Reentrant Synchronization and Pattern Formation in Pacemaker-Entrained Kuramoto Oscillators

TL;DR

This paper investigates how Kuramoto oscillators synchronize and form patterns when influenced by a pacemaker, revealing reentrant synchronization transitions and pacemaker emergence without asymmetric coupling or frequency gaps.

Contribution

It provides analytical and numerical insights into reentrant synchronization and pacemaker-induced pattern formation in Kuramoto oscillators with varying interaction ranges.

Findings

Reentrant synchronization transition as a function of coupling range.

Analytical derivation of entrainment frequency and threshold.

Emergence of pacemaker behavior from natural frequency gradients.

Abstract

We study phase entrainment of Kuramoto oscillators under different conditions on the interaction range and the natural frequencies. In the first part the oscillators are entrained by a pacemaker acting like an impurity or a defect. We analytically derive the entrainment frequency for arbitrary interaction range and the entrainment threshold for all-to-all couplings. For intermediate couplings our numerical results show a reentrance of the synchronization transition as a function of the coupling range. The origin of this reentrance can be traced back to the normalization of the coupling strength. In the second part we consider a system of oscillators with an initial gradient in their natural frequencies, extended over a one-dimensional chain or a two-dimensional lattice. Here it is the oscillator with the highest natural frequency that becomes the pacemaker of the ensemble, sending out circular waves in oscillator-phase space. No asymmetric coupling between the oscillators is needed for this dynamical induction of the pacemaker property nor need it be distinguished by a gap in the natural frequency.