Branching processes and computational collapse of discretized unimodal mappings

Diamond, P and Vladimirov, I (2002) Branching processes and computational collapse of discretized unimodal mappings. International Journal of Bifurcation And Chaos, 12 12: 2847-2867. doi:10.1142/S0218127402006229

Author Diamond, P
Vladimirov, I
Title Branching processes and computational collapse of discretized unimodal mappings
Journal name International Journal of Bifurcation And Chaos   Check publisher's open access policy
ISSN 0218-1274
Publication date 2002
Sub-type Article (original research)
DOI 10.1142/S0218127402006229
Volume 12
Issue 12
Start page 2847
End page 2867
Total pages 21
Editor L.O. Chua
Place of publication Singapore
Publisher World Scientific Publishing Co
Collection year 2002
Language eng
Subject C1
230119 Systems Theory and Control
780101 Mathematical sciences
Abstract In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping.
Keyword Mathematics, Interdisciplinary Applications
Multidisciplinary Sciences
Dynamical Systems
Computational Collapse
Chaotic Dynamical-systems
Q-Index Code C1

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
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Created: Tue, 14 Aug 2007, 17:52:40 EST