Synchronization and scaling properties of chaotic networks with multiple delays

TL;DR

This paper investigates chaotic networks with multiple delays, revealing a hierarchical Lyapunov spectrum that influences synchronization and chaos types, with implications for network design and understanding complex delay systems.

Contribution

It introduces a hierarchical structure of Lyapunov exponents in multi-delay chaotic systems and links this to synchronization properties in hierarchical networks.

Findings

Lyapunov spectrum exhibits hierarchical structure based on delays.

Synchronization within subnetworks depends on short delay scaling.

Long-range synchronization requires maximal Lyapunov exponent to scale with long delays.

Abstract

We study chaotic systems with multiple time delays that range over several orders of magnitude. We show that the spectrum of Lyapunov exponents (LE) in such systems possesses a hierarchical structure, with different parts scaling with the different delays. This leads to different types of chaos, depending on the scaling of the maximal LE. Our results are relevant, in particular, for the synchronization properties of hierarchical networks (networks of networks) where the nodes of subnetworks are coupled with shorter delays and couplings between different subnetworks are realized with longer delay times. Units within a subnetwork can synchronize if the maximal exponent scales with the shorter delay, long range synchronization between different subnetworks is only possible if the maximal exponent scales with the long delay. The results are illustrated analytically for Bernoulli maps and numerically for tent maps.