Generating Fully Bounded Chaotic Attractors

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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 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, discrete mapping given by is bounded for all and unbounded for all and . 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 and To avoid this problem, we will consider maps of the form where is the vector of bifurcation parameters space and 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) where 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 and .