Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Remarks on the nonlinear stability of the Kuramoto model with inertia

Authors: Young-Pil Choi, Seung-Yeal Ha and Se Eun Noh
Journal: Quart. Appl. Math. 73 (2015), 391-399
MSC (2000): Primary 92D25, 74A25, 76N10
DOI: https://doi.org/10.1090/qam/1383
Published electronically: March 31, 2015
MathSciNet review: 3357501
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Abstract: In this short note, we present an a priori nonlinear stability estimate for the Kuramoto model with finite inertia in $ \ell ^{\infty }$-norm under some a priori condition on the size of the phase diameter. As a direct corollary of our nonlinear stability estimate, we show that phase-locked states obtained in Choi, Ha, and Yun (2011) are orbital-stable in $ \ell ^{\infty }$-norm, which means that the perturbed phase-locked state approaches the phase-shift of the given phase-locked state. The phase-shift is explicitly determined by the averages of initial phase and frequency distribution and the strength of inertia $ m$.

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Additional Information

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

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

Se Eun Noh
Affiliation: Department of Mathematics, Myongji University, Yongin 449-728, Republic of Korea
Email: senoh@mju.ac.kr

DOI: https://doi.org/10.1090/qam/1383
Received by editor(s): September 13, 2013
Published electronically: March 31, 2015
Additional Notes: The first author was supported by the Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology (2012R1A6A3A03039496).
The second author was supported by NRF grant 2011-0015388.
The third author was partially supported by the 2012 Research Fund of Myongji University.
Article copyright: © Copyright 2015 Brown University

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