Early-warning signals for bifurcations in random dynamical systems with bounded noise
File(s)
Accepted version
Author(s)
Kuehn, Christian
Malavolta, Giuseppe
Rasmussen, Martin
Type
Journal Article
Abstract
We consider discrete-time one-dimensional random dynamical systems with
bounded noise, which generate an associated set-valued dynamical system. We
provide necessary and sufficient conditions for a discontinuous bifurcation of
a minimal invariant set of the set-valued dynamical system in terms of the
derivatives of the so-called extremal maps. We propose an algorithm for
reconstructing the derivatives of the extremal maps from a time series that is
generated by iterations of the original random dynamical system. We demonstrate
that the derivative reconstructed for different parameters can be used as an
early-warning signal to detect an upcoming bifurcation, and apply the algorithm
to the bifurcation analysis of the stochastic return map of the Koper model,
which is a three-dimensional multiple time scale ordinary differential equation
used as prototypical model for the formation of mixed-mode oscillation
patterns. We apply our algorithm to data generated by this map to detect an
upcoming transition.
bounded noise, which generate an associated set-valued dynamical system. We
provide necessary and sufficient conditions for a discontinuous bifurcation of
a minimal invariant set of the set-valued dynamical system in terms of the
derivatives of the so-called extremal maps. We propose an algorithm for
reconstructing the derivatives of the extremal maps from a time series that is
generated by iterations of the original random dynamical system. We demonstrate
that the derivative reconstructed for different parameters can be used as an
early-warning signal to detect an upcoming bifurcation, and apply the algorithm
to the bifurcation analysis of the stochastic return map of the Koper model,
which is a three-dimensional multiple time scale ordinary differential equation
used as prototypical model for the formation of mixed-mode oscillation
patterns. We apply our algorithm to data generated by this map to detect an
upcoming transition.
Date Issued
2018-08-01
Date Acceptance
2018-03-27
Citation
Journal of Mathematical Analysis and Applications, 2018, 464 (1), pp.58-77
ISSN
0022-247X
Publisher
Elsevier
Start Page
58
End Page
77
Journal / Book Title
Journal of Mathematical Analysis and Applications
Volume
464
Issue
1
Copyright Statement
© 2018 Elsevier Inc. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Commission of the European Communities
Identifier
http://arxiv.org/abs/1803.00382v1
Grant Number
EP/I004165/1
643073
Subjects
math.DS
math.DS
37G35, 37H20, 37C70, 49K21, 70K70
Publication Status
Published
Date Publish Online
2018-04-09