Synchronization of small oscillations

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AUTOMATICA, vol.107, pp.154-161, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 107
  • Publication Date: 2019
  • Doi Number: 10.1016/j.automatica.2019.05.047
  • Journal Name: AUTOMATICA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.154-161
  • Keywords: Synchronization, Matrix-weighted Laplacian matrix, Small oscillations, COUPLED HARMONIC-OSCILLATORS, COMPLEX NETWORKS, CONSENSUS
  • Middle East Technical University Affiliated: Yes


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 depend 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. (C) 2019 Elsevier Ltd. All rights reserved.