Universal scaling and phase transitions of coupled phase oscillator populations

TL;DR

This paper develops a general analytical framework for understanding the critical scaling and phase transitions in coupled oscillator populations, extending classical results of the Kuramoto model to diverse frequency distributions.

Contribution

It introduces a self-consistent approach and characteristic function method to analytically capture the critical scaling and phase transition types in coupled oscillator systems.

Findings

Identifies various phase transition types toward synchronization.

Establishes scaling relations for the order parameter near criticality.

Shows that the geometric properties of the characteristic function determine scaling behavior.

Abstract

The Kuramoto model, which serves as a paradigm for investigating synchronization phenomenon of oscillatory system, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here, we generalize a number of classical results by presenting a general framework for capturing, analytically, the critical scaling of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determines the scaling properties of order parameter above the criticality.