Cluster Synchronization and Phase Cohesiveness of Kuramoto Oscillators via Mean-phase Feedback Control and Pacemakers

TL;DR

This paper introduces two control methods for Kuramoto oscillators to achieve desired cluster synchronization and phase cohesiveness, inspired by brain network behaviors and neurological disorders, using mean-phase feedback and pacemakers.

Contribution

It presents novel control strategies for cluster synchronization and phase cohesiveness, including conditions and optimization techniques, with demonstrated effectiveness through numerical examples.

Findings

Conditions for achieving cluster synchronization and phase cohesiveness.

Optimal feedback gains obtained via convex optimization.

Numerical validation of the proposed control methods.

Abstract

Brain networks typically exhibit characteristic synchronization patterns where several synchronized clusters coexist. On the other hand, neurological disorders are considered to be related to pathological synchronization such as excessive synchronization of large populations of neurons. Motivated by these phenomena, this paper presents two approaches to control the cluster synchronization and the cluster phase cohesiveness of Kuramoto oscillators. One is based on feeding back the mean phases to the clusters, and the other is based on the use of pacemakers. First, we show conditions on the feedback gains and the pacemaker weights for the network to achieve cluster synchronization. Then, we propose a method to find optimal feedback gains through convex optimization. Second, we show conditions on the feedback gains and the pacemaker weights for the network to achieve cluster phase cohesiveness. A numerical example demonstrates the effectiveness of the proposed methods.