Effects of time delay on the collective behavior of globally coupled harmonic oscillators with fluctuating frequency

In the viscoelastic environment, coupling forces between particles often play an essential role. Due to the time delay in the transmission of coupling forces, this paper focuses on investigating the effects of time delay on the collective behaviors of coupled harmonic oscillators, including synchronization, stability, and stochastic resonance. Firstly, we establish a time delay decoupling formula that can complete the accurate solution of the model under a large delay, thus breaking the limitation of previous small delay approximation methods. Secondly, the statistical synchronization between particles is derived, which means the mean -field behavior of the system can be studied through single -particle behavior. Thirdly, the system's stability condition is given, demonstrating that the stability region expands with increasing time delay. Finally, the output response amplitude gain of the system is obtained, and then the stochastic resonance behavior of the system is studied. In the small delay region, noise competes with the ordered driving force, resulting in the classical stochastic resonance (CSR) behavior consistent with previous research findings. In the large delay region, periodic matching between the time delay and driving frequency gives rise to parameter -induced stochastic resonance (PSR) behavior, which has not been observed in previous studies.