**5. Conclusions**

In this paper we performed an analysis of the Kuramoto model with coupling delay paying special attention to the distribution of the oscillator frequencies *ω*. We used the method of Ott and Antonsen which allows one to reduce the system dynamics in the case of infinitely many oscillators. For the rational frequency distributions, the dynamics can be reduced to a set of delay-differential equations whose number equals the degree of the denominator. By the means of bifurcation analysis, we obtained the Andronov-Hopf bifurcation curves indicating a synchronization transition in the population and so constructed the synchronization border in the parameter plane. Our results have revealed

the different role of the delay for different frequency distributions. Thus, for the Lorentzian distribution, the delay always prevents synchronization by increasing the critical coupling strength. In contrast, for the distributions different from Lorentzian, the delay can promote synchronization: For certain delay values, the critical coupling turns out to be lower than in the delay-less case.

In the studies of collective dynamics of heterogeneous populations, it is typical to consider a Lorentzian distribution of local parameters. The reason for this choice is the simplicity of analytical treatment: For example, in our study the Lorentzian distribution with *n* = 1 leads to the reduced system (9) of just a single differential equation for the complex variable *z*1. The Lorentzian distribution is often treated as paradigmatic and qualitatively reflects the properties of an arbitrary unimodal distribution. However, our results show that the particular shape of the distribution can play a significant role in the system behaviour and synchronization. In particular, the role of the coupling delay turns out to be opposite for the Lorentzian and non-Lorentzian distributions.

In the end, we would like to emphasize that the present study is limited to the local stability analysis and does not consider the global stability of the asynchronous state. It means that the stable asynchronous state might coexist with a synchronous state in some parameter regions leading to bistability areas, as was demonstrated in Ref. [17] for the Lorentz frequency distribution. The emergence of bistability is associated with the subcritical Andronov-Hopf bifurcation, while the supercritical bifurcation supports monostability. The type of the bifurcation can be determined by the calculation of the first Lyapunov coefficient [54,55] which could be one of the directions of the further investigation. Other possibilities include consideration of a broader class of frequency distributions, including non-unimodal ones.

**Author Contributions:** Conceptualization, V.V.K.; Methodology, V.V.K.; Software, A.A.Z.; Formal analysis, V.V.K. and A.A.Z.; Investigation, V.V.K. and A.A.Z.; Writing—original draft, V.V.K.; Supervision, V.V.K.; Funding acquisition, V.V.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The derivation of the reduced system was supported by the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-15-2022-287. The bifurcation analysis of the reduced system and the numerical simulations were supported by the Russian Science Foundation, grant no. 19-72-10114.

**Data Availability Statement:** The data that support the findings of this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.
