# Emergent dynamics in delayed attractive-repulsively coupled networks

## TL;DR

This paper studies how delayed attractive and repulsive couplings in oscillator networks lead to various emergent behaviors like oscillation quenching and revival, analyzing transitions and bifurcations in both limit cycle and chaotic systems.

## Contribution

It introduces a reduced low-dimensional model capturing the complex dynamics of delayed coupled networks with symmetry-breaking interactions.

## Key findings

- Identifies three transition scenarios: OS-AD-OD, OS-AD-OS, and OS-OD.
- Demonstrates the effect of time delay on these transitions.
- Shows the reduced model accurately reflects full network dynamics.

## Abstract

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.