Master Stability Islands for Amplitude Death in Networks of Delay-Coupled Oscillators

TL;DR

This paper develops a master stability function approach with the novel concept of master stability islands to analyze and predict the stability of amplitude death in delay-coupled oscillator networks, accounting for delay and coupling strength.

Contribution

It introduces master stability islands (MSIs) as a new tool for predicting amplitude death stability in delay-coupled networks, extending MSF analysis to include delay effects.

Findings

MSFs for delay-coupled networks require two additional inputs: delay and coupling strength.

MSIs are two-dimensional stability regions in delay-coupling space with eigenvalue encoding.

Nonzero delay is generally necessary to stabilize amplitude death states.

Abstract

This paper presents a master stability function (MSF) approach for analyzing the stability of amplitude death (AD) in networks of delay-coupled oscillators. Unlike the familiar MSFs for instantaneously coupled networks, which typically have a single input encoding for the effects of the eigenvalues of the network Laplacian matrix, for delay-coupled networks we show that such MSFs generally require two additional inputs: the time delay and the coupling strength. To utilize the MSF for predicting the stability of AD of arbitrary networks for a chosen nonlinear system (node dynamics) and coupling function, we introduce the concept of master stability islands (MSIs), which are two-dimensional stability islands of the delay-coupling space together with a third dimension ("altitude") encoding for eigenvalues that result in stable AD. We compute the MSFs and show the corresponding MSIs for several common chaotic systems including the Rossler, the Lorenz, and Chen's system, and found that it is generally possible to achieve AD and that a nonzero time delay is necessary for the stabilization of the AD states.