# Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings

## TL;DR

This paper investigates how the interplay between attractive and repulsive couplings in coupled oscillators can lead to spontaneous symmetry breaking, multistability, and various synchronized states, with explicit stability analysis.

## Contribution

It introduces a simple model demonstrating symmetry breaking due to coupling trade-offs and provides explicit expressions and stability analysis for different oscillatory states.

## Key findings

- Repulsive coupling can destabilize in-phase synchronization.
- Symmetry breaking oscillatory states emerge from coupling competition.
- Explicit stability conditions for synchronized and oscillation death states.

## Abstract

Spontaneous symmetry breaking (SSB) is an important phenomenon observed in various fields including physics and biology. In this connection, we here show that the trade-off between attractive and repulsive couplings can induce spontaneous symmetry breaking in a homogeneous system of coupled oscillators. With a simple model of a system of two coupled Stuart-Landau oscillators, we demonstrate how the tendency of attractive coupling in inducing in-phase synchronized (IPS) oscillations and the tendency of repulsive coupling in inducing out-of-phase synchronized (OPS) oscillations compete with each other and give rise to symmetry breaking oscillatory (SBO) states and interesting multistabilities. Further, we provide explicit expressions for synchronized and anti-synchronized oscillatory states as well as the so called oscillation death (OD) state and study their stability. If the Hopf bifurcation parameter (${\lambda}$) is greater than the natural frequency ($\omega$) of the system, the attractive coupling favours the emergence of an anti-symmetric OD state via a Hopf bifurcation whereas the repulsive coupling favours the emergence of a similar state through a saddle-node bifurcation. We show that an increase in the repulsive coupling not only destabilizes the IPS state but also facilitates the re-entrance of the IPS state.