On the occurrence of chaos via different routes to chaos: period doubling and border-collision bifurcations
type: Item
instance of: scholarly article
title: On the occurrence of chaos via different routes to chaos: period doubling and border-collision bifurcations
zbMATH Open document ID: 1187.37083
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-01-19T00:00:00Z
review text: One of the most well-known 2D discrete models is the Hénon mapping [\textit{M. Hénon}, Commun. Math. Phys. 50, 69--76 (1976; Zbl 0576.58018)] defined by the relation \[ H(x,y)=\begin{cases} 1-ax^2+y,\\ bx.\end{cases}\tag{1} \] This article introduces and justifies a rarely observed phenomenon shown by a new mapping obtained via a direct modification of the Hénon mapping (1), which is the occurrence of the same chaotic attractor via two different and distinguishable routes to chaos: the standard period doubling and border-collision bifurcations as typical futures. The new proposed system that realized this phenomenon is the following 2D discrete chaotic mapping: \[ f(x,y)=\begin{cases} 1-a|x^2-y|+y,\\ bx,\end{cases}\tag{2} \] where \(a\) and \(b\) are bifurcation parameters. The study is just reduced to some numerical simulations and computations.
reviewed by: Irina V. Konopleva
Mathematics Subject Classification ID: 37J20
zbMATH DE Number: 5660111
zbMATH Keywords: 2D piecewise smooth discrete-time chaotic mapping
author: Zeraoulia Elhadj
MaRDI profile type: Publication
full work available at URL: s10958-009-9545-5
OpenAlex ID: W2007714662
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DOI: 10.1007/S10958-009-9545-5
description: scientific article; zbMATH DE number 5660111