Synchronization of Heterogeneous Kuramoto Oscillators with Graphs of Diameter Two

TL;DR

This paper investigates the synchronization conditions of heterogeneous Kuramoto oscillators on graphs with diameter two, providing improved analytic criteria and an optimization approach for coupling strengths.

Contribution

It introduces a new analytic synchronization condition for diameter-two graphs and an optimization framework for minimal coupling strengths with heterogeneous links.

Findings

Analytic condition guarantees synchronization on diameter-two graphs.

Existence of a Positively Invariant Set implies frequency synchronization.

Optimization method finds minimal coupling strengths for synchronization.

Abstract

In this article we study synchronization of Kuramoto oscillators with heterogeneous frequencies, and where underlying topology is a graph of diameter two. When the coupling strengths between every two connected oscillators are the same, we find an analytic condition that guarantees an existence of a Positively Invariant Set (PIS) and demonstrate that existence of a PIS suffices for frequency synchronization. For graphs of diameter two, this synchronization condition is significantly better than existing general conditions for an arbitrary topology. If the coupling strengths can be different for different pairs of connected oscillators, we formulate an optimization problem that finds sufficient for synchronization coupling strengths such that their sum is minimal.