Oscillateurs couplés, désordre et synchronisation

Abstract : In this thesis, we study the synchronization Kuramoto model and more generally systems of mean-field interacting diffusions on the circle, in the presence of another source of randomness, called disorder. The main motivation of this work is to study the large-population behavior of the system, for a fixed realization of the disorder (quenched model). This document contains, after the introduction, four chapters. The first one addresses the convergence of the empirical measure of the system of oscillators to a deterministic measure that solves a system of coupled, nonlinear partial differential equations (McKean-Vlasov equation). This convergence is indirectly proved through a large deviation principle in the averaged case and through a direct proof in the quenched case, under weaker assumptions on the disorder. The second chapter is part of a joint work with Giambattista Giacomin and Christophe Poquet and concerns the regularity of solutions of the limiting PDE as well as the stability of its nontrivial synchronized solution in the case of weak disorder. The last two chapters tackle the issue of the influence of the disorder on finite-size populations of rotators and illustrate problematics already observed in physical literature. We prove in the third chapter a central limit theorem associated to the previous law of large numbers: the quenched fluctuation process is shown to converge, in a weak sense, to the solution of a linear SPDE. We study in the last chapter the large-time behavior of this solution, illustrating the fact that fluctuations in the Kuramoto model are not self-averaging.
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Eric Luçon. Oscillateurs couplés, désordre et synchronisation. Probabilités [math.PR]. Université Pierre et Marie Curie - Paris VI, 2012. Français. ⟨tel-00709998⟩

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