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

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Yanchuk,  Serhiy
Potsdam Institute for Climate Impact Research;

Wolfrum,  Matthias
External Organizations;

Pereira,  Tiago
External Organizations;

Turaev,  Dmitry
External Organizations;

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Zitation

Yanchuk, S., Wolfrum, M., Pereira, T., Turaev, D. (2022): Absolute stability and absolute hyperbolicity in systems with discrete time-delays. - Journal of Differential Equations, 318, 323-343.
https://doi.org/10.1016/j.jde.2022.02.026


Zitierlink: https://publications.pik-potsdam.de/pubman/item/item_26836
Zusammenfassung
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.