Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model

Abstract

Abstract. The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N → ∞ limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N → ∞ limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value Kc. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K > Kc.

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

“…Hence, if ones denotes by E 1 (u, v) ∶= ⟨u , (1 − A 1 )v⟩ −1,q0 the Dirichlet form associated to 1 − A 1 , one deduces from [7,Eq. (2.47)] that E 1 is well defined on L 2 and that it is equivalent to E 0 : there exists a constant C > 0 such that…”

Interacting particle systems

“…[37] and references therein) and more recently from a mathematical point of view [4,18,19]). In particular, a striking extension of (2) concerns the active rotator model (in which the local dynamics is more complex than a simple constant rotation [36]) where the emergence of periodic behaviors for large systems of excitable units has been observed [20]).…”

“…the Gibbs measure of the mean-field XY model with spin state S. Adding a nontrivial disorder to the system makes the dynamics irreversible [4] and we fall into the domain of non equilibrium statistical mechanics. This observation motivates in particular the introduction of the scaling parameter δ in (2): the main results will be stated for small δ, as they rely on perturbation arguments from the reversible case δ = 0, studied in the seminal work [4].…”

A Note on Dynamical Models on Random Graphs and Fokker–Planck Equations

“…[27,2]). This fixed point problem admits 0 as a solution, this corresponds to the uniform probability density 1/(2π), but for K > K c = 2 it admits also a positive solution, which we just call r K and corresponds to a nontrivial probability density, corresponding to (partial) synchronization in the system.…”

“…. ., choose an adjacency matrix ξ = {ξ (n) i,j } (i,j)∈{1,...,n} 2 , that is ξ (n) i,j ∈ {0, 1}. Consider the graph G (n) = (V (n) , E (n) ) associated with this sequence, namely V (n) := {1, .…”