Complete synchronization equivalence in asynchronous and delayed coupled maps

TL;DR

This paper demonstrates that asynchronous updating and delayed coupling in coupled map lattices lead to the same fully synchronized states, unifying these two approaches under a common framework for stability analysis.

Contribution

It proves, through numerical and analytical methods, that delayed dynamics and asynchronous updates are equivalent in producing complete synchronization in coupled map lattices.

Findings

Delayed dynamics and asynchronous updates yield identical synchronized states.

A unified stability condition for complete synchronization is established.

The results bridge different updating schemes in coupled map lattice models.

Abstract

Coupled map lattices are paradigmatic models of many collective phenomena. However, quite different patterns can emerge depending on the updating scheme. While in early versions, maps were updated synchronously, there has been in recent years a concern to consider more realistic updating schemes where elements do not change all at once. Asynchronous updating schemes and the inclusion of time delays are seen as more realistic than the traditional parallel dynamics, and, in diverse works, either one or the other has been implemented separately. But are they actually distinct cases? For coupled map lattices with adjustable range of interactions, we prove, using both numerical and analytical tools, that an adequate delayed dynamics leads to the same completely synchronized states as an asynchronous update, providing a unified framework to identify the stability conditions for complete synchronization.