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

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.


Synchronization of linearly and nonlinearly coupled harmonic oscillators
Penbegül, Ali Yetkin; Tuna, Sezai Emre; Department of Electrical and Electronics Engineering (2011)
In this thesis, the synchronization in the arrays of identical and non-identical coupled harmonic oscillators is studied. Both linear and nonlinear coupling is considered. The study consists of two main parts. The first part concentrates on theoretical analysis and the second part contains the simulation results. The first part begins with introducing the harmonic oscillators and the basics of synchronization. Then some theoretical aspects of synchronization of linearly and nonlinearly coupled harmonic osci...
Structural Analysis of Synchronization in Networks of Linear Oscillators
Tuna, Sezai Emre (2022-07-01)
In undirected networks of identical linear oscillators coupled through dissipative and restorative connectors (e.g., pendulums undergoing small vibrations connected via dampers and springs), the relation between asymptotic synchronization and coupling structure is studied. Conditions on the interconnection under which synchronization can be achieved for some selection of coupling strengths are established. How to strengthen those conditions so that synchronization is guaranteed for all admissible parameter ...
Synchronization of oscillators not sharing a common ground
Tuna, Sezai Emre (2023-5-01)
Networks of coupled LC oscillators that do not share a common ground node are studied. Both resistive coupling and inductive coupling are considered. For networks under resistive coupling, it is shown that the oscillator-coupler interconnection has to be bilayer if the oscillator voltages are to asymptotically synchronize. Also, for bilayer architecture (when both resistive and inductive couplers are present) a method is proposed to compute a complex-valued effective Laplacian matrix that represents the ove...
Synchronization analysis of coupled Lienard-type oscillators by averaging
Tuna, Sezai Emre (2012-08-01)
Sufficient conditions for the synchronization of coupled Lienard-type oscillators are investigated via averaging technique. The coupling considered here is fixed, nonsymmetric, and nonlinear. Under the assumption that the interconnection topology defines a connected graph, it is shown that the solutions of oscillators converge arbitrarily close to each other, starting from initial conditions arbitrarily far apart, provided that the frequency of oscillations is large enough and the initial phases of oscillat...
Synchronization of nonlinearly coupled harmonic oscillators
Cai, Chaohong; Tuna, Sezai Emre (2010-07-02)
Synchronization of coupled harmonic oscillators is investigated. Coupling considered here is pairwise, unidirectional, and described by a nonlinear function (whose graph resides in the first and third quadrants) of some projection of the relative distance (between the states of the pair being coupled) vector. Under the assumption that the interconnection topology defines a connected graph, it is shown that the synchronization manifold is semiglobally practically asymptotically stable in the frequency of osc...
Citation Formats
S. E. Tuna, “Synchronization of harmonic oscillators under restorative coupling with applications in electrical networks,” AUTOMATICA, pp. 236–243, 2017, Accessed: 00, 2020. [Online]. Available: