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

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.