Synchronization of Kuramoto Oscillators via Cutset Projections

TL;DR

This paper introduces a new method using cutset projections to derive more accurate and less conservative sufficient conditions for synchronization in Kuramoto oscillator networks, improving upon previous criteria.

Contribution

The authors develop a novel cutset projection operator to establish a new family of synchronization conditions, including the first generally-applicable $\infty$-norm criterion.

Findings

The $\infty$-norm condition outperforms previous tests in IEEE test cases.

The method provides explicit bounds avoiding nonconvex optimization.

The new conditions are less conservative and more accurate for various network topologies.

Abstract

Synchronization in coupled oscillators networks is a remarkable phenomenon of relevance in numerous fields. For Kuramoto oscillators the loss of synchronization is determined by a trade-off between coupling strength and oscillator heterogeneity. Despite extensive prior work, the existing sufficient conditions for synchronization are either very conservative or heuristic and approximate. Using a novel cutset projection operator, we propose a new family of sufficient synchronization conditions; these conditions rigorously identify the correct functional form of the trade-off between coupling strength and oscillator heterogeneity. To overcome the need to solve a nonconvex optimization problem, we then provide two explicit bounding methods, thereby obtaining (i) the best-known sufficient condition for unweighted graphs based on the 2-norm, and (ii) the first-known generally-applicable sufficient condition based on the $\infty$-norm. We conclude with a comparative study of our novel $\infty$-norm condition for specific topologies and IEEE test cases; for most IEEE test cases our new sufficient condition is one to two orders of magnitude more accurate than previous rigorous tests.