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Entropy Increase in Switching Systems
Centro de Investigación Operativa, Universidad Miguel Hernández. Avda. de la Universidad s/n, Elche 03202, Spain
Fachbereich Mathematik, Johan Wolfgang Goethe Universität. Frankfurt am Main 60054, Germany
* Author to whom correspondence should be addressed.
Received: 10 May 2013; in revised form: 3 June 2013 / Accepted: 3 June 2013 / Published: 7 June 2013
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
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Cite This Article
MDPI and ACS Style
Amigó, J.M.; Kloeden, P.E.; Giménez, Á. Entropy Increase in Switching Systems. Entropy 2013, 15, 2363-2383.
Amigó JM, Kloeden PE, Giménez Á. Entropy Increase in Switching Systems. Entropy. 2013; 15(6):2363-2383.
Amigó, José M.; Kloeden, Peter E.; Giménez, Ángel. 2013. "Entropy Increase in Switching Systems." Entropy 15, no. 6: 2363-2383.