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 𝛼 and the strength 𝛾 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 𝛾 and 𝛼, 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 𝛼 with period $2\pi /\omega$ where 𝜔 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.
Volume 95, 2021
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