Abstract
We study the kinetic Kuramoto model for coupled oscillators with coupling constant below the synchronization threshold. We manage to prove that, for any analytic initial datum, if the interaction is small enough, the order parameter of the model vanishes exponentially fast, and the solution is asymptotically described by a free flow. This behavior is similar to the phenomenon of Landau damping in plasma physics. In the proof we use a combination of techniques from Landau damping and from abstract Cauchy–Kowalewskaya theorem.
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Acebrón, J.A., Bonilla, L.L., Pérez Vicente, C.J., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137 (2005)
Bedrossian, J., Masmoudi, N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ. Math. Inst. Hautes Études Sci. 122, 195–300 (2015)
Bedrossian, J., Masmoudi, N., Mouhot, C. Landau damping: paraproducts and Gevrey regularity. arXiv:1311.2870, 2013
Benedetto, D., Caglioti, E., Montemagno, U.: Dephasing of Kuramoto oscillators in kinetic regime towards a fixed asymptotically free state. Rend. Mat. Appl. VII(35), 189–206 (2014)
Benedetto, D., Caglioti, E., Montemagno, U.: On the complete phase synchronization for the Kuramoto model in the mean-field limit. Commun. Math. Sci. 13(7), 1775–1786 (2015)
Caflish, R.E.: A simplified version of the abstract Cauchy-Kowalewski Theorem with weak singularities Bull. Am. Math. Soc. 23(2), 495–500 (1990)
Caglioti, E., Maffei, C.: Time asymptotics for solutions of Vlasov-Poisson equation in a circle. J. Stat. Phys. 92(1–2), 301–323 (1998)
Carrillo, J.A., Choi, Y.P., Ha, S.Y., Kang, M.J., Kim, Y.: Contractivity of transport distances for the kinetic Kuramoto equation. J. Stat. Phys. 156(2), 395–415 (2014)
Chiba, H.: A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model. Ergod. Theory Dyn. Syst. 35, 762–834 (2015)
Choi, Y.P., Ha, S.Y., Jung, S., Kim, Y.: Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. Physica D 241(7), 735–754 (2012)
Dietert, H. Stability and bifurcation for the Kuramoto model. arXiv:1411.3752 (2014)
Dong, J.G., Xue, X.: Synchronization analysis of Kuramoto oscillators. Commun. Math. Sci. 11(2), 465–480 (2013)
Faou, E., Rousset, F.: Landau damping in Sobolev spaces for the Vlasov-HMF model. Arch. Ration. Mech. Anal. (2015). doi:10.1007/s00205-015-0911-9
Fernandez B., Gérard-Varet D., and Giacomin G. Landau damping in the Kuramoto model. arXiv:1410.6006 (2014)
Ha, S.Y., Ha, T., Kim, J.H.: On the complete synchronization of the Kuramoto phase model. Phys. D 239(17), 1692–1700 (2010)
Hwang, H.J., Velázquez, J.J.L.: On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem. Indiana Univ. Math. J 58, 2623–2660 (2009)
Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: Araki, H. (ed.) International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol. 39, pp. 420–422. Springer, Berlin Heidelberg (1975)
Lancellotti, C.: On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transp. Theory Stat. Phys. 34(7), 523–535 (2005)
Mirollo, R.E.: The asymptotic behavior of the order parameter for the infinite-N Kuramoto model. Chaos: An Interdisciplinary. J. Nonlinear Sci. 22(4), 043118 (2012)
Mirollo, R., Strogatz, S.H.: The Spectrum of the Partially Locked State for the Kuramoto Model. J. Nonlinear Sci. 17(4), 309–347 (2007)
Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207(1), 29–201 (2011)
Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143(1–4), 1–20 (2000). Bifurcations, patterns and symmetry
Strogatz, S.H., Mirollo, R.E.: Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63(3–4), 613–635 (1991)
Strogatz, S.H., Mirollo, R.E., Matthews, P.C.: Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping. Phys. Rev. Lett. 68, 2730–2733 (1992)
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Benedetto, D., Caglioti, E. & Montemagno, U. Exponential Dephasing of Oscillators in the Kinetic Kuramoto Model. J Stat Phys 162, 813–823 (2016). https://doi.org/10.1007/s10955-015-1426-3
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DOI: https://doi.org/10.1007/s10955-015-1426-3