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.

References in corpus (1)

• Noise-Induced Synchronization, Desynchronization, and Clustering in Globally Coupled Nonidentical Oscillators

Cited by in corpus (1)

• Synchronisation in Invertible Random Dynamical Systems on the Circle