Basin stability measure of different steady states in coupled oscillators

TL;DR

This paper studies the stability of different steady states in coupled oscillators, using basin stability measures to analyze complex multistable states like amplitude and oscillation death, supported by bifurcation analysis.

Contribution

It introduces basin stability as a tool to quantify multistable states in coupled oscillators, including amplitude and oscillation death, with analytical and bifurcation analysis.

Findings

Basin stability effectively measures stability of multistable states.

Oscillation death states exhibit multi-clustered configurations.

Stability depends on mean-field density and coupling strength.

Abstract

In this report, we investigate the stabilization of saddle fixed points in coupled oscillators where individual oscillators exhibit the saddle fixed points. The coupled oscillators may have two structurally different types of suppressed states, namely amplitude death and oscillation death. The stabilization of saddle equilibrium point refers to the amplitude death state where oscillations are ceased and all the oscillators converge to the single stable steady state via inverse pitchfork bifurcation. Due to multistability features of oscillation death states, linear stability theory fails to analyze the stability of such states analytically, so we quantify all the states by basin stability measurement which is an universal nonlocal nonlinear concept and it interplays with the volume of basins of attractions. We also observe multi-clustered oscillation death states in a random network and measure them using basin stability framework. To explore such phenomena we choose a network of coupled Duffing-Holmes and Lorenz oscillators which are interacting through mean-field coupling. We investigate how basin stability for different steady states depends on mean-field density and coupling strength. We also analytically derive stability conditions for different steady states and confirm by rigorous bifurcation analysis.