Analysis of Nonlinear Synchronization Dynamics of Oscillator Networks by Laplacian Spectral Methods

TL;DR

This paper investigates how the topology of oscillator networks influences their synchronization behavior, using Laplacian spectral methods to analyze dynamics on scale-free networks with varying clustering coefficients.

Contribution

It introduces a spectral approach using Laplacian eigenvectors to analyze nonlinear synchronization dynamics far from the manifold, highlighting the impact of network clustering.

Findings

High clustering inhibits full synchronization due to low-lying Laplacian modes.

Low clustering facilitates partial synchronization through high-eigenvalue modes.

Laplacian spectrum features correlate with synchronization patterns.

Abstract

We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronizationat low coupling noted elsewhere is associated with high-eigenvalue modes.