Synchronization of unidirectional time delay chaotic networks and the greatest common divisor

TL;DR

This paper explores how the synchronization patterns in unidirectional chaotic networks are governed by the greatest common divisor of loop delays, revealing conditions for zero-lag and sublattice synchronization.

Contribution

It establishes a link between network delay structures and synchronization regimes, providing analytical and simulation evidence for the role of GCD in chaotic network synchronization.

Findings

GCD=1 leads to zero-lag chaos synchronization

GCD=m>1 results in m-sublattice synchronization

Homogeneous delays enable complete synchronization

Abstract

We present the interplay between synchronization of unidirectional coupled chaotic nodes with heterogeneous delays and the greatest common divisor (GCD) of loops composing the oriented graph. In the weak chaos region and for GCD=1 the network is in chaotic zero-lag synchronization, whereas for GCD=m>1 synchronization of m-sublattices emerges. Complete synchronization can be achieved when all chaotic nodes are influenced by an identical set of delays and in particular for the limiting case of homogeneous delays. Results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps.