The emergence of coherence in complex networks of heterogeneous dynamical systems

TL;DR

This paper develops a general theory for the emergence of coherence in heterogeneous dynamical systems on complex networks, extending classical models to more realistic scenarios with diverse individual dynamics and network structures.

Contribution

It introduces a unified framework to determine the critical coupling strength for synchronization in heterogeneous networks, generalizing the Kuramoto model to complex, realistic systems.

Findings

Critical coupling depends on individual dynamics and network eigenvalues.

The theory applies to networks with large, heterogeneous connections.

It extends synchronization analysis beyond phase oscillators to complex systems.

Abstract

We present a general theory for the onset of coherence in collections of heterogeneous maps interacting via a complex connection network. Our method allows the dynamics of the individual uncoupled systems to be either chaotic or periodic, and applies generally to networks for which the number of connections per node is large. We find that the critical coupling strength at which a transition to synchrony takes place depends separately on the dynamics of the individual uncoupled systems and on the largest eigenvalue of the adjacency matrix of the coupling network. Our theory directly generalizes the Kuramoto model of equal strength, all-to-all coupled phase oscillators to the case of oscillators with more realistic dynamics coupled via a large heterogeneous network.