Emergent Dynamics for the Kinetic Kuramoto Equation

  • Debora AmadoriEmail author
  • Jinyeong Park
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 236)


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 [13]), 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, [2]). 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, [8]).


Kinetic Kuramoto model Approximate BV solutions Large time behavior 


  1. 1.
    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
  2. 2.
    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
  3. 3.
    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
  4. 4.
    N.J. Balmforth, R. Sassi, A shocking display of synchrony. Phys. D 143, 21–55 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D. Benedetto, E. Caglioti, U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit. Commun. Math. Sci. 13, 1775–1786 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J.A. Carrillo, Y.-P. Choi, S.-Y. Ha, M.-J. Kang, Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation. J. Stat. Phys. 156, 395–415 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. Chiba, Continuous limit of the moments system for the globally coupled phase oscillators. Discret. Contin. Dyn. Syst. 33, 1891–903 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    S.-Y. Ha, Y.-H. Kim, J. Morales, J. Park, Emergence of phase concentration for the Kuramoto–Sakaguchi equation, (submitted)
  9. 9.
    S.-Y. Ha, D. Ko, J. Park, X. Zhang, Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3(2), 209–267 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S.-Y. Ha, H.K. Kim, J. Park, Remarks on the complete synchronization of Kuramoto oscillators. Nonlinearity 28, 1441–1462 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984)CrossRefGoogle Scholar
  12. 12.
    Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics, Lecture notes in theoretical physics, vol. 30 (1975), p. 420Google Scholar
  13. 13.
    C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transp. Theory Stat. Phys. 34, 523–535 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic theories and the Boltzmann equation. Lecture Notes in Mathematics, vol. 1048 (Springer, Berlin)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of L’AquilaL’AquilaItaly
  2. 2.Universidad de GranadaGranadaSpain

Personalised recommendations