{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:39:41Z","timestamp":1772293181568,"version":"3.50.1"},"reference-count":17,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2013,6,7]],"date-time":"2013-06-07T00:00:00Z","timestamp":1370563200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u201cmacroscopic\u201d and \u201cmicroscopic\u201d 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\u2019s paradox.<\/jats:p>","DOI":"10.3390\/e15062363","type":"journal-article","created":{"date-parts":[[2013,6,10]],"date-time":"2013-06-10T03:32:11Z","timestamp":1370835131000},"page":"2363-2383","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Entropy Increase in Switching Systems"],"prefix":"10.3390","volume":"15","author":[{"given":"Jos\u00e9","family":"Amig\u00f3","sequence":"first","affiliation":[{"name":"Centro de Investigaci\u00f3n Operativa, Universidad Miguel Hern\u00e1ndez. Avda. de la Universidad s\/n, Elche 03202, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Kloeden","sequence":"additional","affiliation":[{"name":"Fachbereich Mathematik, Johan Wolfgang Goethe Universit\u00e4t. Frankfurt am Main 60054, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u00c1ngel","family":"Gim\u00e9nez","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n Operativa, Universidad Miguel Hern\u00e1ndez. Avda. de la Universidad s\/n, Elche 03202, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,6,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Amig\u00f3, J.M., Kloeden, P.E., and Gim\u00e9nez, A. (2013). Switching systems and entropy. J. Differ. Equ. Appl.","DOI":"10.1080\/10236198.2013.788166"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kloeden, P.E., and Rasmussen, M. (2011). Nonautonomous Dynamical Systems, American Mathematical Society.","DOI":"10.1090\/surv\/176"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Liberzon, D. (2003). Switching in Systems and Control, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-0017-8"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1137\/05063516X","article-title":"Stability criteria for switched and hybrid systems","volume":"49","author":"Shorten","year":"2007","journal-title":"SIAM Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.physd.2004.10.003","article-title":"Can two chaotic systems give rise to order?","volume":"200","author":"Almeida","year":"2005","journal-title":"Phys. D: Nonlinear Phenom."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.physd.2005.07.015","article-title":"Randomly chosen chaotic maps can give rise to nearly ordered behavior","volume":"210","author":"Boyarsky","year":"2005","journal-title":"Phys. D: Nonlinear Phenom."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.physd.2006.05.004","article-title":"Dynamic Parrondo\u2019s paradox","volume":"218","author":"Linero","year":"2006","journal-title":"Phys. D: Nonlinear Phenom."},{"key":"ref_8","first-page":"206","article-title":"Parrondo\u2019s paradox","volume":"14","author":"Harner","year":"1999","journal-title":"Stat. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1080\/10236190008808212","article-title":"Pullback attractors in nonautonomous difference equations","volume":"6","author":"Kloeden","year":"2000","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_10","first-page":"125","article-title":"The relationship between pullback, forward and global attractors of nonautonomous dynamical systems","volume":"2","author":"Cheban","year":"2002","journal-title":"Nonlinear Dyn. Syst. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1142\/S0219493703000632","article-title":"Pullback attractors of nonautonomous semidynamical systems","volume":"3","author":"Kloeden","year":"2003","journal-title":"Stoch. Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1142\/S0218127405012454","article-title":"Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems","volume":"15","author":"Kloeden","year":"2005","journal-title":"Int. J. Bifurcat. Chaos Appl. Sci. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1080\/14689360500446262","article-title":"Nonautonomous attractors of switching systems","volume":"21","author":"Kloeden","year":"2006","journal-title":"Dyn. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Amig\u00f3, J.M. (2010). Permutation Complexity in Dynamical Systems, Springer.","DOI":"10.1007\/978-3-642-04084-9"},{"key":"ref_15","unstructured":"Walters, P. (2000). An Introduction to Ergodic Theory, Springer."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Peitgen, H.O., J\u00fcrgens, H., and Saupe, D. (2004). Chaos and Fractals: New Frontiers of Science, Springer. [2nd ed.].","DOI":"10.1007\/b97624"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"026205","DOI":"10.1103\/PhysRevE.67.026205","article-title":"Estimating topological entropy via a symbolic data compression technique","volume":"67","author":"Hirata","year":"2003","journal-title":"Phys. Rev. E"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/6\/2363\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:47:15Z","timestamp":1760219235000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/6\/2363"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,6,7]]},"references-count":17,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2013,6]]}},"alternative-id":["e15062363"],"URL":"https:\/\/doi.org\/10.3390\/e15062363","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,6,7]]}}}