Amplitude death and restoration in networks of oscillators with random-walk diffusion

TL;DR

This paper investigates how random-walk diffusion in oscillator networks can induce amplitude death and subsequently restore oscillations, revealing a novel stabilization mechanism different from traditional diffusion coupling.

Contribution

It introduces a new phenomenon where random-walk diffusion stabilizes fixed points and restores oscillations, supported by numerical simulations and mean-field analysis.

Findings

Random-walk diffusion can induce amplitude death in oscillator networks.

Increasing coupling strength can restore oscillations after death.

The phenomena are robust in large, heterogeneous networks.

Abstract

We study the death and restoration of collective oscillations in networks of oscillators coupled through random-walk diffusion. Differently than the usual diffusion coupling used to model chemical reactions, here the equilibria of the uncoupled unit is not a solution of the coupled ensemble. Instead, the connectivity modifies both, the original unstable fixed point and the stable limit-cycle, making them node-dependent. Using numerical simulations in random networks we show that, in some cases, this diffusion induced heterogeneity stabilizes the initially unstable fixed point via a Hopf bifurcation. Further increasing the coupling strength the oscillations can be restored as well. Upon numerical analysis of the stability properties we conclude that this is a novel case of amplitude death. Finally we use a heterogeneous mean-field reduction of the system in order to proof the robustness of this phenomena upon increasing the system size.