Dependence of the dynamics of a model of coupled oscillators on the number of oscillators

In this paper we study the nonlocal dynamics of a model describing $N$ coupled oscillators with delay. Studying the asymptotics of solutions of the original system is reduced to studying the dynamics of a simpler mapping. It is shown that, for positive values of the coupling parameter in the considered model, the oscillators are synchronized. For negative values of the coupling parameter, the asymptotics of the solutions of the system depends significantly on the parity of the number $N$: for even $N$, two-cluster synchronization is observed, and, for odd $N$, the dynamics of the model is more complicated.