Uniform stability and uniform-in-time mean-field limit of the thermodynamic Kuramoto model
Authors:
Seung-Yeal Ha, Myeongju Kang, Hansol Park, Tommaso Ruggeri and Woojoo Shim
Journal:
Quart. Appl. Math. 79 (2021), 445-478
MSC (2020):
Primary 70F99; Secondary 92B25
DOI:
https://doi.org/10.1090/qam/1588
Published electronically:
February 22, 2021
MathSciNet review:
4288593
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Abstract: We consider the thermodynamic Kuramoto model proposed in \cite{H-P-R-S}. For each oscillator in thermodynamic Kuramoto model, there is a coupling effect between the phase and the temperature field. For such a model, we study a uniform stability and uniform-in-time mean-field limit to the corresponding kinetic equation. For this, we first derive a uniform $\ell ^p$-stability of the thermodynamic Kuramoto model with respect to initial data by directly estimating the temporal evolution of $\ell ^p$-distance between two admissible solutions to the particle thermodynamic Kuramoto model. In a large-oscillator limit, the Vlasov type mean-field equation can be rigorously derived using the BBGKY hierarchy, uniform stability estimate, and particle-in-cell method. We construct unique global-in-time measure-valued solutions to the derived kinetic equation and also derive a uniform-in-time stability estimate and emergent estimates.
References
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- S.-Y. Ha, M. Kang, H. Park, T. Ruggeri, and W. Shim, Emergent behaviors of the continuum thermodynamic Kuramoto model in a large coupling regime, Submitted.
- Seung-Yeal Ha, Hansol Park, Tommaso Ruggeri, and Woojoo Shim, Emergent behaviors of thermodynamic Kuramoto ensemble on a regular ring lattice, J. Stat. Phys. 181 (2020), no. 3, 917–943. MR 4160916, DOI 10.1007/s10955-020-02611-2
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- Seung-Yeal Ha and Marshall Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations 251 (2011), no. 10, 2685–2695. MR 2831709, DOI 10.1016/j.jde.2011.04.004
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- Charles S. Peskin, Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975. Notes based on a course given at New York University during the year 1973/74. MR 0414135
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- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382, DOI 10.1016/S0167-2789(00)00094-4
- J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys 72 (1993), 145-166.
- Cédric Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
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References
- J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005), 137-185.
- Dario Benedetto, Emanuele Caglioti, and Umberto Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys. 162 (2016), no. 4, 813–823. MR 3456977, DOI 10.1007/s10955-015-1426-3
- Dario Benedetto, Emanuele Caglioti, and Umberto Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci. 13 (2015), no. 7, 1775–1786. MR 3393174, DOI 10.4310/CMS.2015.v13.n7.a6
- Jared C. Bronski, Lee DeVille, and Moon Jip Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos 22 (2012), no. 3, 033133, 17. MR 3388627, DOI 10.1063/1.4745197
- J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562-564.
- José A. Carrillo, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang, and Yongduck Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys. 156 (2014), no. 2, 395–415. MR 3215629, DOI 10.1007/s10955-014-1005-z
- Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung, and Yongduck Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), no. 7, 735–754. MR 2897541, DOI 10.1016/j.physd.2011.11.011
- Nikhil Chopra and Mark W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control 54 (2009), no. 2, 353–357. MR 2491964, DOI 10.1109/TAC.2008.2007884
- Felipe Cucker and Steve Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852–862. MR 2324245, DOI 10.1109/TAC.2007.895842
- Pierre Degond and Sébastien Motsch, Macroscopic limit of self-driven particles with orientation interaction, C. R. Math. Acad. Sci. Paris 345 (2007), no. 10, 555–560 (English, with English and French summaries). MR 2374464, DOI 10.1016/j.crma.2007.10.024
- Pierre Degond and Sébastien Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys. 131 (2008), no. 6, 989–1021. MR 2407377, DOI 10.1007/s10955-008-9529-8
- Pierre Degond and Sébastien Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008), no. suppl., 1193–1215. MR 2438213, DOI 10.1142/S0218202508003005
- Jiu-Gang Dong and Xiaoping Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci. 11 (2013), no. 2, 465–480. MR 3002560, DOI 10.4310/CMS.2013.v11.n2.a7
- Florian Dörfler and Francesco Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 3, 1070–1099. MR 2837521, DOI 10.1137/10081530X
- Florian Dörfler and Francesco Bullo, Synchronization in complex networks of phase oscillators: a survey, Automatica J. IFAC 50 (2014), no. 6, 1539–1564. MR 3214901, DOI 10.1016/j.automatica.2014.04.012
- Seung-Yeal Ha, Eunhee Jeong, and Moon-Jin Kang, Emergent behaviour of a generalized Viscek-type flocking model, Nonlinearity 23 (2010), no. 12, 3139–3156. MR 2739419, DOI 10.1088/0951-7715/23/12/008
