Delay-Induced Switched States in a slow-fast system

TL;DR

This paper investigates delay-induced switching phenomena in a two-component slow-fast delay system, providing conditions for their existence and analyzing the effects of small parameters on these states.

Contribution

It introduces new conditions for the existence of delay-induced switched states and explores the limiting behavior as parameters approach zero.

Findings

Conditions for existence of switched states are established.

Analysis of the system's behavior as parameters tend to zero.

Insights into the delay-induced switching mechanism.

Abstract

We consider the two-component delay system $\varepsilon x^{\prime}(t)=-x(t)-y(t)+f(x(t-1)),$ $y^{\prime}(t)=\eta x(t)$ with small parameters $\varepsilon,\eta$, and positive feedback function $f$. Previously, such systems have been reported to model switching in optoelectronic experiments, where each switching induces another one after approximately one delay time, related to one round trip of the signal. In this paper, we study these delay-induced switched states. We provide conditions for their existence and show how the formal limits $\varepsilon\to0$ and/or $\eta\to0$ facilitate our understanding of this phenomenon.