WIAS Preprint No. 2824, (2021)

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


  • Yanchuk, Serhiy
  • Wolfrum, Matthias
    ORCID: 0000-0002-4278-2675
  • Pereira, Tiago
  • Turaev, Dmitry

2020 Mathematics Subject Classification

  • 34K20 34K06 34K08


  • Delay differential equations, absolute stability




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 timedelays. 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.

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