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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fixed points imply chaos for a class of differential inclusions that arise in economic models


Authors: Brian E. Raines and David R. Stockman
Journal: Trans. Amer. Math. Soc. 364 (2012), 2479-2492
MSC (2010): Primary 34A60, 54H20, 37B20, 37D45
Published electronically: January 19, 2012
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Abstract: We consider multi-valued dynamical systems with continuous time of the form $ \dot {x}\in F(x)$, where $ F(x)$ is a set-valued function. Such models have been studied recently in mathematical economics. We provide a definition for chaos, $ \omega $-chaos and topological entropy for these differential inclusions that is in terms of the natural $ \mathbb{R}$-action on the space of all solutions of the model. By considering this more complicated topological space and its $ \mathbb{R}$-action we show that chaos is the `typical' behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic, $ \omega $-chaotic, and has infinite topological entropy.


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Additional Information

Brian E. Raines
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798
Email: brian_raines@baylor.edu.

David R. Stockman
Affiliation: Department of Economics, University of Delaware, Newark, Delaware 19716
Email: stockman@udel.edu.

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05377-3
PII: S 0002-9947(2012)05377-3
Keywords: Chaos, fixed point, multi-valued dynamical systems
Received by editor(s): August 13, 2009
Received by editor(s) in revised form: April 16, 2010
Published electronically: January 19, 2012
Additional Notes: The first author was supported by NSF grant 0604958
The second author would like to thank the Lerner College of Business & Economics for its generous summer research support.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.