Uniform-in-time asymptotic limits of generalized Kuramoto models
Hangjun Cho, Seung-Yeal Ha, Myeongju Kang, Chan Ho Min
TL;DR
This work addresses uniform-in-time connections between particle-based generalized Kuramoto (GK) oscillators and their continuum and kinetic limits. By developing a robust uniform stability framework for GK dynamics, the authors derive whole-time continuum and mean-field limits: lattice GK converges to a continuum GK model in supremum norm, and the GK particle system yields a measure-valued solution to the kinetic GK equation, with precise W_q stability. The results include an L^∞ contraction for the continuum model and a uniform-in-time mean-field limit, under structured assumptions on the coupling, natural frequencies, and initial data. Collectively, these findings provide rigorous, time-uniform links between discrete GK ensembles, their continuum description, and their kinetic formulation, improving upon prior finite-time results and offering a solid basis for further analysis of generalized synchronization phenomena.
Abstract
We study two uniform-in-time asymptotic limits for generalized Kuramoto (GK) models. For these GK type models, we first derive the uniform stability estimates with respect to initial data, natural frequency and communication network under a suitable framework, and then as direct applications of this uniform stability estimate, we establish two asymptotic limits which are valid in the whole time interval, namely uniform-in-time continuum and mean-filed limit to the continuum and kinetic GK models, respectively. In the mean-field limit setting (the number of particles tends to infinity), we show global-in-time existence of measure-valued solutions to the corresponding kinetic equation. On the other hand, in a continuum limit setting (the lattice size tends to zero), we show that the lattice GKM solutions converge to a classical solution to the continuum GK model in supremum norm. Two asymptotic limits improve earlier results for the generalized GK type models.