Temporal dissipative solitons in the Morris--Lecar model with time-delayed feedback

Abstract

We study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability we use the Morris--Lecar model. In the limit of large delay soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show, how a homoclinic orbit flip of a single pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris--Lecar model.