Dynamics of the system of delay differential equations with nonlinearity having a simple behavior at infinity

In this paper, we study the dynamics of a system of nonlinear differential equations with delay. We find stable equilibrium states and regions of attraction to them in the phase space of the system, as well as stable and unstable homogeneous and inhomogeneous cycles. We find conditions on the parameters of the system for multistability. We show that the coupling parameter has a decisive influence on the dynamics of the system. We find regions of the parameters of the system and extensive sets of initial conditions such that if we take these values of the parameters and any initial conditions from these sets, the system will have simple dynamics.