In this note, we study the emergent dynamics of the kinetic Kuramoto equation, which is a mean-field limit of the Kuramoto synchronization model. For this equation, also referred to as the Kuramoto–Sakaguchi equation Lancellotti (Transp Theory Stat Phys 34:523–535, 2005 ), we present two approaches for the analysis on its dynamics. First, for the system of identical oscillators, we apply a wave-front-tracking algorithm which is used for scalar conservation laws. This method gives a quantitative estimate on the approximate BV solution to the kinetic model Amadori et al. (J Differ Equ 262:978–1022, 2017, ). Second, we study the emergence of phase concentration phenomena by directly analyzing the dynamics of the order parameters. This can show the asymptotic behavior of the system from generic initial data Ha et al. (J Park, 2016, ).
Kinetic Kuramoto model Approximate BV solutions Large time behavior
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J.A. Acebron, L.L. Bonilla, C.J.P. Pérez Vicente, F. Ritort, R. Spigler, The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)CrossRefGoogle Scholar
D. Amadori, S.-Y. Ha, J. Park, On the global well-posedness of BV weak solutions for the Kuramoto–Sakaguchi equation. J. Differ. Equ. 262, 978–1022 (2017)MathSciNetCrossRefGoogle Scholar
D. Amadori, S.-Y. Ha, J. Park, in Innovative Algorithms and Analysis A nonlocal version of wave-front tracking motivated by the Kuramoto–Sakaguchi equation (Springer INdAM Series, 2017)Google Scholar