Synchronized chaos in networks of simple units

TL;DR

This paper investigates how synchronization in networks of simple units with arbitrary, possibly asymmetric and weighted connections can lead to emergent complex behaviors like chaos, contrasting with the units' individual dynamics.

Contribution

It introduces a synchronization criterion based on the spectrum of the generalized graph Laplacian, applicable to diverse network topologies and dynamics.

Findings

Chaos can emerge in synchronized states despite simple individual dynamics.

Chaotic units can synchronize to produce simple collective behavior.

The criterion links network structure and unit dynamics to synchronization outcomes.

Abstract

We study synchronization of non-diffusively coupled map networks with arbitrary network topologies, where the connections between different units are, in general, not symmetric and can carry both positive and negative weights. We show that, in contrast to diffusively coupled networks, the synchronous behavior of a non-diffusively coupled network can be dramatically different from the behavior of its constituent units. In particular, we show that chaos can emerge as synchronized behavior although the dynamics of individual units are very simple. Conversely, individually chaotic units can display simple behavior when the network synchronizes. We give a synchronization criterion that depends on the spectrum of the generalized graph Laplacian, as well as the dynamical properties of the individual units and the interaction function. This general result will be applied to coupled systems of tent and logistic maps and to two models of neuronal dynamics. Our approach yields an analytical understanding of how simple model neurons can produce complex collective behavior through the coordination of their actions.