Generating Fully Bounded Chaotic Attractors

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. 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 twodimensional, C discrete mapping given by x y a x by x , sin , ( ) → − + ( ) 1 is bounded for all b < 1 and unbounded for all b > 1 . This map is capable to generating “multifold” strange attractors via period-doubling bifurcation routes to chaos. This partial boundedness of the above map is due to the presence of the terms by and x. To avoid this DOI: 10.4018/jalr.2011070104 International Journal of Artificial Life Research, 2(3), 36-42, July-September 2011 37 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. problem, we will consider maps of the form x y f X g X , , , , ( ) → ( ) ( ) ( ) m m where m Î R is the vector of bifurcation parameters space and X R Î is the vector of the state space. The simplest form of this map is obtained when the functions f and g are nonlinear and 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: x y f a a x y x y g a a x       → ( ) 0 3 0 3 , , sin , cos , cos , sin , , sin , cos , cos , sin y x y ( ) 