Filippov flows and mean-field limits in the kinetic singular Kuramoto model

Abstract

The agent-based singular Kuramoto model was proposed in [60] as a singular version of the Kuramoto model of coupled oscillators that is consistent with Hebb's rule of neuroscience. In such paper, the authors studied its well-posedness via the concept of Filippov solutions. Interestingly, they found some new emergent phenomena in the paradigm of Kuramoto model: clustering into subgroups and emergence of global phase synchronization taking place at finite time.This paper aims at introducing the associated kinetic singular Kuramoto model along T × R, that is reminiscent of the classical Kuramoto-Sakaguchi equation. Our main goal is to propose a well-posedness theory of measure-valued solutions that remains valid after eventual phase collisions. The results will depend upon the specific regime of singularity: subcritical, critical and supercritical. The cornerstone is the existence of Filippov characteristic flows for interaction kernels with jump discontinuities. Our second goal is to study stability with respect to initial data, that in particular will provide quantitative estimates for the mean-field limit of the agent-based model towards the kinetic equation in quadratic Wasserstein-type distances. Finally, we will recover global phase synchronization at the macroscopic scale under appropriate generic assumptions on the initial data. The most singular regime will be tackled separately with an alternative method, namely, a singular hyperbolic limit on a regularized second order model with inertia. Note that we will work within the manifold T × R and will avoid resorting on Euclidean approximations. Also, no gradient-type structure will be needed, as opposed to some preceding literature.

On the trend to global equilibrium for Kuramoto Oscillators

“…Indeed, we do not necessarily need an underlying gradient structure, but only require that the interaction kernel is one-sided Lipschitz-continuous. This was proved in [38,Theorem 4.7] for the Kuramoto model with weakly singular weights, that in our case provides the following stability estimate for W 2 (6.4) W 2 (f t , ft ) ≤ e 2K+ 1 2 t W 2 (f 0 , f0 ), which holds for any two measured valued solution to (1.2). Notice that units are not correct in the above inequality, and this is again due to the fact that W 2 is not dimensionally correct in this problem (recall 3.2).…”

“…A similar result was explored in [38,Theorem 4.4]. There, the author used the definition of W 2,g in (3.6) for general measures that may enjoy atoms eventually.…”

“…See also [22, Appendix A] for an overview in the context of Kuramoto-Sakaguchi with identical oscillators. -A fibered Wasserstein distance W 2,g presented independently in [32] and [38]. Such a distance is well adapted to the nonlinear problem.…”

“…Indeed, inspired by the arguments in [36] on their proof of the logarithmic Sobolev and Talagrand inequalities, we derive analogous inequalities that yield the exponential convergence result and uniqueness of the global equilibrium. Since our system is not a Wasserstein gradient flow, we derive such inequalities for a fibered transportation distance W 2,g , independently introduced in [32] and [38], which is well adapted to the nonlinear problem. The proof of such inequalities is the content of the next section.…”

“…This metric is constructed through a gluing procedure of the standard quadratic Wasserstein distance in T between conditional probabilities at any fiber ω ∈ R. Since it behaves in a fiber-wise way, we call it the fibered quadratic Wasserstein distance. See also [32] and [38], where it was introduced independently by both authors. For the reader convenience, we recall it here and introduce some of the main properties that will be used throughout the paper.…”