Matrix coupling and generalized frustration in Kuramoto oscillators

TL;DR

This paper generalizes the Kuramoto model by introducing a coupling matrix, revealing new synchronization transitions and dynamic behaviors, including oscillatory and static phases, analyzed through the Ott-Antonsen ansatz.

Contribution

It introduces a matrix-based coupling generalization of the Kuramoto model, analyzing phase transitions and dynamic states using eigenvalue analysis and the Ott-Antonsen approach.

Findings

Eigenvalues determine the direction of the order parameter.

Complex eigenvalues cause oscillatory behavior of the order parameter.

Changing natural frequency averages induces additional phase transitions.

Abstract

The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a {\it frustrated} version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model, but written in terms of unit vectors. Replacing the coupling constant by a coupling matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with highest eigenvalue, when the eigenvalues are real. For complex eigenvalues the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for Lorentzian distribution of frequencies using the Ott-Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.