Linearization error in synchronization of Kuramoto oscillators

摘要

Synchronization among a set of networked nodes has attracted much attention in different fields. This paper thoroughly investigates linear formulation of the Kuramoto model, with and without frustration, for an arbitrarily weighted undirected network where all nodes may have different intrinsic frequencies. We develop a mathematical framework to estimate errors of the linear approximation for globally and locally coupled networks. We mathematically prove that the eigenvector corresponding to the largest eigenvalue of the network’s Laplacian matrix is enough for examining synchrony alignment and that the functionality of this vector depends on the corresponding eigenvalue. Moreover, we prove that if a globally coupled network with frustration has perfect phase synchronization when its coupling strength tends to infinity, it is a regular network. Finally, the effect of correlation between frustration values and degrees (or frequencies) on the synchronizability of the network is investigated.