%0 Journal Article
%T Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map
%A Avrutin Viktor
%A Futter Ben
%A Gardini Laura
%A Schanz Michael
%J ESAIM : Proceedings
%D 2012
%I EDP Sciences
%R 10.1051/proc/201236011
%X 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. Dans ce travail nous consid¨¦rons l¡¯application discontinue de tente ¨¤ haut plat qui repr¨¦sente un exemple d¡¯application r¨¦guli¨¨re par morceaux discontinue, o¨´ la fonction du syst¨¨me est constante sur un intervalle. De telles applications montrent plusieurs aspects caus¨¦s par cette valeur constante qui ne sont toujours pas suffisamment compris. Dans ce travail nous d¨¦montrons que pour l¡¯applicatiion discontinue de tente ¨¤ haut plat toutes les orbites p¨¦riodiques instables peuvent devenir un attracteur de Milnor. De plus, il appara t 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 pr¨¦cis¨¦e du sc¨¦nario de bifurcation auto-similaire d¨¦couvert r¨¦cemment pour l¡¯applicatiion discontinue de tente ¨¤ haut plat, le sc¨¦nario d¡¯incr¨¦ment de p¨¦riode nich¨¦e.
%K piecewise-smooth maps
%K discontinuous flat top tent map
%K maps with a horizontal part
%K nested period incrementing
%K Milnor attractors
%K U-sequence
%K applications r¨¦guli¨¨res par morceaux
%K application discontinue de tente ¨¤ haut plat
%K applications avec une partie horizontale
%K incr¨¦ment de p¨¦riode nich¨¦e
%K attracteur de Milnor
%K U-suite
%U http://dx.doi.org/10.1051/proc/201236011