On the asymptotic stability of solutions of nonlinear systems with delay (Q1760514)

The authors consider two systems of differential equations: ordinary \[ \dot X(t)=F(X(t)),\quad F(0)=0\eqno(1) \] and retarded \[ \dot X(t)=F(X(t-\tau)),\eqno(2) \] where \(X\) is an \(n\)-dimensional vector and the components of \(F(X)\) are homogeneous functions of order \(\mu\geq1\). In the case \(\mu>1\), they prove that the trivial solution of (2) is asymptotically stable for all \(\tau>1\) if the zero solution of equations (1) is asymptotically stable. They 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.