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arXiv · 1408.5599

Necessary and Sufficient Conditions for Stable Synchronisation in Random Dynamical Systems

Julian Newman
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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.

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Julian Newman. 2014-08-24. Necessary and Sufficient Conditions for Stable Synchronisation in Random Dynamical Systems. https://arxiv.org/abs/1408.5599

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