Contractivity of Transport Distances for the Kinetic Kuramoto Equation

José A. Carrillo; Young Pil Choi; Seung Yeal Ha; Moon Jin Kang; Yongduck Kim

doi:10.1007/s10955-014-1005-z

Contractivity of Transport Distances for the Kinetic Kuramoto Equation

José A. Carrillo, Young Pil Choi, Seung Yeal Ha, Moon Jin Kang, Yongduck Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

35 Citations (Scopus)

Abstract

We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. In the case of non-identical oscillators, we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein p-distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous L¹-contraction properties of the Kuramoto phase model.

Original language	English
Pages (from-to)	395-415
Number of pages	21
Journal	Journal of Statistical Physics
Volume	156
Issue number	2
DOIs	https://doi.org/10.1007/s10955-014-1005-z
Publication status	Published - 2014 Jul

Bibliographical note

Funding Information:
Acknowledgments JAC was partially supported by the project MTM2011-27739-C04-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. JAC acknowledges support from the Royal Society by a Wolfson Research Merit Award. YPC was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (ref. 2012R1A6A3A03039496). JAC and YPC were supported by Engineering and Physical Sciences Research Council Grants with references EP/K008404/1 (individual Grant) and EP/I019111/1 (platform Grant). The work of SYHA is supported by NRF Grant (2011-0015388).

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s10955-014-1005-z

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title = "Contractivity of Transport Distances for the Kinetic Kuramoto Equation",

abstract = "We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. In the case of non-identical oscillators, we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein p-distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous L1-contraction properties of the Kuramoto phase model.",

author = "Carrillo, {Jos{\'e} A.} and Choi, {Young Pil} and Ha, {Seung Yeal} and Kang, {Moon Jin} and Yongduck Kim",

note = "Funding Information: Acknowledgments JAC was partially supported by the project MTM2011-27739-C04-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. JAC acknowledges support from the Royal Society by a Wolfson Research Merit Award. YPC was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (ref. 2012R1A6A3A03039496). JAC and YPC were supported by Engineering and Physical Sciences Research Council Grants with references EP/K008404/1 (individual Grant) and EP/I019111/1 (platform Grant). The work of SYHA is supported by NRF Grant (2011-0015388).",

year = "2014",

month = jul,

doi = "10.1007/s10955-014-1005-z",

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AU - Carrillo, José A.

AU - Choi, Young Pil

AU - Ha, Seung Yeal

AU - Kang, Moon Jin

AU - Kim, Yongduck

N1 - Funding Information: Acknowledgments JAC was partially supported by the project MTM2011-27739-C04-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. JAC acknowledges support from the Royal Society by a Wolfson Research Merit Award. YPC was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (ref. 2012R1A6A3A03039496). JAC and YPC were supported by Engineering and Physical Sciences Research Council Grants with references EP/K008404/1 (individual Grant) and EP/I019111/1 (platform Grant). The work of SYHA is supported by NRF Grant (2011-0015388).

PY - 2014/7

Y1 - 2014/7

N2 - We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. In the case of non-identical oscillators, we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein p-distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous L1-contraction properties of the Kuramoto phase model.

AB - We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. In the case of non-identical oscillators, we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein p-distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous L1-contraction properties of the Kuramoto phase model.

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Contractivity of Transport Distances for the Kinetic Kuramoto Equation

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