Issue |
ESAIM: Proc.
Volume 36, April 2012
European Conference on Iteration Theory 2010
|
|
---|---|---|
Page(s) | 170 - 179 | |
DOI | https://doi.org/10.1051/proc/201236013 | |
Published online | 28 August 2012 |
Coexisting cycles in a class of 3-D discrete maps
Dept. Economic and Social Science, Catholic
University, via Emilia Parmense
84, 49100
Piacenza,
Italy
e-mail: anna.agliari@unicatt.it
In this paper we consider the class of three-dimensional discrete maps M (x, y, z) = [φ(y), φ(z), φ(x)], where φ : ℝ → ℝ is an endomorphism. We show that all the cycles of the 3-D map M can be obtained by those of φ(x), as well as their local bifurcations. In particular we obtain that any local bifurcation is of co-dimension 3, that is three eigenvalues cross simultaneously the unit circle. As the map M exhibits coexistence of cycles when φ(x) has a cycle of period n ≥ 2, making use of the Myrberg map as endomorphism, we describe the structure of the basins of attraction of the attractors of M and we study the effect of the flip bifurcation of a fixed point.
Résumé
Dans ce papier nous considérons la classe des applications trois-dimensionnelles discrètes M (x, y, z) = [φ(y), φ(z), φ(x)], où φ : ℝ → ℝ est un endomorphisme. Nous montrons que tous les cycles de l’application 3-D M peuvent être obtenus par ceux de φ(x), ainsi que leurs bifurcations locales. En particulier, nous obtenons que toute bifurcation locale est de co-dimension 3, c’est-à-dire que trois valeurs propres franchissent simultanément le cercle unité. Comme l’application M exhibe une coexistence de cycles lorsque φ(x) a un cycle de période n ≥ 2, en utilisant l’application de Myrberg comme endomorphisme, nous décrivons la structure des bassins d’attraction des attracteurs de M et nous étudions les effets d’une bifurcation de doublement de période d’un point fixe.
Mathematics Subject Classification: 37B99 / 37G35 / 37M05
Key words: 3-D discrete maps / Periodic orbits / Bifurcations of co-dimension 3
Mots clés : Transformation à temps discret de dimension 3 / Orbites périodiques / Bifurcations de codimension 3
© EDP Sciences, SMAI 2012
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