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Topological bifurcations of minimal invariant sets for set-valued dynamical systems

Authors: Jeroen S. W. Lamb, Martin Rasmussen and Christian S. Rodrigues
Journal: Proc. Amer. Math. Soc. 143 (2015), 3927-3937
MSC (2010): Primary 37G35, 37H20, 37C70, 49K21; Secondary 37B25, 34A60
Published electronically: April 2, 2015
MathSciNet review: 3359583
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Abstract: We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are naturally satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.

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Additional Information

Jeroen S. W. Lamb
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom

Martin Rasmussen
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom

Christian S. Rodrigues
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany

Received by editor(s): November 26, 2013
Received by editor(s) in revised form: May 12, 2014
Published electronically: April 2, 2015
Additional Notes: The first author was supported by an FAPESP-Brazil Visiting Professorship (2009-18338-2)
The first and second author gratefully acknowledge partial support by EU IRSES project DynEurBraz and the warm hospitality of IMECC UNICAMP during the development of this paper
The second author was supported by an EPSRC Career Acceleration Fellowship and a Marie Curie Intra-European Fellowship of the European Community
The third author has received ERC funding under EU’s Seventh Framework Programme FP7 (grant agreement number: 267087)
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

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