Non-rough Relaxation Solutions of a System with Delay and Sign-Changing Nonlinearity

Abstract

In this paper we study nonlocal dynamics of a system of two coupled delay differential equations with a sign-changing compactly supported nonlinearity. The main assumptions in the problem are that nonlinearity is multiplied by a large parameter and a coupling coefficient is sufficiently small. Using asymptotic methods we investigate the existence of relaxation periodic solutions of a given system. We choose a special set in the phase space of the initial system. Then we calculate asymptotics of all solutions of the considered system with initial conditions from a chosen set. By this asymptotics we build a special mapping. Cycles of this mapping correspond to periodic asymptotic (by the discrepancy) solutions of the initial system. Constructed mapping has a two-parameter families of non-rough cycles. Thus, the initial system has two-parameter families of non-rough inhomogeneous relaxation periodic asymptotic (by the discrepancy) solutions.