Reviving Delayed Dynamics

Abstract

We introduce a delay differential equation that manifests a distinctive dynamical behavior. Specifically, the transient dynamics of this equation demonstrate a unique ``reviving" amplitude phenomenon within certain ranges of delay values. In this intriguing phenomenon, the amplitude initially decreases towards a fixed point until a specific time point, after which it ultimately diverges. Our analysis encompasses both analytical and numerical approaches, incorporating an approximation using the Lambert W function. The derived approximate solution effectively captures the qualitative aspects of the reviving dynamics across various delay values.