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Chaotic Attractor with Bounded Function Proceedings of Engineering & Technology (PET) Copyright IPCO-2016 pp. 880-886 Chaotic Attractor With Bounded Function Nahed Aouf Souayed kais Bouallegue Electronical and micro-electronical laboratory, Departement of Electrical Engineering, Faculty of science of Monastir, Higher Institute Of Applied Sciences and Technology of Sousse, University of Monastir University of Sousse, Monastir 5000,Tunisia 4003 Sousse Tunisa Email: [email protected] Nader Belghith Mohsen Machhout Departement of Electrical Engineering, Electronical and micro-electronical laboratory, Higher Institute Of Applied Sciences and Technology of Sousse, Faculty of science of Monastir, University of Sousse, University of Monastir 4003 Sousse Tunisa Monastir 5000,Tunisia Abstract—In this paper we propose a new chaotic attractor researchers on chaos concerning two wanting as- with bounded behavior by combining Julia’s Process with the pects of the Lorenz equations (Lorenz,1963). The Chua’s attractor and a bounded function hyperbolic tangent. We use many forms to apply this combination and obtain different existence of the chaotic attractor from the chua figures of these new behaviors. circuit was confirmed numerically by Matsumoto I. INTRODUCTION (1984) , and observed experimentally by by Zhong During the last four decades[1],the behavior of and Ayrom (1985), and proved strictly by Chua chaotic attractor has been competently studied . It and al(1986). The basic approach of the proof is reaches many natural and artificial dynamic systems illustrated in a guided exercise on Chuas circuit such as human heart, mechanical system, electronic in the well-known textbook by Hirsch, Smale and circuits, etc [2]. There are so many classical attrac- Devaney (2003).[4] This system has become one of tors are known until now such us Lorenz , Rossler the models in the research of chaos and is described , Chua , Chen, and others... Our approach in this by paper is to generate a new behavior of chaotic attractor using Chua attractor combined with Julia 8 Process and a bounded function. To create a chaotic > x_ 1 = x2 > system with multi-scroll attractor became a desired <> goal for many engineering application [11].in or- x_ 2 = x3 (1) der to modify a multi-scroll attractor we apply > > a bounded function.we choose to use hyperbolic : x_ 3 = a(−x1 − x2 − x3 + f(x1)) tangent for this application and after that we will check the different modification. The rest of this paper is organized as follows. In section 2, we present the chaotic attractor and the Julia process. where x_,y_ and z_ are the first time derivatives and a is Then we explain the formalism mathematic . Our a real parameter. Where f(x1) is a statured function proposed scheme, experimental results and analysis as follows : is then presented in section 3. Finally, the paper is concluded in section4. II. Chaotic Attractor A. Chua attractor 8 < k; ifx > 1 In 1983, chua invented the chua circuit in order to f(x) = kx; ifjxj < 1 (2) response to two unsuccessful quest between many : −k; ifx < −1 C. Bounded Function In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other way, there exists a real number M such that jf(x)j ≤ M (4) Fig. 1. classical Chua attractor with 2 scrolls In our case we have chosen hyperbolic tangent B. julia process as a bounded function which is continuous on its In recent years, there have been a lot of develop- domain, bounded, and symmetric, namely odd, since ments in Julia sets, including qualitative characters, we have tanh(-x) = -tanh(x). applications and controls. In this section, the use jtanh(x)j ≤ 1 (5) of algorithms inspired from Julia processes, will be presented. To generate Julia processes, some of the The derivative: properties are well known: [tanh(x)]’ = 1/cosh2(x). • The Julia set is a repeller. D. Mathematic Formulation • The Julia set is invariant In this section,we study two chaotic attractor • An orbit on Julia set is either periodic or using transformation by julia’s process as cited in chaotic. [5] • All unstable periodic points are on Julia set. (yi+1; xi+1) = Pj(xi; yi). • The Julia set is either wholly