  We investigate different emergent dynamics namely oscillation quenching and
revival of oscillation in a global network of identical oscillators coupled
with diffusive (positive) delay coupling as it is perturbed by symmetry
breaking localized repulsive delayed interaction. Starting from the oscillatory
states (OS) we systematically identify three types of transition phenomena in
the parameter space: (1) The system may reach inhomogeneous steady states
(IHSS) from the homogeneous steady state (HSS) sometimes called as the
transition from amplitude death (AD) to oscillation death (OD) state i.e.
OS-AD-OD scenario, (2) Revival of oscillation (OS) from the AD state (OS-AD-OS)
and (3) Emergence of OD state from oscillatory state (OS) without passing
through AD i.e. OS-OD. The dynamics of each node in the network is assumed to
be governed either by identical limit cycle Stuart-Landau system or by chaotic
Rossler system. Based on clustering behavior observed in oscillatory network we
derive a reduced low-dimensional model of the large network. Using the reduced
model, we investigate the effect of time delay on these transitions and
demarcate OS, AD and OD regimes in the parameter space. We also explore and
characterize the bifurcation transitions present in both systems. The generic
behavior of the low dimensional model and full network are found to match
satisfactorily.
