Synchronization of oscillators with long range power law interactions

TL;DR

This paper analyzes how oscillators with long-range power law interactions synchronize, identifying a critical exponent at which a transition occurs, supported by analytical and numerical methods.

Contribution

It introduces a critical power law exponent for synchronization transition and extends the analysis to higher dimensions using spin-wave and coarse graining techniques.

Findings

Critical exponent for synchronization transition: 1.5

Synchronization depends on the power law exponent $\alpha$

Results applicable to higher dimensions

Abstract

We present analytical calculations and numerical simulations for the synchronization of oscillators interacting via a long range power law interaction on a one dimensional lattice. We have identified the critical value of the power law exponent $\alpha_c$ across which a transition from a synchronized to an unsynchronized state takes place for a sufficiently strong but finite coupling strength in the large system limit. We find $\alpha_c=3/2$. Frequency entrainment and phase ordering are discussed as a function of $\alpha \geq 1$. The calculations are performed using an expansion about the aligned phase state (spin-wave approximation) and a coarse graining approach. We also generalize the spin-wave results to the {\it d}-dimensional problem.