Synchronization of linear oscillators coupled through dynamic networks with interior nodes

TL;DR

This paper investigates the conditions under which a network of identical linear oscillators, connected via a dynamic network with interior nodes, achieves synchronization, focusing on the spectral properties of the Laplacian matrix.

Contribution

It provides a necessary and sufficient spectral condition involving the Schur complement of the Laplacian matrix for synchronization in networks with interior nodes.

Findings

Oscillators synchronize if the Schur complement of the Laplacian has a single eigenvalue on the imaginary axis.

The study extends synchronization analysis to networks with interior nodes and mixed coupling types.

Spectral conditions are derived for asymptotic synchronization in complex dynamic networks.

Abstract

Synchronization is studied in an array of identical linear oscillators of arbitrary order, coupled through a dynamic network comprising dissipative connectors (e.g., dampers) and restorative connectors (e.g., springs). The coupling network is allowed to contain interior nodes, i.e., those that are not directly connected to an oscillator. It is shown that the oscillators asymptotically synchronize if and only if the Schur complement (with respect to the boundary nodes) of the complex-valued Laplacian matrix representing the coupling has a single eigenvalue on the imaginary axis.