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 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 $$p$$p-distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous $$L^1$$L1-contraction properties of the Kuramoto phase model.

[1]  Giuseppe Toscani,et al.  Contractive Metrics for a Boltzmann Equation for Granular Gases: Diffusive Equilibria , 2005 .

[2]  G. Burton.  TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

[3]  Seung‐Yeal Ha,et al.  Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model , 2012 .

[4]  Juan P. Torres,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[5]  Neil J. Balmforth,et al.  A shocking display of synchrony , 2000 .

[6]  Seung-Yeal Ha,et al.  Exponential synchronization of finite-dimensional Kuramoto model at critical coupling strength , 2013 .

[7]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[8]  J. Buck,et al.  Biology of Synchronous Flashing of Fireflies , 1966, Nature.

[9]  C. Villani.  Topics in Optimal Transportation , 2003 .

[10]  P. Raviart.  An analysis of particle methods , 1985 .

[11]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[12]  S. Strogatz,et al.  Stability of incoherence in a population of coupled oscillators , 1991 .

[13]  J. Crawford,et al.  Synchronization of globally coupled phase oscillators: singularities and scaling for general couplings , 1997 .

[14]  Giuseppe Toscani,et al.  WASSERSTEIN METRIC AND LARGE-TIME ASYMPTOTICS OF NONLINEAR DIFFUSION EQUATIONS , 2005 .

[15]  Huzihiro Araki,et al.  International Symposium on Mathematical Problems in Theoretical Physics , 1975 .

[16]  Helmut Neunzert,et al.  An introduction to the nonlinear Boltzmann-Vlasov equation , 1984 .

[17]  G. Toscani,et al.  Long-Time Asymptotics of Kinetic Models of Granular Flows , 2004 .

[18]  Arkady Pikovsky,et al.  A universal concept in nonlinear sciences , 2006 .

[19]  Seung-Yeal Ha,et al.  A simple proof of the Cucker-Smale flocking dynamics and mean-field limit , 2009 .

[20]  Carlo Cercignani,et al.  Kinetic Theories and the Boltzmann Equation , 1984 .

[21]  Steven H. Strogatz,et al.  The Spectrum of the Partially Locked State for the Kuramoto Model , 2007, J. Nonlinear Sci..

[22]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[23]  Seung-Yeal Ha,et al.  On the complete synchronization of the Kuramoto phase model , 2010 .

[24]  A. Winfree.  Biological rhythms and the behavior of populations of coupled oscillators. , 1967, Journal of theoretical biology.

[25]  A. Jadbabaie,et al.  On the stability of the Kuramoto model of coupled nonlinear oscillators , 2005, Proceedings of the 2004 American Control Conference.

[26]  Jos'e A. Carrillo,et al.  A well-posedness theory in measures for some kinetic models of collective motion , 2009, 0907.3901.

[27]  Mark W. Spong,et al.  On Exponential Synchronization of Kuramoto Oscillators , 2009, IEEE Transactions on Automatic Control.

[28]  C. Lancellotti.  On the Vlasov Limit for Systems of Nonlinearly Coupled Oscillators without Noise , 2005 .

[29]  E. Caglioti,et al.  A kinetic equation for granular media , 2009 .

[30]  Hayato Chiba.  Continuous limit and the moments system for the globally coupled phase oscillators , 2010, 1003.5024.

[31]  Jesús Rosado,et al.  Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model , 2010, SIAM J. Math. Anal..

[32]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[33]  M. Slemrod,et al.  A fast–slow dynamical systems theory for the Kuramoto type phase model , 2011 .

[34]  Seung‐Yeal Ha,et al.  Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators , 2013 .