Small-world networks of Kuramoto oscillators

TL;DR

This paper investigates how small-world network structures influence the synchronization and pattern formation of Kuramoto oscillators, revealing the role of long-range connections in shaping stable states and spatial patterns.

Contribution

It introduces a detailed analysis of q-twisted states in the Kuramoto model on small-world graphs, combining stability analysis and numerical methods to understand pattern formation.

Findings

Long-range connections promote synchronization.

Increase in long-range links leads to plateau patterns.

Analysis of q-twisted states explains spatial pattern formation.

Abstract

The Kuramoto model of coupled phase oscillators on small-world (SW) graphs is analyzed in this work. When the number of oscillators in the network goes to infinity, the model acquires a family of steady state solutions of degree q, called q-twisted states. We show that this class of solutions plays an important role in the formation of spatial patterns in the Kuramoto model on SW graphs. In particular, the analysis of q-twisted elucidates the role of long-range random connections in shaping the attractors in this model. We develop two complementary approaches for studying q-twisted states in the coupled oscillator model on SW graphs: the linear stability analysis and the numerical continuation. The former approach shows that long-range random connections in the SW graphs promote synchronization and yields the estimate of the synchronization rate as a function of the SW randomization parameter. The continuation shows that the increase of the long-range connections results in patterns consisting of one or several plateaus separated by sharp interfaces. These results elucidate the pattern formation mechanisms in nonlocally coupled dynamical systems on random graphs.