Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Kuramoto oscillators with inertia: A fast-slow dynamical systems approach

Authors: Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung and Marshall Slemrod
Journal: Quart. Appl. Math. 73 (2015), 467-482
MSC (2010): Primary 91G80, 97M30
DOI: https://doi.org/10.1090/qam/1380
Published electronically: June 18, 2015
MathSciNet review: 3400753
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Abstract: We present a fast-slow dynamical systems theory for a Kuramoto type model with inertia. The fast part of the system consists of $ N$-decoupled pendulum equations with constant friction and torque as the phase of individual oscillators, whereas the slow part governs the evolution of order parameters that represent the amplitude and phase of the centroid of the oscillators. In our new formulation, order parameters serve as orthogonal observables in the framework of Artstein-Kevrekidis-Slemrod-Titi's unified theory of singular perturbation. We show that Kuramoto's order parameters become stationary regardless of the coupling strength. This generalizes an earlier result (Ha and Slemrod (2011)) for Kuramoto oscillators without inertia.

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Young-Pil Choi
Affiliation: Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
Email: young-pil.choi@imperial.ac.uk

Seung-Yeal Ha
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
Email: syha@snu.ac.kr

Sungeun Jung
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Email: mirinsil@naver.com

Marshall Slemrod
Affiliation: Department of Mathematics, University of Wisconsin-Madison – and – Department of Mathematics, Weizmann Institute of Science, Rehobot 76100, Israel
Email: slemrod@math.wisc.edu

DOI: https://doi.org/10.1090/qam/1380
Keywords: Fast-slow dynamical system, Kuramoto oscillators, mean-field limit, order parameter, synchronization
Received by editor(s): August 19, 2013
Published electronically: June 18, 2015
Additional Notes: The work of the second author was partially supported by KRF-2011-0015388, and the work of the fourth author was supported by KMRS-KAIST
Article copyright: © Copyright 2015 Brown University

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