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The role of restorative coupling on synchronization of coupled identicalharmonic oscillators is studied. Necessary and sufficient conditions, underwhich the individual systems' solutions converge to a common trajectory, arepresented. Through simple physical examples, the meaning and limitations of thetheorems are expounded. Also, to demonstrate their versatility, the results areextended to cover LTI passive electrical networks. One of the extensionsgeneralizes the well-known link between the asymptotic stability of thesynchronization subspace and the second smallest eigenvalue of the Laplacianmatrix.

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