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Entropy 2013, 15(6), 2363-2383; doi:10.3390/e15062363

Entropy Increase in Switching Systems

1
Centro de Investigación Operativa, Universidad Miguel Hernández. Avda. de la Universidad s/n, Elche 03202, Spain
2
Fachbereich Mathematik, Johan Wolfgang Goethe Universität. Frankfurt am Main 60054, Germany
*
Author to whom correspondence should be addressed.
Received: 10 May 2013 / Revised: 3 June 2013 / Accepted: 3 June 2013 / Published: 7 June 2013
(This article belongs to the Special Issue Dynamical Systems)
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Abstract

The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox.
Keywords: non-autonomous dynamical systems; switching systems; set-valued pullback attractors; topological entropy; complexity non-autonomous dynamical systems; switching systems; set-valued pullback attractors; topological entropy; complexity
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Amigó, J.M.; Kloeden, P.E.; Giménez, Á. Entropy Increase in Switching Systems. Entropy 2013, 15, 2363-2383.

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