Broadcasting solutions on networked systems of phase oscillators

TL;DR

This paper introduces a method to analyze complex networked systems of Kuramoto oscillators by reducing multi-level networks to simpler models, enabling easier solution finding and stability analysis.

Contribution

It presents a novel approach using network join construction to reduce multi-level oscillator networks, facilitating analysis of their dynamics and stability.

Findings

Reduced models accurately represent original network dynamics

Solutions from reduced systems can be broadcasted to original networks

Stability analysis is simplified through the reduced representation

Abstract

Networked systems have been used to model and investigate the dynamical behavior of a variety of systems. For these systems, different levels of complexity can be considered in the modeling procedure. On one hand, this can offer a more realistic and rich modeling option. On the other hand, it can lead to intrinsic difficulty in analyzing the system. Here, we present an approach to investigate the dynamics of Kuramoto oscillators on networks with different levels of connections: a network of networks. To do so, we utilize a construction in network theory known as the join of networks, which represents ``intra-area" and ``inter-area" connections. This approach provides a reduced representation of the original, multi-level system, where both systems have equivalent dynamics. Then, we can find solutions for the reduced system and broadcast them to the original network of networks. Moreover, using the same idea we can investigate the stability of these states, where we can obtain information on the Jacobian of the multi-level system by analyzing the reduced one. This approach is general for arbitrary connection schemes between nodes within the same area. Finally, our work opens the possibility of studying the dynamics of networked systems using a simpler representation, thus leading to a better understanding of the dynamical behavior of these systems.