  An equilibrium of a delay differential equation (DDE) is absolutely stable,
if it is locally asymptotically stable for all delays. We present criteria for
absolute stability of DDEs with discrete time-delays. In the case of a single
delay, the absolute stability is shown to be equivalent to asymptotic stability
for sufficiently large delays. Similarly, for multiple delays, the absolute
stability is equivalent to asymptotic stability for hierarchically large
delays. Additionally, we give necessary and sufficient conditions for a linear
DDE to be hyperbolic for all delays. The latter conditions are crucial for
determining whether a system can have stabilizing or destabilizing bifurcations
by varying time delays.
