Entrainment of Coupled Oscillators on Regular Networks by Pacemakers

TL;DR

This paper analyzes how pacemakers influence synchronization in coupled Kuramoto oscillators on regular networks, deriving critical conditions for entrainment and showing how topology and size affect synchronization thresholds.

Contribution

It provides analytical expressions for the critical pacemaker frequency and demonstrates the impact of network topology and size on synchronization in regular lattice networks.

Findings

Changing topology from chain to ring induces synchronization.

Critical pacemaker frequency decreases as a power of system size.

For infinite systems, critical frequency decreases exponentially with size.

Abstract

We study Kuramoto oscillators, driven by one pacemaker, on $d$-dimensional regular topologies with nearest neighbor interactions. We derive the analytical expressions for the common frequency in the case of phase-locked motion and for the critical frequency of the pacemaker, placed at an arbitrary position in the lattice, so that above the critical frequency no phase-locked motion is possible. We show that the mere change in topology from an open chain to a ring induces synchronization for a certain range of pacemaker frequencies and couplings, while keeping the other parameters fixed. Moreover we demonstrate numerically that the critical frequency of the pacemaker decreases as a power of the linear size of the lattice with an exponent equal to the dimension of the system. This leads in particular to the conclusion that for infinite-dimensional topologies the critical frequency for having entrainment decreases exponentially with the size of the system, or, more generally, with the depth of the network, that is the average distance of the oscillators from the pacemaker.