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

Jeroen S. W. Lamb, Martin Rasmussen, and Christian S. Rodrigues

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Submission date: 25. May. 2011
Pages: 17
published in: Proceedings of the American Mathematical Society, 143 (2015) 9, p. 3927-3937 
DOI number (of the published article): 10.1090/S0002-9939-2015-12544-0
Bibtex
MSC-Numbers: 37G35, 37H20, 37C70, 49K21, 37B25, 34A60
Keywords and phrases: Set-valued dynamical systems, random dynamical systems, Control
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Abstract:
We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are often 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.