Robustness and asymptotic stability for the Winfree model on a general network under the effect of time-delay

Abstract

We study the robustness and asymptotic stability of the unique equilibrium for the Winfree model on a general network under the effect of small time-delay. Time-delay is a ubiquitous mechanism in classical and quantum many-body systems, and it is due to the finite speed of communication. In this paper, we further investigate the robustness of the equilibrium under the effects of small time-delay and general network topology. For this, we present several sufficient frameworks formulated in terms of the network topology, coupling strength, time-delay, and initial configuration. In the proposed framework, we present a positively invariant arc which contains the unique equilibrium independent of strength of time-delay. By using energy estimates, we show that the unique equilibrium in the invariant arc is asymptotically stable. We also provide several numerical simulations and compare them with analytical results.