connected or Our method is to apply the methodology cited in wholly disconnected. paper [5] we conceder this two systems which are All sets generated only with Julia sets combination defined for two different chaotic attractor respec- have fractal structure[13]. Real and imaginary parts tively: of the complex numbers are separately calculated. the first chaotic attractor 2 2 xi+1 = xi − yi + xc 8 (3) x_ = f (x ; x ; x ) + α yi+1 = xiyi + yc < 1 1 1 2 3 1 x_ 2 = f2(x1; x2; x3) + β1 (6) : x_ 3 = f3(x1; x2; x3) Algorithm 1 (xi+1; yi+1) = Pj(xi; yi) We use the same chaotic attractor and we modify 1: if x < 0 then i q p 2 2 (xi) the value of α and β So we consider the second one 2: xi+1 = ( ((xi) + (yi) ) + 2 ) yi as follows: 3: yi+1 = 2xi+1 8 4: end if < y_1 = f1(x1; x2; x3) + α2 5: x = 0 if i then y_2 = f2(x1; x2; x3) + β2 (7) 6: if xi > 0 then : y_ = f (x ; x ; x ) yi 3 3 1 2 3 7: xi+1 = 2yi+1 8: end if we treat the first system with a julia’s fractal 9: if xi < 0 then Process. 10: yi+1 = 0 the results are obtained as follows: 11: end if (u; v) = Pj(_x1; x_ 2). 12: end if 13: if x > 0 then (u; v) = Pj(f1(x1; x2) + α1; f2(x1; x2) + β1). i q p 2 2 (xi) the second system as follows: 14: yi+1 = ( ((xi) + (yi) ) − 2 ) yi (p; q) = P (_y ; y_ ). 15: xi+1 = j 1 2 2xi+1 16: if yi < 0 then (p; q) = Pj(f1(x1; x2) + α2; f2(x1; x2) + β2). 17: yi+1 = −yi+1 after,we generate two outputs(XG;YG) 18: end if (XG;YG) = Pj((u − p); (v − q)). 19: end if Pj switches between two cases: The first one when xg1 > 0 the second case is so the (XG;YG) takes this results when xg1 < 0 if x_ > 0,the outputs are computed as follows 8 rq 1 < 2 2 (u−p) XG = (u − p) + (v − q) + 2 q (v−q) ( p x_ u = x_ 2 +x _ 2 + 1 : YG = 1 2 2 (8) 2(u−p) x_ (14) v = 2 2x _ 1 III. RESULTS OF IMPLEMENTATION if x_ 1 < 0,the outputs are computed as follows In our approach we apply a combination of julia’s ( u = x_ 2 2x _ 2 process [5] with chua attractor as follow: q (9) p 2 2 x_ 1 Let E be the complete metric unit, Φ a fractal v = x_ 1 +x _ 2 + 2 processes system of E in E such as: the second system is as follow: E!E (p; q) = Pj(_y1; y_2). Φ:(f1; f2) ! (XG;YG) if y_1 > 0,the outputs are computed as follows Pj(Xk;Yk) = (Xk+1;Yk+1) and we apply the hyperbolic tangent each time to a ( q p = py_2 +y _2 + y_1 variable. 1 2 2 (10) q = y_2 2y _2 A. A combination between Julia Process and Chua attractor if y_1 < 0 ,the outputs are computed as follows The simulation results allows us to obtain a chaotic attractor in a convex space. The number ( p = y_1 of scrolls are increased as shown in the figures 2y _2 q (11) bellow: Let E be the complete metric unit, Φ a q = py_2 +y _2 + y_1 1 2 2 fractal processes system of E in E such as: We concider (XG;YG), the output vector generated E!E as follows: Φ:(f1; f2) ! (XG;YG) (X ;Y ) = P ((u − p); (v − q)). G G j (XG;YG) = Pj((y1 + β1); (y2 + β2): (15) In this stage, we have four cases of switching: case 1: when (_x1 > 0) and (_y1 > 0) case 2: when (_x1 > 0) and (_y1 < 0) case 3: when (_x1 < 0) and (_y1 > 0) case 4: when (_x1 < 0) and (_y1 < 0) we analyse only the first one: when (_x1 > 0) and (_y1 > 0) so (u; v) takes this result Fig. 2. CHUA attractor with 4 scrolls q ( p x_ u = x_ 2 +x _ 2 + 1 1 2 2 (12) Let E be the complete metric unit, Φ a fractal v = x_ 2 2x _ 1 processes system of E in E such as: and (p; q) takes this result E!E Φ:(f1; f2) ! (XG;YG) ( q p = py_2 +y _2 + y_2 1 2 2 (13) (U1;V1) = Pj((y1 + β1); (y2 + β2): q = y_2 (16) 2y _2 (XG;YG) = Pj(U1;V1): Fig. 3. CHUA attractor with 8 scrolls E Φ Let be the complete metric unit, a fractal Fig. 5. CHUA attractor with 32 scrolls processes system of E in E such as: Let E be the complete metric unit, Φ a fractal E!E processes system of E in E such as: Φ:(f ; f ) ! (X ;Y ) 1 2 G G E!E Φ:(f ; f ) ! (X ;Y ) 8 1 2 G G (U1;V1) = Pj((y1 + β1); (y2 + β2)): 8 < > (U1;V1) = Pj((y1 + β1); (y2 + β2): (U2;V2) = Pj(U1;V1): (17) > > (U2;V2) = Pj(U1;V1): : (XG;YG) = Pj(U2;V2): < (U3;V3) = Pj(U2;V2): (19) > (U4;V4) = Pj(U3;V3): > : (XG;YG) = Pj(U4;V4): Fig. 4. CHUA attractor with 16 scrolls Let E be the complete metric unit, Φ a fractal processes system of E in E such as: Fig.
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