the oscillators can synchronize their frequencies and exhibit a stable pattern of oscillations. Time-delay coupling can also lead to frequency modulation and chaos in nonlinear dynamical systems [ Wernecke:2019 ]. This can happen when the delay is long enough to cause significant changes in the oscillators’ phases and amplitudes. In this case， Studying the delayed Kuramoto model allows for a more realistic representation of many real-world distributed systems. Analyzing the effects of time delays on the properties。

and bifurcation behavior of the synchronization provides insights into the dynamics of these systems and can thus help us to understand and predict their behaviors under more realistic conditions. Time-delay coupling can have significant effects on the frequency of oscillations[ Hsia:2020 ]. Time-delay coupling can induce oscillations that are not present in the system without delay [ Pyragas:2003 ，。

in which the frequency of one oscillator is entrained to that of another oscillator with a slightly different frequency. This can happen when the delay is tuned to a specific value that corresponds to the natural frequency difference between the oscillators. In this case， a single oscillator with delayed feedback can exhibit oscillations with a frequency that is higher than or lower than the natural frequency of the oscillator. These delay-induced oscillations can be stable or unstable， Deco:2009 ]. For example， stability， the oscillations can become irregular and exhibit a wide range of frequencies. New insights of the dependence of collective frequency with time-delay and its distribution over the oscillator networks can help us further understand the synchronization patterns that can emerge in distributed systems like the neuronal systems in the brain in various short-term and long-term conditions [ Pathak:2022 ]. ， depending on the system’s parameters. Time-delay coupling can also lead to resonance and frequency entrainment。

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