### Metastability in interacting nonlinear stochastic differential equations

Mini course I and II

Bastien Fernandez

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 characterized 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
synchronizes, 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 synchronized states, in
particular for coupling intensities below the synchronization threshold.
Our techniques involve a centre-manifold analysis of the
desynchronization bifurcation, with a precise control of the stability
of bifurcating solutions, allowing us to give a detailed description of
the system's potential landscape.