Model reduction for Kuramoto models with complex topologies

Synchronisation of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics, to social networks and technology. A fundamental and long-time goal in the study of synchronisation has been to find low-order descriptions of complex oscillator networks and thei...

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Bibliographic Details
Published in:arXiv.org 2018-06
Main Authors: Hancock, Edward J, Gottwald, Georg A
Format: Article
Language:English
Subjects:
Computer simulation
Mathematical models
Model reduction
Network topologies
Oscillators
Social networks
Synchronism
Time synchronization
Online Access:Get full text
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Summary:Synchronisation of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics, to social networks and technology. A fundamental and long-time goal in the study of synchronisation has been to find low-order descriptions of complex oscillator networks and their collective dynamics. However, for the Kuramoto model - the most widely used model of coupled oscillators - this goal has remained surprisingly challenging, in particular for finite-size networks. Here, we propose a model reduction framework that effectively captures synchronisation behaviour in complex network topologies. This framework generalises a collective coordinates approach for all-to-all networks [Gottwald (2015) Chaos 25, 053111] by incorporating the graph Laplacian matrix in the collective coordinates. We first derive low dimensional evolution equations for both clustered and non-clustered oscillator networks. We then demonstrate in numerical simulations for Erdos-Renyi (ER) networks that the collective coordinates capture the synchronisation behaviour in both finite-size networks as well as in the thermodynamic limit, even in the presence of interacting clusters.
ISSN:2331-8422
DOI:10.48550/arxiv.1804.07444