Generating Fully Bounded Chaotic Attractors

Generating Fully Bounded Chaotic Attractors

Zeraoulia Elhadj (University of Tébessa, Algeria)
DOI: 10.4018/978-1-4666-3890-7.ch013
OnDemand PDF Download:
No Current Special Offers


Generating chaotic attractors from nonlinear dynamical systems is quite important because of their applicability in sciences and engineering. This paper considers a class of 2-D mappings displaying fully bounded chaotic attractors for all bifurcation parameters. It describes in detail the dynamical behavior of this map, along with some other dynamical phenomena. Also presented are some phase portraits and some dynamical properties of the given simple family of 2-D discrete mappings.
Chapter Preview

1. Introduction

Generating chaotic attractors from nonlinear dynamical system is quite important, because of their applicability in sciences and engineering. The discreet mathematical models are gotten directly via scientific experiences, or by the use of the Poincaré section for the study of continuous-time models. This type of applications is used in secure communications using the notions of chaos (Tsonis, 1992; Andreyev, Belsky, Dmitriev, & Kuminov, 1996; Newcomb & Sathyan, 1983). Many papers have described chaotic systems, one of the most famous being a two-dimensional discrete map which models the original Hénon map 978-1-4666-3890-7.ch013.m01 studied in (Hénon, 1976; Benedicks & Carleson, 1991; Sprott, 1993; Zeraoulia & Sprott, 2008). This map has been widely studied because it is the simplest example of a dissipative map with chaotic solutions. It has a single quadratic nonlinearity and a constant area contraction over the orbit in the xy-plane. However, the Hénon map is unbounded for the almost values of its bifurcation parameters. Thus, constructing a fully bounded chaotic map is a very important result. In the literature, there is some cases where the boundedness of a map was proved rigorously in some regions of the bifurcation parameters space, for example in (Zeraoulia & Sprott, 2008) it was proved that the two-dimensional, 978-1-4666-3890-7.ch013.m02 discrete mapping given by 978-1-4666-3890-7.ch013.m03 is bounded for all 978-1-4666-3890-7.ch013.m04 and unbounded for all and 978-1-4666-3890-7.ch013.m05 This map is capable to generating “multi- fold” strange attractors via period-doubling bifurcation routes to chaos. This partial boundedness of the above map is due to the presence of the terms 978-1-4666-3890-7.ch013.m06 and 978-1-4666-3890-7.ch013.m07 To avoid this problem, we will consider maps of the form 978-1-4666-3890-7.ch013.m08 where 978-1-4666-3890-7.ch013.m09 is the vector of bifurcation parameters space and978-1-4666-3890-7.ch013.m10 is the vector of the state space. The simplest form of this map is obtained when the functions f and g are linear and the resulting map displays chaotic attractors.

In this paper we present some phase portrait and some dynamical properties of the following simple family of 2-D discrete mappings:

(1) where978-1-4666-3890-7.ch013.m12 makes a part of the bifurcation parameters space and f and g are linear functions in their corresponding arguments. Equation (1) is an interesting minimal system, similar to the 2-D linear quadratic mapping but with the functions 978-1-4666-3890-7.ch013.m13 and 978-1-4666-3890-7.ch013.m14.

Complete Chapter List

Search this Book: