Comparing the Locking Threshold for Rings and Chains of Oscillators

TL;DR

This paper investigates how the topology of oscillator arrays, specifically rings versus chains, influences their ability to synchronize, revealing that rings generally synchronize more easily but with some exceptions, supported by theoretical bounds.

Contribution

The study provides rigorous bounds on the ratio of locking thresholds between rings and chains for a class of Kuramoto-like models, highlighting topology's role in synchronization.

Findings

Rings typically have higher locking thresholds than chains.

Theoretical bounds are derived for the ratio of thresholds.

Synchronization behavior depends on boundary conditions and natural frequency distribution.

Abstract

We present a case study of how topology can affect synchronization. Specifically, we consider arrays of phase oscillators coupled in a ring or a chain topology. Each ring is perfectly matched to a chain with the same initial conditions and the same random natural frequencies. The only difference is their boundary conditions: periodic for a ring, and open for a chain. For both topologies, stable phase-locked states exist if and only if the spread or "width" of the natural frequencies is smaller than a critical value called the locking threshold (which depends on the boundary conditions and the particular realization of the frequencies). The central question is whether a ring synchronizes more readily than a chain. We show that it usually does, but not always. Rigorous bounds are derived for the ratio between the locking thresholds of a ring and its matched chain, for a variant of the Kuramoto model that also includes a wider family of models.