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Necessary and sufficient conditions for stable synchronization in random dynamical systems

Published online by Cambridge University Press:  24 January 2017

JULIAN NEWMAN Open the ORCID record for JULIAN NEWMAN [Opens in a new window]
JULIAN NEWMAN*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, UK email [email protected]
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Abstract

For a composition of independent and identically distributed 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 (‘synchronization’). Namely, we find that synchronization 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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1857 - 1875
Copyright
© Cambridge University Press, 2017 

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