Asymptotic analysis of the behavior of solutions to equations with large delay

Asymptotic analysis of the behavior of solutions to equations with large delays has been reported. Equations with delays are characterized by the presence of several specific effects and phenomena, which are caused by the infinite-dimensionality of the phase space. The behavior in a neighborhood of the zero equilibrium state of the solutions to the differential-difference equation was studied under the assumption that the delay time is sufficiently large. A method was developed for studying local dynamics in a neighborhood of the zero equilibrium state. It was established that the local dynamics for the equation with large delays is largely determined by the linearized equation.