TL;DR
This paper investigates the conditions for synchronization in an array of identical oscillators with small vibrations, using a complex matrix approach to analyze eigenvalues related to the coupling matrices.
Contribution
It introduces a novel method combining real Laplacian matrices into a complex matrix to determine synchronization conditions.
Findings
Synchronization depends on eigenvalues of the complex matrix on the imaginary axis.
Refined conditions are provided for cases with weak or no restorative coupling.
The approach offers a new perspective on oscillator synchronization analysis.
Abstract
Synchronization is studied in an array of identical oscillators undergoing small vibrations. The overall coupling is described by a pair of matrix-weighted Laplacian matrices; one representing the dissipative, the other the restorative connectors. A construction is proposed to combine these two real matrices in a single complex matrix. It is shown that whether the oscillators synchronize in the steady state or not depends on the number of eigenvalues of this complex matrix on the imaginary axis. Certain refinements of this condition for the special cases, where the restorative coupling is either weak or absent, are also presented.
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