# Dynamics of oscillators globally coupled via two mean fields

## TL;DR

This paper explores the complex dynamics of coupled oscillators influenced by two competing mean fields, revealing new stability behaviors, bifurcation phenomena, and the emergence of modulated quasiperiodic states.

## Contribution

It introduces a novel Kuramoto-type model with two mean fields and analyzes the stability and bifurcation properties of its solutions, including nontrivial and hybrid states.

## Key findings

- Identification of normal and anomalous bifurcation branches.
- Discovery of hybrid states combining properties of different solutions.
- Observation of modulated quasiperiodic dynamics when no stable traveling wave exists.

## Abstract

Many studies of synchronization properties of coupled oscillators, based on the classical Kuramoto approach, focus on ensembles coupled via a mean field. Here we introduce a setup of Kuramoto-type phase oscillators coupled via two mean fields. We derive stability properties of the incoherent state and find traveling wave solutions with different locking patterns; stability properties of these waves are found numerically. Mostly nontrivial states appear when the two fields compete, i.e. one tends to synchronize oscillators while the other one desynchronizes them. Here we identify normal branches which bifurcate from the incoherent state in a usual way, and anomalous branches, appearance of which cannot be described as a bifurcation. Furthermore, hybrid branches combining properties of both are described. In the situations where no stable traveling wave exists, modulated quasiperiodic in time dynamics is observed. Our results indicate that a competition between two coupling channels can lead to a complex system behavior, providing a potential generalized framework for understanding of complex phenomena in natural oscillatory systems.