[Paper Review] Filippov flows and mean-field limits in the kinetic singular Kuramoto model

David Poyato|arXiv (Cornell University)|Mar 4, 2019

Nonlinear Dynamics and Pattern Formation68 references4 citations

TL;DR

This paper establishes a well-posedness theory for the kinetic singular Kuramoto model on the manifold $\mathbb{T} \times \mathbb{R}$ using Filippov flows to handle discontinuous interaction kernels in subcritical, critical, and supercritical regimes. It proves stability in quadratic Wasserstein distances, derives a rigorous mean-field limit from the agent-based model, and demonstrates finite-time global phase synchronization under generic initial conditions, particularly in the critical and supercritical cases via a regularized second-order 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 $\mathbb{T} imes\mathbb{R}$, 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 $\mathbb{T} imes\mathbb{R}$ and will avoid resorting on Euclidean approximations. Also, no gradient-type structure will be needed, as opposed to some preceding literature.

Motivation & Objective

• To develop a measure-valued solution theory for the kinetic singular Kuramoto model on $\mathbb{T} \times \mathbb{R}$ that remains valid after phase collisions.
• To establish stability and uniqueness of solutions using a fibered quadratic Wasserstein distance, enabling rigorous mean-field limits.
• To prove finite-time global phase synchronization of identical oscillators under generic initial data, especially in the critical and supercritical regimes.
• To handle the most singular regime via a singular hyperbolic limit on a regularized second-order model with inertia, avoiding Euclidean approximations.

Proposed method

• Formal derivation of the Vlasov-type equation on the manifold $\mathbb{T} \times \mathbb{R}$ using tangent transport fields and measure-valued solutions.
• Application of Filippov flows to manage interaction kernels with jump discontinuities, ensuring existence and uniqueness of solutions in the critical and supercritical regimes.
• Use of compactness arguments and a priori estimates for empirical measures to pass to the limit and construct weak measure-valued solutions.
• Introduction of a fibered quadratic Wasserstein distance to quantify stability and derive quantitative mean-field limits.
• Regularization of the second-order model to handle the supercritical regime, followed by a singular hyperbolic limit to recover the kinetic equation.
• Use of Riemannian geometry tools, including geodesic exponential maps and Dini derivatives, to analyze the distance function and its directional derivatives on the manifold.

Experimental results

Research questions

• RQ1 How can a well-posedness theory be constructed for the kinetic singular Kuramoto model on $\mathbb{T} \times \mathbb{R}$ when the interaction kernel has jump discontinuities?
• RQ2 What is the quantitative rate of convergence of the agent-based model to the kinetic equation in Wasserstein-type distances?
• RQ3 Can global phase synchronization occur in finite time at the macroscopic scale, and under what conditions on the initial data?
• RQ4 How does the solution theory differ across subcritical, critical, and supercritical regimes of singularity?
• RQ5 What alternative method can be used to analyze the most singular (supercritical) regime, and how does it recover synchronization?

Key findings

• The paper establishes existence and uniqueness of measure-valued solutions via Filippov flows for interaction kernels with jump discontinuities in the critical and supercritical regimes.
• A rigorous mean-field limit is proven with quantitative estimates in the fibered quadratic Wasserstein distance, showing convergence of the agent-based model to the kinetic equation.
• Finite-time global phase synchronization is recovered at the macroscopic level for identical oscillators under generic initial data, particularly in the critical and supercritical regimes.
• In the supercritical regime, a regularized second-order model is introduced, and a singular hyperbolic limit is used to derive the kinetic equation and prove synchronization.
• The analysis avoids Euclidean approximations and does not require a gradient-type structure, making the framework more general and geometrically intrinsic.