- Seung-Yeal Ha, Sungeun Jung, and Marshall Slemrod, Fast-slow dynamics of planar particle models for flocking and swarming, J. Differential Equations 252 (2012), no. 3, 2563–2579. MR 2860630, DOI 10.1016/j.jde.2011.09.014
- Seung-Yeal Ha, Jeongho Kim, Chan Ho Min, Tommaso Ruggeri, and Xiongtao Zhang, Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model, Quart. Appl. Math. 77 (2019), no. 1, 131–176. MR 3897922, DOI 10.1090/qam/1517
- Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, and Xiongtao Zhang, Uniform stability and mean-field limit for the augmented Kuramoto model, Netw. Heterog. Media 13 (2018), no. 2, 297–322. MR 3811564, DOI 10.3934/nhm.2018013
- Seung-Yeal Ha, Jeongho Kim, and Tommaso Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal. 50 (2018), no. 3, 3092–3121. MR 3814022, DOI 10.1137/17M111064X
- Seung-Yeal Ha, Hwa Kil Kim, and Sang Woo Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci. 14 (2016), no. 4, 1073–1091. MR 3491817, DOI 10.4310/CMS.2016.v14.n4.a10
- Seung-Yeal Ha, Jeongho Kim, and Xiongtao Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models 11 (2018), no. 5, 1157–1181. MR 3810860, DOI 10.3934/krm.2018045
- Seung-Yeal Ha and Jian-Guo Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297–325. MR 2536440
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- Seung-Yeal Ha, Zhuchun Li, Marshall Slemrod, and Xiaoping Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit, Quart. Appl. Math. 72 (2014), no. 4, 689–701. MR 3291822, DOI 10.1090/S0033-569X-2014-01350-5
- S.-Y. Ha, M. Kang, H. Park, T. Ruggeri, and W. Shim, Emergent behaviors of the continuum thermodynamic Kuramoto model in a large coupling regime, Submitted.
- Seung-Yeal Ha, Hansol Park, Tommaso Ruggeri, and Woojoo Shim, Emergent behaviors of thermodynamic Kuramoto ensemble on a regular ring lattice, J. Stat. Phys. 181 (2020), no. 3, 917–943. MR 4160916, DOI 10.1007/s10955-020-02611-2
- Seung-Yeal Ha and Tommaso Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1397–1425. MR 3594359, DOI 10.1007/s00205-016-1062-3
- Seung-Yeal Ha and Marshall Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations 251 (2011), no. 10, 2685–2695. MR 2831709, DOI 10.1016/j.jde.2011.04.004
- Seung-Yeal Ha and Eitan Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008), no. 3, 415–435. MR 2425606, DOI 10.3934/krm.2008.1.415
- Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Springer, Berlin, 1975, pp. 420–422. Lecture Notes in Phys., 39. MR 0676492
- Carlo Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory Statist. Phys. 34 (2005), no. 7, 523–535. MR 2265477, DOI 10.1080/00411450508951152
- H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic theories and the Boltzmann equation (Montecatini, 1981) Lecture Notes in Math., vol. 1048, Springer, Berlin, 1984, pp. 60–110. MR 740721, DOI 10.1007/BFb0071878
- Charles S. Peskin, Mathematical aspects of heart physiology, notes based on a course given at New York University during the year 1973/74, Courant Institute of Mathematical Sciences, New York University, New York, 1975. MR 0414135
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044, DOI 10.1017/CBO9780511755743
- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators: Bifurcations, patterns and symmetry, Phys. D 143 (2000), no. 1-4, 1–20. MR 1783382, DOI 10.1016/S0167-2789(00)00094-4
- J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys 72 (1993), 145-166.
- Cédric Villani, Optimal transport: Old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. MR 2459454, DOI 10.1007/978-3-540-71050-9
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967), 15-42.
- Arthur T. Winfree, The geometry of biological time, Biomathematics, vol. 8, Springer-Verlag, Berlin-New York, 1980. MR 572965
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826; and Korea Institute for Advanced Study, Hoegiro 85, 02455, Seoul, Republic of Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Myeongju Kang
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
ORCID:
0000-0002-5081-442X
Email:
bear0117@snu.ac.kr
Hansol Park
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
MR Author ID:
1356829
ORCID:
0000-0002-1075-6472
Email:
hansol960612@snu.ac.kr
Tommaso Ruggeri
Affiliation:
Department of Mathematics and Alma Mater Research Center on Applied Mathematics AM$^2$, University of Bologna, Italy
MR Author ID:
151655
ORCID:
0000-0002-7588-2074
Email:
tommaso.ruggeri@unibo.it
Woojoo Shim
Affiliation:
The Research Institute of Basic Sciences, Seoul National University, Seoul 08826, Republic of Korea
MR Author ID:
1337213
ORCID:
0000-0003-3051-9420
Email:
cosmo.shim@gmail.com
Keywords:
The Kuramoto model,
mean-field limit,
synchronization,
thermodynamics,
uniform stability,
kinetic equation
Received by editor(s):
October 25, 2020
Received by editor(s) in revised form:
January 23, 2021
Published electronically:
February 22, 2021
Additional Notes:
The work of the first author was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881), the work of the second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP)(2016K2A9A2A13003815), the work of the third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1I1A1A01059585), and the work of the fourth author was supported National Group of Mathematical Physics GNFM-INdAM.
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