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

Abstract

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. The system shows a metastable behaviour, which is characterised by the location and stability of its equilibrium points. We show that as the coupling strength increases, the number of equilibrium points decreases 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 particles 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.

Metastability in interacting nonlinear stochastic differential equations: II. Large-Nbehaviour

“…For small γ, H( γ) grows like the square-root of γ. This is compatible with the weakcoupling behaviour H = (1/4 + 3/2γ + O(γ 2 ))/N obtained in [BFG06a], if one takes into account the scaling of γ.…”

“…In the previous paper [BFG06a], we showed that the number of stationary configurations increases from 3 to 3 N as the coupling intensity γ decreases from a critical value γ 1 of order N 2 to 0. This transition from strong to weak coupling involves a sequence of successive symmetry-breaking bifurcations, and we analysed in particular the first of these bifurcations, which corresponds to desynchronisation.…”

“…Finally, Theorem 2.3 is proved in an analogous way as Theorems 2.7 and 2.8 in [BFG06a], using results from [FW98] (see also [Kif81,Sug96]). …”

“…Understanding the dynamics for small noise essentially requires knowing the graph G = (S 0 , E), in which two vertices x , y ∈ S 0 are connected by an edge if and only if there is a 1-saddle s ∈ S 1 whose unstable manifolds converge to x and y . The system behaves essentially like a Markovian jump process on G. The mean transition time from x to y is of order e 2H/σ 2 , where H is the potential difference between x and the lowest saddle leading to y (see [FW98], as well as [BFG06a] for more details).…”

“…In this paper, we continue our analysis of the metastable dynamics of a periodic chain of coupled bistable elements, initiated in [BFG06a]. In contrast with similar models involving discrete on-site variables, or "spins", whose metastable behaviour has been studied extensively (see for instance [dH04,OV05]), our model involves continuous local variables, and is therefore described by a set of interacting stochastic differential equations.…”