Synchronization of Kuramoto Oscillators: Inverse Taylor Expansions

TL;DR

This paper introduces a power series expansion method for efficiently approximating the synchronized state in Kuramoto oscillator networks, providing a hierarchy of tests for synchronization with proven convergence and improved accuracy.

Contribution

It presents a novel transcription of the equilibrium equation and develops a power series approach with convergence guarantees, advancing synchronization analysis methods.

Findings

The series approximation accurately predicts synchronization states.

The hierarchy of tests generalizes existing synchronization criteria.

Numerical results show improved efficiency over iterative methods.

Abstract

Synchronization in networks of coupled oscillators is a widely studied topic with extensive scientific and engineering applications. In this paper, we study the frequency synchronization problem for networks of Kuramoto oscillators with arbitrary topology and heterogeneous edge weights. We propose a novel equivalent transcription for the equilibrium synchronization equation. Using this transcription, we develop a power series expansion to compute the synchronized solution of the Kuramoto model as well as a sufficient condition for the strong convergence of this series expansion. Truncating the power series provides (i) an efficient approximation scheme for computing the synchronized solution, and (ii) a simple-to-check, statistically-correct hierarchy of increasingly accurate synchronization tests. This hierarchy of tests provides a theoretical foundation for and generalizes the best-known approximate synchronization test in the literature. Our numerical experiments illustrate the accuracy and the computational efficiency of the truncated series approximation compared to existing iterative methods and existing synchronization tests.