Next Article in Journal
Entropy of Shortest Distance (ESD) as Pore Detector and Pore-Shape Classifier
Next Article in Special Issue
A Method for Choosing an Initial Time Eigenstate in Classical and Quantum Systems
Previous Article in Journal
Quantum Contextuality with Stabilizer States
Previous Article in Special Issue
A Unification between Dynamical System Theory and Thermodynamics Involving an Energy, Mass, and Entropy State Space Formalism
Article Menu

Export Article

Open AccessArticle
Entropy 2013, 15(6), 2363-2383;

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 / Revised: 3 June 2013 / Accepted: 3 June 2013 / Published: 7 June 2013
(This article belongs to the Special Issue Dynamical Systems)
Full-Text   |   PDF [156 KB, uploaded 24 February 2015]   |  


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. View Full-Text
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

Figure 1

This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
Printed Edition Available!
A printed edition of this Special Issue is available here.

Share & 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.

Show more citation formats Show less citations formats

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top