Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators


Authors: Seung-Yeal Ha and Moon-Jin Kang
Journal: Quart. Appl. Math. 71 (2013), 707-728
MSC (2010): Primary 92D25, 74A25, 76N10
DOI: https://doi.org/10.1090/S0033-569X-2013-01302-0
Published electronically: August 29, 2013
MathSciNet review: 3136992
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Abstract: We present asymptotic relaxation estimates to bi-cluster configurations for the ensemble of Kuramoto oscillators with two different natural frequencies which have been observed in numerical simulations. We provide a set of initial configurations with a positive Lebesgue measure in $ \mathbb{T}^N$ leading to bi-(point) cluster configurations consisting of linear combinations of two Dirac measures in super-threshold and threshold-coupling regimes. In a super-threshold regime where the coupling strength is larger than the difference of two natural frequencies, we use the $ \ell _1$-contraction property of the Kuramoto model to derive exponential convergence toward bi-cluster configurations. The exact location of bi-cluster configurations is explicitly computable using the coupling strength, the difference of natural frequencies, and the total phase. In contrast, for the threshold-coupling regime where the coupling strength is exactly equal to the difference of natural frequencies, the mixed ensemble of Kuramoto oscillators undergoes two dynamic phases. First, the initial configuration evolves to the segregated phase (two segregated subconfigurations consisting of the same natural frequency) in a finite time. After this segregation phase, each subconfiguration relaxes to the asymptotic phase algebraically slowly. Our analytical results provide a rigorous framework for the observed numerical simulations.


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Additional Information

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

Moon-Jin Kang
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email: hiofte@snu.ac.kr

DOI: https://doi.org/10.1090/S0033-569X-2013-01302-0
Keywords: The Kuramoto model, bi-cluster configurations, $\ell_1$-contraction, natural frequency
Received by editor(s): November 21, 2011
Published electronically: August 29, 2013
Additional Notes: The work of S.-Y. Ha is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (2011-0015388), and the work of M.-J. Kang is supported by Hi Seoul Science/Humanities Fellowship funded by Seoul Scholarship Foundation
Article copyright: © Copyright 2013 Brown University


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