SYNCHRONIZATION IN NETWORKS OF COUPLED OSCILLATORS

Abstract

We study different aspects of synchronization in networks of coupled oscillators:

We adapt a previous model and analysis method (the master stability function),

extensively used for

studying the stability of the synchronous state of networks of identical chaotic oscillators,

to the case of oscillators that are similar but not exactly identical.

We find that bubbling induced desynchronization bursts occur for some parameter values. These

bursts have spatial patterns, which can be predicted from the

network connectivity matrix and the unstable periodic orbits embedded in the attractor.

We test the analysis of bursts by comparison with numerical experiments.

In the case that no bursting occurs, we discuss the deviations from the exactly synchronous state caused

by the mismatch between oscillators.

We present a method to determine the relative parameter mismatch in a collection of nearly identical

chaotic oscillators by measuring large deviations from the synchronized state.

We demonstrate our method with an ensemble of slightly different circle maps.

We discuss how to apply our method when there is noise, and show an example where the noise intensity is comparable to the mismatch.

We consider a ring of identical or near identical coupled periodic oscillators in which the connections

have randomly heterogeneous strength. We use the

master stability function method to determine the possible patterns at the desynchronization transition that occurs

as the coupling strengths are

increased. We demonstrate Anderson localization of the modes of instability, and show that such localized

instability generates waves of desynchronization that spread to the whole array. Similar results should apply

to other networks with regular topology and heterogeneous connection strengths.

We study the transition from incoherence to coherence in large networks of coupled phase oscillators.

We present various approximations that describe the behavior of an appropriately defined order parameter

past the transition, and generalize recent results for the critical coupling strength.

We find that, under appropriate conditions,

the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix.

We show how, with an additional assumption, a mean field approximation recently proposed

is recovered from our results. We test our theory with numerical simulations, and find that it describes the transition when

our assumptions are satisfied.

We find that our theory describes the transition well in situations in which the mean field approximation fails.

We study the finite size effects caused by nodes with small degree and find that they cause the critical coupling strength

to increase.