**Alexandra Kashchenko**

Mathematical Department, P.G. Demidov Yaroslavl State University, Yaroslavl 150003, Russia; sa-ahr@yandex.ru Received: 31 August 2020; Accepted: 12 October 2020; Published: 15 October 2020

**Abstract:** In this paper, we study the nonlocal dynamics of a system of delay differential equations with large parameters. This system simulates coupled generators with delayed feedback. Using the method of steps, we construct asymptotics of solutions. By these asymptotics, we construct a special finite-dimensional map. This map helps us to determine the structure of solutions. We study the dependence of solutions on the coupling parameter and show that the dynamics of the system is significantly different in the case of positive coupling and in the case of negative coupling.

**Keywords:** relaxation mode; delay differential equation; large parameter; asymptotics

**MSC:** 34K13; 34K25
