| Issue |
ESAIM: Proc.
Volume 46, November 2014
ECIT 2012, 19th European Conference on Iteration Theory
|
|
|---|---|---|
| Page(s) | 98 - 110 | |
| DOI | https://doi.org/10.1051/proc/201446009 | |
| Published online | 05 December 2014 | |
Invariant Curves of Quadratic Maps of the Plane from the One-Parameter Family Containing the Trace Map*
1 Lobachevsky Nizhny Novgorod State
University (Research University), Nizhny Novgorod, Russia
2 Lobachevsky Nizhny Novgorod State
University (Research University), Nizhny Novgorod, Russia
3 LAAS-CNRS, INSA, University of
Toulouse, France
The rigorous proofs are given: (1) for the existence of the unbounded invariant curves, containing the fixed point – source (μ + 1; 1), of the maps from the one-parameter family Fμ(x,y) = (xy, (x − μ)2), μ ∈ [0, 2]; (2) for the birth of the closed invariant curve from the elliptic fixed point (μ − 1; 1) for μ = 3 / 2. Numerical results are presented for the main steps of the evolution of this invariant curve, when μ changes in the interval (3 / 2, 2).
Résumé
Les démonstrations rigoureuses sont données : (1) pour l’existence des courbes invariantes non bornes contenant le point fixe - source (μ + 1; 1) des cartes de la famille a un parametre Fμ(x,y) = (xy, (x − μ)2), μ ∈ [0, 2]; (2) pour la naissance de la courbe fermée invariante du point fixe elliptique (μ − 1; 1) pour μ = 3 / 2. Numerical resultats sont presentes pour les principales étapes de l’évolution de cette courbe invariante, quand μ des changements dans l’intervalle (3 / 2, 2).
Mathematics Subject Classification: 37Exx / 34K18, 34K19
Key words: Quadratic map / fixed point / invariant curve
Mots clés : Carte quadratique / point fixe / courbe invariante
© EDP Sciences, SMAI 2014
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