Metastability in interacting nonlinear stochastic differential equations I: From weak coupling to synchronisation

Nils Berglund, Bastien Fernandez and Barbara Gentz
Nonlinearity 20:2551-2581 (2007)

We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. We show that as the coupling strength increases, the number of equilibrium points of the system changes from 3 ^N to 3. While for weak coupling, the system behaves like an Ising model with spin-flip dynamics, for strong coupling (of the order N ² ), it synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. We derive the exponential asymptotics for the transition times, and describe the most probable transition paths between synchronised states, in particular for coupling intensities below the synchronisation threshold. Our techniques involve a centre-manifold analysis of the desynchronisation bifurcation, with a precise control of the stability of bifurcating solutions, allowing us to give a detailed description of the system's potential landscape, in which the metastable behaviour is encoded.

Mathematical Subject Classification: 37H20, 37L60 (primary), 37G40, 60K35 (secondary)

Keywords and phrases: Spatially extended systems, lattice dynamical systems, open systems, stochastic differential equations, interacting diffusions, transitions times, most probable transition paths, large deviations, Wentzell-Freidlin theory, diffusive coupling, synchronisation, metastability, symmetry groups.

 

Journal Homepage Published article: 10.1088/0951-7715/20/11/006 MR2361246 Zbl1135.60056

PDF file (952 Kb) hal-00115416 arXiv/math.PR/0611647