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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Conditioned Lyapunov exponents for random dynamical systems


Authors: Maximilian Engel, Jeroen S. W. Lamb and Martin Rasmussen
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37A50, 37H15; Secondary 37H10
DOI: https://doi.org/10.1090/tran/7803
Published electronically: May 20, 2019
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Abstract: We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context.

The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the asymptotic stability behaviour of trajectories that remain within a bounded domain and, in particular, that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced, and its main characteristics are established.


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Additional Information

Maximilian Engel
Affiliation: Zentrum Mathematik der TU München, Boltzmannstr. 3, D-85748 Garching bei München, Germany
Email: maximilian.engel@tum.de

Jeroen S. W. Lamb
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: jsw.lamb@imperial.ac.uk

Martin Rasmussen
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: m.rasmussen@imperial.ac.uk

DOI: https://doi.org/10.1090/tran/7803
Received by editor(s): June 30, 2018
Received by editor(s) in revised form: December 17, 2018
Published electronically: May 20, 2019
Additional Notes: The first author would like to thank the Department of Mathematics at Imperial College London and the SFB Transregio 109 “Discretization in Geometry and Dynamics”, sponsored by the German Research Foundation (DFG), for support during the course of this research.
The second and third authors would like to thank the EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems for support during the course of this research.
The first author is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society