Synchronization of Coupled Oscillators: The Taylor Expansion of the Inverse Kuramoto Map

TL;DR

This paper introduces an algorithm to compute the Taylor expansion of the inverse Kuramoto map, enabling the development of efficient approximate tests for synchronization thresholds and manifolds in coupled oscillator networks.

Contribution

It presents a novel method for deriving the Taylor series of the inverse Kuramoto map, providing a hierarchy of low-complexity synchronization tests.

Findings

Hierarchical approximate synchronization tests with increasing accuracy

Efficient estimation of synchronization thresholds

Determination of the synchronization manifold position

Abstract

Synchronization in the networks of coupled oscillators is a widely studied topic in different areas. It is well-known that synchronization occurs if the connectivity of the network dominates heterogeneity of the oscillators. Despite extensive study on this topic, the quest for sharp closed-form synchronization tests is still in vain. In this paper, we present an algorithm for finding the Taylor expansion of the inverse Kuramoto map. We show that this Taylor series can be used to obtain a hierarchy of increasingly accurate approximate tests with low computational complexity. These approximate tests are then used to estimate the threshold of synchronization as well as the position of the synchronization manifold of the network.