Coherence in complex networks of oscillators

TL;DR

This paper investigates the stability of synchronized states in complex networks of chaotic oscillators, analyzing different topologies and coupling schemes, and finds that coherence depends on network structure and coupling heterogeneity.

Contribution

It provides a comprehensive analytical and numerical analysis of synchronization stability in various scale-free network topologies of chaotic oscillators, highlighting the role of eigenvalues and coupling heterogeneity.

Findings

Synchronization stability scales with node connectivity.

Heterogeneous coupling enables coherence in deterministic networks.

Transition to coherence is of first-order.

Abstract

We study fully synchronized (coherent) states in complex networks of chaotic oscillators, reviewing the analytical approach of determining the stability conditions for synchronizability and comparing them with numerical criteria. As an example, we present detailed results for networks of chaotic logistic maps having three different scale-free topologies: random scale-free topology, deterministic pseudo-fractal scale-free network and Apollonian network. For random scale-free topology we find that the lower boundary of the synchronizability region scales approximately as $k^{-\mu}$, where $k$ is the outgoing connectivity and $\mu$ depends on the local nonlinearity. For deterministic scale-free networks coherence is observed only when the coupling is heterogeneous, namely when it is proportional to some power of the neighbor connectivity. In all cases, stability conditions are determined from the eigenvalue spectrum of the Laplacian matrix and agree well with numerical results based on histograms of coherent states in parameter space. Additionally, we show that almost everywhere in the synchronizability region the basin of attraction of coherent states fills the entire phase space, and that the transition to coherence is of first-order.