Absolute stability and absolute hyperbolicity in systems with discrete time-delays (Q2670016)

It is well known that, under variations of the delay time, the equilibria of a delay differential equation (DDE) do not change, but their stability properties may change. If the stability of an equilibrium remains unchanged for all delays, the equilibrium of a DDE is called being absolutely stable. The paper considers linear DDEs \[ \dot{x}(t)=A_0x(t)+\sum^m_{k=1}A_kx(t-\tau_k), \] and obtains that the absolute stability is equivalent to asymptotic stability for hierarchically large delays \(1\ll \tau_1\ll...\ll\tau_m\). Additionally, the authors give necessary and sufficient conditions of hyperbolicity for the DDEs, which implies the bifurcation condition of stability along with varying time delays.