Hopf-Induced Desynchronization

TL;DR

This paper introduces a novel type of transition between synchrony and incoherence in coupled oscillators, characterized by a Hopf bifurcation, expanding understanding of phase transitions in complex systems.

Contribution

It uncovers a new transition mechanism in the Kuramoto model involving Hopf bifurcation, with analytical and numerical analysis providing insights into synchronization phenomena.

Findings

Discovered a new transition type where synchrony does not connect to incoherence.

Identified Hopf bifurcation as the destabilizing mechanism.

Developed a quaternion order parameter for analysis.

Abstract

The emergence of synchrony essentially underlies the functionality of many systems across physics, biology and engineering. In all established synchronization phase transitions so far, a stable synchronous state is connected to a stable incoherent state: For continuous transitions, stable synchrony directly connects to stable incoherence at a critical point, whereas for discontinuous transitions, stable synchrony is connected to stable incoherence via an additional unstable branch. Here we present a novel type of transition between synchrony and incoherence where the synchronous state does not connect to the state of incoherence. We uncover such transitions in the complexified Kuramoto model with their variables and coupling strength parameter analytically continued. Deriving a self-consistency equation for a quaternion order parameter that we propose helps to mathematically pin down the mechanisms underlying this transition type. Local numerical analysis suggests that the transition is linked to a Hopf bifurcation destabilizing synchrony, in contrast to branching point bifurcations established for the transition between synchrony and incoherence so far.