Necessary and Sufficient Conditions for Stable Synchronisation in Random Dynamical Systems
Author(s)
Newman, JMI
Type
Journal Article
Abstract
For a composition 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 non-empty 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.
Date Acceptance
2016-09-06
Citation
Ergodic Theory and Dynamical Systems
ISSN
1469-4417
Publisher
Cambridge University Press (CUP)
Journal / Book Title
Ergodic Theory and Dynamical Systems
Subjects
General Mathematics
0101 Pure Mathematics
Publication Status
Accepted