Synchronization of harmonic oscillators under restorative coupling with applications in electrical networks

TL;DR

This paper investigates how restorative coupling influences the synchronization of identical harmonic oscillators, providing conditions for convergence and extending results to electrical networks, with practical examples illustrating the concepts.

Contribution

It offers necessary and sufficient conditions for synchronization under restorative coupling and extends these results to linear passive electrical networks, linking stability to Laplacian eigenvalues.

Findings

Conditions for synchronization convergence are established.

Results are extended to electrical networks with stability linked to Laplacian eigenvalues.

Physical examples illustrate the theoretical results.

Abstract

The role of restorative coupling on synchronization of coupled identical harmonic oscillators is studied. Necessary and sufficient conditions, under which the individual systems' solutions converge to a common trajectory, are presented. Through simple physical examples, the meaning and limitations of the theorems are expounded. Also, to demonstrate their versatility, the results are extended to cover LTI passive electrical networks. One of the extensions generalizes the well-known link between the asymptotic stability of the synchronization subspace and the second smallest eigenvalue of the Laplacian matrix.