response amplitude is periodic on the parameter alpha with period 2 pi/omega where omega is the angular frequency of the external periodic force. We show the occurrence of multiple branches of the response amplitude curve with and without hysteresis.