Dynamics of Temporal Dissipative Solitons in Time-Delayed Systems with Large Delay

Abstract

Dynamical effects in time-delayed systems with large delays exhibit a wide range of intriguing phenomena. In particular, the emergence of pulse-like solutions, known as Temporal Dissipative Solitons, has been a central point of research, as they are highly relevant in various domains such as optical and neural systems. This thesis is dedicated to understanding the bifurcations and instabilities of these kinds of solutions. By scrutinizing the singular limit of large delay, it is possible to derive a desingularized equation with an advanced argument, in which the solitons appear as homoclinic solutions. In our investigation, we show how classical homoclinic bifurcation theory can be employed as an effective tool to gain insights into the dynamics of these soliton solutions. We demonstrate our results with examples of the Morris-Lecar model under time-delayed feedback and a delay-algebraic model of the Kerr-Gires-Tournois interferometer. Our studies are supported by a numerical analysis using the path-continuation package DDE-BIFTOOL.