Synchronization in repulsively coupled oscillators

TL;DR

This paper introduces a general coupling condition for achieving complete synchronization in purely repulsively coupled oscillators, validated through stability analysis and applicable to large ensembles.

Contribution

It proposes a novel coupling profile based on a linear matrix condition that enables synchronization in repulsively coupled identical oscillators, a problem previously considered difficult.

Findings

The coupling condition successfully induces synchronization in various chaotic oscillators.

Linear stability analysis confirms the effectiveness of the proposed coupling scheme.

The method scales to large ensembles of oscillators.

Abstract

A long-standing expectation is that two repulsively coupled oscillators tend to oscillate in opposite directions. It has been difficult to achieve complete synchrony in coupled identical oscillators with purely repulsive coupling. Here, we introduce a general coupling condition based on the linear matrix of dynamical systems for the emergence of the complete synchronization in pure repulsively coupled oscillators. The proposed coupling profiles (coupling matrices) define a bidirectional cross-coupling link that plays the role of indicator for the onset of complete synchrony between identical oscillators. We illustrate the proposed coupling scheme on several paradigmatic two-coupled chaotic oscillators and validate its effectiveness through the linear stability analysis of the synchronous solution based on the master stability function approach. We further demonstrate that the proposed general condition for the selection of coupling profiles to achieve synchronization even works perfectly for a large ensemble of oscillators.