Contractivity of Transport Distances for the Kinetic Kuramoto Equation
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 m...
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| Published in | Journal of statistical physics Vol. 156; no. 2; pp. 395 - 415 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
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Springer US
01.07.2014
Springer |
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| ISSN | 0022-4715 1572-9613 |
| DOI | 10.1007/s10955-014-1005-z |
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| 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
L
1
-contraction properties of the Kuramoto phase model. |
|---|---|
| AbstractList | 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 [L.sup.1]-contraction properties of the Kuramoto phase model. Keywords Kuramoto model * Complete synchronization * Wasserstein distance * Contraction Mathematics Subject Classification 92D25 * 74A25 * 76N10 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 L 1 -contraction properties of the Kuramoto phase model. |
| Audience | Academic |
| Author | Kim, Yongduck Carrillo, José A. Choi, Young-Pil Ha, Seung-Yeal Kang, Moon-Jin |
| Author_xml | – sequence: 1 givenname: José A. surname: Carrillo fullname: Carrillo, José A. organization: Department of Mathematics, Imperial College London – sequence: 2 givenname: Young-Pil surname: Choi fullname: Choi, Young-Pil email: youngpilc@gmail.com, youngpil.choi@imperial.ac.uk organization: Department of Mathematics, Imperial College London – sequence: 3 givenname: Seung-Yeal surname: Ha fullname: Ha, Seung-Yeal organization: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University – sequence: 4 givenname: Moon-Jin surname: Kang fullname: Kang, Moon-Jin organization: Department of Mathematics, The University of Texas at Austin – sequence: 5 givenname: Yongduck surname: Kim fullname: Kim, Yongduck organization: Department of Mathematical Sciences, Seoul National University |
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| Cites_doi | 10.1016/S0167-2789(98)00235-8 10.1017/CBO9780511755743 10.1109/TAC.2008.2007884 10.1103/RevModPhys.77.137 10.1016/j.physd.2011.11.011 10.1007/978-3-642-84371-6 10.1007/s00205-004-0307-8 10.1007/s00332-006-0806-x 10.1016/S0167-2789(00)00094-4 10.1137/090757290 10.1007/978-3-642-69689-3 10.1007/BF01029202 10.1016/0022-5193(67)90051-3 10.1142/S0218202511005131 10.4310/CMS.2013.v11.n2.a3 10.1007/s10955-004-8785-5 10.4310/CMS.2009.v7.n2.a2 10.1090/S0033-569X-2013-01302-0 10.1016/S0167-2789(00)00095-6 10.1080/00411450508951152 10.1016/j.physd.2010.05.003 10.1038/211562a0 10.1016/j.jde.2011.04.004 10.1051/m2an/1997310506151 10.23919/ACC.2004.1383983 10.1007/BFb0071878 10.1007/BFb0074532 |
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| Keywords | Complete synchronization Kuramoto model 92D25 74A25 Wasserstein distance Contraction 76N10 |
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| Title | Contractivity of Transport Distances for the Kinetic Kuramoto Equation |
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