*Article* **Kuramoto Model with Delay: The Role of the Frequency Distribution**

**Vladimir V. Klinshov 1,2,3,4,\* and Alexander A. Zlobin 1,2,3**


**Abstract:** The Kuramoto model is a classical model used for the describing of synchronization in populations of oscillatory units. In the present paper we study the Kuramoto model with delay with a focus on the distribution of the oscillators' frequencies. We consider a series of rational distributions which allow us to reduce the population dynamics to a set of several delay differential equations. We use the bifurcation analysis of these equations to study the transition from the asynchronous to synchronous state. We demonstrate that the form of the frequency distribution may play a substantial role in synchronization. In particular, for Lorentzian distribution the delay prevents synchronization, while for other distributions the delay can facilitate synchronization.

**Keywords:** Kuramoto model; time delay; synchronization

**MSC:** 34K24
