Harmonic synchronization under all three types of coupling: position, velocity, and acceleration

TL;DR

This paper investigates the synchronization of harmonic oscillators coupled through position, velocity, and acceleration, providing spectral conditions and Laplacian constructions for achieving asymptotic synchronization.

Contribution

It generalizes spectral tests for synchronization to include all three types of coupling and offers new Laplacian matrix constructions under structural conditions.

Findings

Synchronization occurs if the complex Laplacian has a single eigenvalue on the imaginary axis.

New spectral conditions for synchronization with combined coupling types.

Simpler Laplacian constructions work under specific structural conditions.

Abstract

Synchronization of identical harmonic oscillators interconnected via position, velocity, and acceleration couplings is studied. How to construct a complex Laplacian matrix representing the overall coupling is presented. It is shown that the oscillators asymptotically synchronize if and only if this matrix has a single eigenvalue on the imaginary axis. This result generalizes some of the known spectral tests for synchronization. Some simpler Laplacian constructions are also proved to work provided that certain structural conditions are satisfied by the coupling graphs.