  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.
