Analysis of Oscillator Ensemble with Dynamic Couplings

Abstract

Challenges to analyze networks of weakly coupled oscillators are considered. The model of the oscillator ensemble with the frequency-dependent couplings has been developed. The construction of the model is based on special locking functions obtained by the analysis of synchronized oscillator under external excitation. It is shown that the set of locking functions completely define coupling functions of well-known Kuramoto model. The ensemble interconnections are presented by linear time-invariant dynamical systems that correspond to the frequency-dependent transfer functions included in the model. The order of the resulting system is equal to the number of oscillators, and it does not depend on orders of transfer functions. Two approaches to take into account the frequency dependence of couplings are proposed—by the average frequency and by instantaneous frequencies. The simplified form of the model for weakly nonlinear sinusoidal oscillators is derived. Presented numerical examples illustrate the usage of the model and demonstrate some new effects arising in the oscillator ensemble with the dynamic couplings.