Synchronization Patterns in Networks of Kuramoto Oscillators: A Geometric Approach for Analysis and Control

TL;DR

This paper analyzes how synchronization patterns form in networks of Kuramoto oscillators using a geometric approach, providing conditions for pattern formation and a control method for minimal network modifications.

Contribution

It introduces a linear algebra and graph theory-based framework to characterize and control synchronization patterns in heterogeneous oscillator networks.

Findings

Derived conditions for cluster synchronization in directed, weighted networks.

Developed a control method for minimal network perturbations to achieve desired patterns.

Validated results with numerical simulations.

Abstract

Synchronization is crucial for the correct functionality of many natural and man-made complex systems. In this work we characterize the formation of synchronization patterns in networks of Kuramoto oscillators. Specifically, we reveal conditions on the network weights and structure and on the oscillators' natural frequencies that allow the phases of a group of oscillators to evolve cohesively, yet independently from the phases of oscillators in different clusters. Our conditions are applicable to general directed and weighted networks of heterogeneous oscillators. Surprisingly, although the oscillators exhibit nonlinear dynamics, our approach relies entirely on tools from linear algebra and graph theory. Further, we develop a control mechanism to determine the smallest (as measured by the Frobenius norm) network perturbation to ensure the formation of a desired synchronization pattern. Our procedure allows us to constrain the set of edges that can be modified, thus enforcing the sparsity structure of the network perturbation. The results are validated through a set of numerical examples.