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BIRS Workshop Lecture Videos

Necessary and Sufficient Conditions for Stable Synchronisation in Random Dynamical Systems Newman, Julian


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 ⇢ 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 su\0cient) replaces the intricate vector field conditions assumed in Baxendale’s similar result of 1991, where (working on a compact manifold) su\0cient conditions are given for synchronisation to occur in a SDE with negative Lyapunov exponents.

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