# Hierarchical frequency clusters in adaptive networks of phase oscillators

## TL;DR

This paper explores how slow adaptation in coupled phase oscillators leads to hierarchical frequency multi-clusters and metastable states, providing insights into the dynamics of adaptive networks.

## Contribution

It extends previous work by analyzing the impact of time-scale separation on cluster formation and stability in adaptive oscillator networks.

## Key findings

- Slow adaptation is necessary for multi-cluster emergence.
- Double antipodal clusters are metastable and long-lived.
- Heteroclinic orbits explain metastability phenomena.

## Abstract

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency-multi-clusters by investigating the effect of the time-scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multi-clusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit.