Spatio-temporal representation of long-delayed systems: an alternative approach

TL;DR

This paper explores an alternative approach to representing long-delayed dynamical systems in a spatio-temporal framework, improving physical interpretation and simplifying the derivation of normal forms across various examples.

Contribution

It extends a recent alternative interpretation method, enhancing the understanding and analysis of long-delayed systems in a spatio-temporal context.

Findings

New interpretation improves physical relevance of variables

Simplifies normal form derivation for delayed systems

Applicable to multiple paradigmatic examples

Abstract

Dynamical systems with long delay feedback can exhibit complicated temporal phenomena, which once re-organized in a two-dimensional space are reminiscent of spatio-temporal behavior. In this framework, normal forms description have been developed to reproduce the dynamics and the opportunity to treat the corresponding variables as true space and time has been since established. However, recently an alternative approach has been proposed in Ref. \cite{Marino2018} with a different interpretation of the variables involved, which takes better into account their physical character and allows for an easier determination of the normal forms. In this paper, we extend such idea and apply it to a number of paradigmatic examples, paving the way to a re-thinking of the concept of spatio-temporal representation of long-delayed systems.