A Family of Non-Rough Cycles in a System of Two Coupled Delayed Generators

Abstract

In this paper we consider the nonlocal dynamics of the model of two coupled oscillators with delayed feedback. This model has the form of a system of two differential equations with delay. The feedback function is non-linear, compactly supported and smooth. The main assumption in the problem is that the coupling between the generators is sufficiently small. With the help of asymptotic methods we investigate the existence of relaxation periodic solutions of a given system. For this purpose, a special set is constructed in the phase space of the original system. Then we build an asymptotics of the solutions of the given system with initial conditions from this set. Using this asymptotics, a special mapping is constructed. Dynamics of this map describes the dynamics of the original problem in general. It is proved that all solutions of this mapping are non-rough cycles of period two. As a result, we formulate conditions for the coupling parameter such that the initial system has a two-parameter family of non-rough inhomogeneous relaxation periodic asymptotic (with respect to the residual) solutions.