Multi-clusters in networks of adaptively coupled phase oscillators networks

TL;DR

This paper studies adaptive phase oscillator networks, revealing conditions for multi-cluster formations with various phase patterns and demonstrating high multistability, relevant to systems like neuronal and power networks.

Contribution

It provides explicit existence criteria and stability conditions for multi-cluster solutions with diverse phase arrangements in adaptively coupled oscillators.

Findings

Multi-clusters can have different phase patterns such as antipodal and splay.

Existence criteria for multi-cluster solutions are derived.

High multistability is observed in the system.

Abstract

Dynamical systems on networks with adaptive couplings appear naturally in real-world systems such as power grid networks, social networks as well as neuronal networks. We investigate a paradigmatic system of adaptively coupled phase oscillators inspired by neuronal networks with synaptic plasticity. One important behaviour of such systems reveals splitting of the network into clusters of oscillators with the same frequencies, where different clusters correspond to different frequencies. Starting from one-cluster solutions we provide existence criteria for multi-cluster solutions and present their explicit form. The phases of the oscillators within one cluster can be organized in different patterns: antipodal, double antipodal, and splay type. Interestingly, multi-clusters are shown to exist where different clusters exhibit different patterns. For instance, an antipodal cluster can coexist with a splay cluster. We also provide stability conditions for one- and multi-cluster solutions. These conditions, in particular, reveal a high level of multistability.