# Symmetry broken states in an ensemble of globally coupled pendulums

## TL;DR

This paper analyzes the rotational dynamics of globally coupled pendulums, revealing various symmetry-broken states, their stability, and bifurcations, extending the Kuramoto model to include inertia and nonlinearity.

## Contribution

It introduces a generalized model of coupled pendulums with inertia, analyzing symmetry-breaking, stability, and bifurcations both analytically and numerically.

## Key findings

- Identification of instability boundaries for in-phase rotations
- Discovery of multiple out-of-phase regimes and their stability
- Detailed bifurcation analysis revealing regular and chaotic states

## Abstract

We consider the rotational dynamics in an ensemble of globally coupled identical pendulums. This model is essentially a generalization of the standard Kuramoto model, which takes into account the inertia and the intrinsic nonlinearity of the community elements. There exists the wide variety of in-phase and out-of-phase regimes. Many of these states appear due to broken symmetry. In the case of small dissipation our theoretical analysis allows one to find the boundaries of the instability domain of in-phase rotational mode for ensembles with arbitrary number of pendulums, describe all arising out-of-phase rotation modes and study in detail their stability. For the system of three elements parameter sets corresponding to the unstable in-phase rotations we find a number of out-of-phase regimes and investigate their stability and bifurcations both analytically and numerically. As a result, we obtain a sufficiently detailed picture of the symmetry breaking and existence of various regular and chaotic states.