The Bellerophon state: a novel coherent phase of globally coupled oscillators

TL;DR

The paper introduces the Bellerophon state, a new coherent phase in globally coupled oscillators characterized by quantized, non-phase-locked clusters with periodic behavior, occurring near the transition from explosive to continuous synchronization.

Contribution

It reveals and characterizes the Bellerophon state, a novel coherent phase in coupled oscillators, with analytical and numerical insights into its microscopic and macroscopic properties.

Findings

Oscillators form quantized, non-phase-locked clusters.

Clusters exhibit periodic behavior with identical average speeds.

The Bellerophon state occurs near the transition from explosive to continuous synchronization.

Abstract

From rhythmic physiological processes to the collective behaviors of technological and natural networks, coherent phases of interacting oscillators are the foundation of the events' coordination leading a system to behave cooperatively. We unveil the existence of a new of such states, occurring in globally coupled nonidentical oscillators in the proximity of the point where the transition from the system's incoherent to coherent phase converts from explosive to continuous. In such a state, oscillators form quantized clusters, where they are neither phase- nor frequency-locked. Oscillators' instantaneous speeds are different within the clusters, but they form a characteristic cusped pattern and, more importantly, they behave periodically in time so that their average values are the same. Given its intrinsic specular nature with respect to the recently introduced Chimera states, the phase is termed the {\it Bellerophon} state. We provide analytical and numerical description of the microscopic and macroscopic details of {\it Bellerophon} states, thus furnishing practical hints on how to seek for the new phase in a variety of experimental and natural systems.