Issue |
ESAIM: Proc.
Volume 36, April 2012
European Conference on Iteration Theory 2010
|
|
---|---|---|
Page(s) | 126 - 158 | |
DOI | https://doi.org/10.1051/proc/201236011 | |
Published online | 28 August 2012 |
Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map
1
Institute of Parallel and Distributed Systems, University of
Stuttgart, Germany
Viktor.Avrutin@ipvs.uni-stuttgart.de
2
Institute of Parallel and Distributed Systems, University of
Stuttgart, Germany
Ben.Futter@ipvs.uni-stuttgart.de
3
Department of Economics, Society and Politics, University of
Urbino, Italy
Laura.Gardini@uniurb.it
4
Institute of Parallel and Distributed Systems, University of
Stuttgart, Germany
Michael.Schanz@ipvs.uni-stuttgart.de
In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns out that there exists a strong connection between stable and unstable orbits and that the appearance of a single unstable orbit may cause an infinite number of stable orbits to appear. Based on this connection we provide a more precise explanation of the recently discovered self-similar bifurcation scenario occurring in the discontinuous flat top tent map denoted as the nested period incrementing scenario.
Résumé
Dans ce travail nous considérons l’application discontinue de tente à haut plat qui représente un exemple d’application régulière par morceaux discontinue, où la fonction du système est constante sur un intervalle. De telles applications montrent plusieurs aspects causés par cette valeur constante qui ne sont toujours pas suffisamment compris. Dans ce travail nous démontrons que pour l’applicatiion discontinue de tente à haut plat toutes les orbites périodiques instables peuvent devenir un attracteur de Milnor. De plus, il apparaît qu’il y a un forte connexion entre les orbites stables et instables et que l’apparition d’une seule orbite instable peut provoquer l’apparition d’un nombre infini d’orbites stables. Sur la base de cette connexion nous proposons une explication précisée du scénario de bifurcation auto-similaire découvert récemment pour l’applicatiion discontinue de tente à haut plat, le scénario d’incrément de période nichée.
Mathematics Subject Classification: 37E05 / 37E15 / 37G35
Key words: piecewise-smooth maps / discontinuous flat top tent map / maps with a horizontal part / nested period incrementing / Milnor attractors / U-sequence
Mots clés : applications régulières par morceaux / application discontinue de tente à haut plat / applications avec une partie horizontale / incrément de période nichée / attracteur de Milnor / U-suite
© EDP Sciences, SMAI 2012
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