Dynamics of one model with delay and a large parameter

In this paper, we consider a system of two differential equations with delay and a finite nonlinearity. Using a special asymptotic method, we examine the existence and stability of relaxation periodic solutions of the system under the assumption that the positive coefficient of the finite nonlinearity is sufficiently large. This method allows one to reduce the problem on the behavior of solutions whose initial conditions lie in a certain set of the phase space of the original infinite-dimensional system to the study of the dynamics of a certain three-dimensional mapping. We prove that rough cycles of this mapping correspond to relaxation periodic solutions of the original system with the same stability. By stable cycles of the mapping constructed, we find exponentially orbitally stable relaxation cycles of the original system.