Model reduction for networks of coupled oscillators

TL;DR

This paper introduces a collective coordinate method to significantly reduce the complexity of Kuramoto oscillator networks, accurately capturing synchronization phenomena and finite size effects.

Contribution

The paper presents a novel collective coordinate approach that reduces high-dimensional oscillator networks to low-dimensional models, accurately describing synchronization transitions.

Findings

Successfully reproduces local and global synchronization onset.

Describes interaction of partially synchronized clusters.

Accurately captures finite size scaling of critical coupling.

Abstract

We present a collective coordinate approach to describe coupled phase oscillators. We apply the method to study synchronisation in a Kuramoto model. In our approach an N-dimensional Kuramoto model is reduced to an n-dimensional ordinary differential equation with n<<N, constituting an immense reduction in complexity. The onset of both local and global synchronisation is reproduced to good numerical accuracy, and we are able to describe both soft and hard transitions. By introducing 2 collective coordinates the approach is able to describe the interaction of two partially synchronised clusters in the case of bimodally distributed native frequencies. Furthermore, our approach allows us to accurately describe finite size scalings of the critical coupling strength. We corroborate our analytical results by comparing with numerical simulations of the Kuramoto model with all-to-all coupling networks for several distributions of the native frequencies.