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

TL;DR

This paper establishes criteria for absolute stability and hyperbolicity in delay differential equations with discrete delays, linking stability for all delays to asymptotic stability at large delays and providing conditions for delay-independent hyperbolicity.

Contribution

It introduces new criteria for absolute stability and hyperbolicity in DDEs, connecting stability across all delays to asymptotic stability at large delays and offering delay-independent hyperbolicity conditions.

Findings

Absolute stability is equivalent to asymptotic stability for large delays.

Conditions for delay-independent hyperbolicity are provided.

Criteria help predict bifurcations caused by delay variations.

Abstract

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.