On the asymptotic stability of solutions of nonlinear systems with delay

Under study are systems of homogeneous differential equations with delay. We assume that in the absence of delay the trivial solutions to the systems under consideration are asymptotically stable. Using the direct Lyapunov method and Razumikhin's approach, we show that if the order of homogeneity of the right-hand sides is greater than 1 then asymptotic stability persists for all values of delay. We estimate the time of transitions, study the influence of perturbations on the stability of the trivial solution, and prove a theorem on the asymptotic stability of a complex system describing the interaction of two nonlinear subsystems.