Necessary and Sufficient Conditions for Stable Synchronisation in Random Dynamical Systems

Abstract

For a product of i.i.d. random maps or a memoryless stochastic flow on a compact space \(X\), we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories ("synchronisation"). Namely, we find that synchronisation occurs and is stable if and only if the system exhibits the following properties: (i) there is a smallest deterministic invariant set \(K \subset X\), (ii) any two points in \(K\) are capable of being moved closer together, and (iii) \(K\) admits asymptotically stable trajectories. Our first condition (for which unique ergodicity of the one-point transition probabilities is sufficient) replaces the intricate vector field conditions assumed in Baxendale's similar result of 1991, where (working on a compact manifold) sufficient conditions are given for synchronisation to occur in a SDE with negative Lyapunov exponents.