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The Mean-Field Ott-Antonsen Manifold is an Unstable Manifold in the Continuum Limit

Christian Kuehn, Giacomo Landi

TL;DR

The paper addresses how invariant structures in the continuum-limit (CL) and mean-field limit (MFL) descriptions of Kuramoto-type interacting particle systems relate. It develops an explicit mapping from the Ott-Antonsen (OA) manifold of the MFL to a two-dimensional unstable manifold of the CL by representing CL solutions via the inverse CDF $F_t^{-1}$ and exploiting a CCDF-based parametrization; the unstable CL manifold is given by $W^u(y_q(\xi))=\{F^{-1}_{\alpha,\beta}(\xi+C(\alpha,\beta)+q)\}$ with $\dot{\beta}=\tfrac{1}{2}\beta(1-\beta^2)$ and $\dot{\alpha}=0$. The authors provide closed-form expressions for the CCDF $F_{\alpha,\beta}$ and its inverse (Appendix) and demonstrate how OA corresponds to a dynamical analogue of the CL manifold, enriching the understanding of invariant manifolds across IPS limits. This constructive link offers a framework to transfer geometric insights between microscopic, mean-field, and continuum descriptions and suggests directions for extending these correspondences to broader interaction kernels and non-exchangeable particle models.

Abstract

We study interacting particle systems of Kuramoto-type. Our focus is on the dynamical relation between the partial differential equation (PDE) arising in the continuum limit (CL) and the one obtained in the mean-field limit (MFL). Both equations arise when we are considering the limit of infinitely many interacting particles but the classes of PDEs are structurally different. The CL tracks particles effectively pointwise, while the MFL is an evolution for a typical particle. First, we briefly discuss the relation between solutions of the CL and the MFL showing how to generate solutions of the CL starting from solutions of the MFL. Our main result concerns a dynamical relation between important invariant manifolds of the CFL and the MFL. In particular, we give an explicit proof that the unstable manifold of the homogeneous steady state of the CL is the direct dynamical analogue of the famous Ott-Antonsen manifold for the MFL.

The Mean-Field Ott-Antonsen Manifold is an Unstable Manifold in the Continuum Limit

TL;DR

The paper addresses how invariant structures in the continuum-limit (CL) and mean-field limit (MFL) descriptions of Kuramoto-type interacting particle systems relate. It develops an explicit mapping from the Ott-Antonsen (OA) manifold of the MFL to a two-dimensional unstable manifold of the CL by representing CL solutions via the inverse CDF and exploiting a CCDF-based parametrization; the unstable CL manifold is given by with and . The authors provide closed-form expressions for the CCDF and its inverse (Appendix) and demonstrate how OA corresponds to a dynamical analogue of the CL manifold, enriching the understanding of invariant manifolds across IPS limits. This constructive link offers a framework to transfer geometric insights between microscopic, mean-field, and continuum descriptions and suggests directions for extending these correspondences to broader interaction kernels and non-exchangeable particle models.

Abstract

We study interacting particle systems of Kuramoto-type. Our focus is on the dynamical relation between the partial differential equation (PDE) arising in the continuum limit (CL) and the one obtained in the mean-field limit (MFL). Both equations arise when we are considering the limit of infinitely many interacting particles but the classes of PDEs are structurally different. The CL tracks particles effectively pointwise, while the MFL is an evolution for a typical particle. First, we briefly discuss the relation between solutions of the CL and the MFL showing how to generate solutions of the CL starting from solutions of the MFL. Our main result concerns a dynamical relation between important invariant manifolds of the CFL and the MFL. In particular, we give an explicit proof that the unstable manifold of the homogeneous steady state of the CL is the direct dynamical analogue of the famous Ott-Antonsen manifold for the MFL.

Paper Structure

This paper contains 4 sections, 4 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Conclusion
  4. Appendix: Closed formula for $F^{-1}_{\alpha,\beta}(\xi)$

Key Result

Proposition 2.2

Assume the previous hypothesis and define $x_t(\xi)$ as with $F^{-1}_t$ the inverseWe have omitted the dependency on the measure for brevity, so we identify $F_{\mu_t}$ with $F_{t}$ and the same holds for $F^{-1}_t$ of the CDF relative to $f_t$, solution of MFL. Then we have that $x_t(\xi)$ solves

Figures (1)

  • Figure 1: Ideal graphic representation of the unstable manifolds $W^\textnormal{u}(y_q(\xi))$, for three different values of $q$, and their common asymptotic curve $\Gamma$.

Theorems & Definitions (11)

  • Remark 1.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 1 more