A NEW CHAOTIC ATTRACTOR FROM 2D DISCRETE MAPPING VIA BORDER-COLLISION PERIOD-DOUBLING SCENARIO

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1 A NEW CHAOTIC ATTRACTOR FROM D DISCRETE MAPPING VIA BORDER-COLLISION PERIOD-DOUBLING SCENARIO ZERAOULIA ELHADJ Received 1 April 5 The following map is studied: (, ) (1 + a( )+,b). It is proved numericall that this model can displa two different chaotic attractors, one is new and the other is a Lozi-tpe attractor. The new chaotic attractor is allowed via a border-collision perioddoubling scenario, which is differentfrom the classical period-doubling bifurcation. 1. Introduction The discreet mathematical models are gotten directl via scientific eperiences, or b the use of the Poincaré section for the stud of a continuous model. One of these models is the Henon map. Man papers have described chaotic sstems, one of the most famous being a two-dimensional discrete map which models the original Henon map [3, 4, 5, 7, 8]. Moreover, it is possible to change the form of the Henon map for obtaining others chaotic attractors [, 6], this tpe of applications is used in secure communications using the notions of chaos. The Lozi map is D noninvertible iterated map proposed b Lozi [6]asfollow: (, ) ( 1 a +,b ). (1.1) This model gives a chaotic attractor called Lozi attractor and its shape is resembled to the one shown in Figure.1.. The proposed model In this article, we essentiall stud the following modified Lozi map: (, ) ( 1+a ( ) +,b ). (.1) We show numericall that the new model (.1) can displa two different chaotic attractors; includingthe classicallozi-tpe attractoras shown infigure.1 to Figure 4.. Copright 5 Hindawi Publishing Corporation Discrete Dnamics in Nature and Societ 5:3 (5) DOI: /DDNS.5.35

2 36 Comparison and routes to chaos Figure.1. A Lozi-tpe chaotic attractor obtained from sstem (.1)fora = 1.8, b = Figure.. A Lozi-tpe chaotic attractor obtained from sstem (.1)fora = 1.8, b = Comparison with the Lozi sstem The sstem (.1) has the same compleit as the Lozi sstem (1.1), the are both twodimensional discrete noninvertible maps. However, the two models are topologicall not equivalent because the Lozi sstem (1.1) is a piecewise-linear map,but model(.1) is a nonlinear sstem. In addition; it can be rigorousl proved that a non-singular homeomorphism that transforms each sstem to other does not eist. The proof needed some straightforward but tedious algebra which leads to a sstem of algebraic equations without solutions. 4. Route to chaos It is well known that while varing the parameter a,thehénon attractor is obtained via a period-doubling bifurcation route to chaos as a tpical future [3], unless for Lozi map, no

3 Zeraoulia Elhadj Figure 4.1. A tpical orbit of sstem (.1) obtained for a = 1.35, b = Figure 4.. The new chaotic attractor obtained for a = 1.4, b =.3. period doubling route to chaos is allowed, and the attractor goes directl from a bordercollision bifurcation developed from a stable periodic orbit [4], and for the same case the new chaotic attractor given b sstem (.1) is obtained from a border-collision perioddoubling bifurcation scenario [1]; this scenario (see Figure 4.3) isformedbasequence of pairs of bifurcations, whereb each pair consists of a border-collision bifurcation and a pitchfork bifurcation. Thus, the three chaotic attractors go via different and distinguishable route to chaos as a tpical future. 5. Conclusion This paper reports the finding of a new two-dimensional discrete chaotic attractor obtained via direct modification of the Lozi mapping. The new chaotic attractor is allowed via a border-collision period-doubling scenario. More detailed analsis about the structure, dnamics and bifurcations of the new sstem (.1) will be providedin near future.

4 38 Comparison and routes to chaos a Figure 4.3. Border-collision period-doubling scenario route to chaos observed for sstem (.1). References [1] V.AvrutinandM.Schanz,Border-collision period-doubling scenario,phs.rev.e(3)7 (4), no., 6, 11. [] M. A. Aziz-Alaoui, C. Robert, and C. Grebogi, Dnamics of a Hénon-Lozi-tpe map, Chaos Solitons Fractals 1 (1), no. 1, [3] M. Benedicks and L. Carleson, The dnamics of the Hénon map, Ann. of Math. () 133 (1991), no. 1, [4] Y. Cao and Z. Liu, Strange attractors in the orientation-preserving Lozi map, Chaos Solitons Fractals 9 (1998), no. 11, [5] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phs. 5 (1976), no. 1, [6] R. Lozi, Un attracteur étrange du tpe attracteur de Hénon, J.Phs.ColloqueC539 (1978), Supplément au no. 8, 9 1 (French). [7] F. R. Marotto, Chaotic behavior in the Hénon mapping, Comm. Math. Phs. 68 (1979), no., [8] M. Misiurewicz, Strange attractors for the Lozi mappings, Nonlinear Dnamics (Internat. Conf., New York, 1979) (R. G. Heman, ed.), Ann. New York Acad. Sci., vol. 357, New York Academ of Sciences, New York, 198, pp Zeraoulia Elhadj: Department of Mathematics, Universit of Tébéssa, 1 Tébéssa, Algeria address: zelhadj1@ahoo.fr

5 Mathematical Problems in Engineering Special Issue on Modeling Eperimental Nonlinear Dnamics and Chaotic Scenarios Call for Papers Thinking about nonlinearit in engineering areas, up to the 7s, was focused on intentionall built nonlinear parts in order to improve the operational characteristics of a device or sstem. Keing, saturation, hsteretic phenomena, and dead zones were added to eisting devices increasing their behavior diversit and precision. In this contet, an intrinsic nonlinearit was treated just as a linear approimation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analtical tools from Qualitative Theor of Differential Equations, allowing more precise analsis and snthesis, in order to produce new vital products and services. Bifurcation theor, dnamical sstems and chaos started to be part of the mandator set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theor, relating it with nonlinear and chaotic dnamics for natural and engineered sstems. Ideas of how this dnamics can be captured through precisel tailored real and numerical eperiments and understanding b the combination of specific tools that associate dnamical sstem theor and geometric tools in a ver clever, sophisticated, and at the same time simple and unique analtical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic cop of their complete manuscript through the journal Manuscript Tracking Sstem at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Poltechnic School, The Universit of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 17-1 São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Department of Phsics, King s College, Universit of Aberdeen, Aberdeen AB4 3UE, UK; grebogi@abdn.ac.uk Manuscript Due Februar 1, 9 First Round of Reviews Ma 1, 9 Publication Date August 1, 9 Hindawi Publishing Corporation