Exponential synchronization rate of Kuramoto oscillators in the presence of a pacemaker

TL;DR

This paper investigates the exponential synchronization rate of Kuramoto oscillators with a pacemaker, providing new conditions for synchronization and phase locking, and deriving bounds that depend on pacemaker strength and coupling.

Contribution

It improves existing synchronization results by removing phase constraints and offers bounds on the synchronization rate related to pacemaker strength.

Findings

Synchronization can be achieved without phase constraints for identical frequencies.

The exponential rate increases with pacemaker strength.

Phase differences can be trapped to reduce synchronization error.

Abstract

The exponential synchronization rate is addressed for Kuramoto oscillators in the presence of a pacemaker. When natural frequencies are identical, we prove that synchronization can be ensured even when the phases are not constrained in an open half-circle, which improves the existing results in the literature. We derive a lower bound on the exponential synchronization rate, which is proven to be an increasing function of pacemaker strength, but may be an increasing or decreasing function of local coupling strength. A similar conclusion is obtained for phase locking when the natural frequencies are non-identical. An approach to trapping phase differences in an arbitrary interval is also given, which ensures synchronization in the sense that synchronization error can be reduced to an arbitrary level.