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Journal Article

Dynamics of a time-delayed relay system


Illing,  Lucas
External Organizations;

Ryan,  Pierce
External Organizations;


Amann,  Andreas
Potsdam Institute for Climate Impact Research;

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Illing, L., Ryan, P., Amann, A. (2024): Dynamics of a time-delayed relay system. - Physical Review E, 109, 1, 014223.

Cite as: https://publications.pik-potsdam.de/pubman/item/item_29576
We study the dynamics of a piecewise-linear second-order delay differential equation that is representative of feedback systems with relays (switches) that actuate after a fixed delay. The system under study exhibits strong multirhythmicity, the coexistence of many stable periodic solutions for the same values of the parameters. We present a detailed study of these periodic solutions and their bifurcations. Starting from an integrodifferential model, we show how to reduce the system to a set of finite-dimensional maps. We then demonstrate that the parameter regions of existence of periodic solutions can be understood in terms of discontinuity-induced bifurcations and their stability is determined by smooth bifurcations. Using this technique, we are able to show that slowly oscillating solutions are always stable if they exist. We also demonstrate the coexistence of stable periodic solutions with quasiperiodic solutions.