Proceedings of Engineering & Technology (PET) Copyright IPCO-2016 pp. 880-886 Chaotic 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  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   k, ifx > 1 In 1983, chua invented the chua circuit in order to f(x) = kx, if|x| < 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

|f(x)| ≤ 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 |tanh(x)| ≤ 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 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 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 (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  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(x ˙ 1, 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  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(y ˙1, 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 (x ˙ 1 > 0) and (y ˙1 > 0) case 2: when (x ˙ 1 > 0) and (y ˙1 < 0) case 3: when (x ˙ 1 < 0) and (y ˙1 > 0) case 4: when (x ˙ 1 < 0) and (y ˙1 < 0) we analyse only the first one: when (x ˙ 1 > 0) and (y ˙1 > 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 )  1 2 G G (U1,V1) = Pj((y1 + β1), (y2 + β2)).    (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. 6. CHUA attractor with 64 scrolls

E → E 1) Combination between Julia Process and Chua attractor with different forms: Φ:(f , f ) → (X ,Y ) 1 2 G G In this section we present a new implementation when we combine th Chua attractor and Julia pro-   (U1,V1) = Pj((y1 + β1), (y2 + β2). cess whith constant the Figure 7bellow show us the  (U ,V ) = P (U ,V ). result. 2 2 j 1 1 (18) (U3,V3) = Pj(U2,V2). Let E be the complete metric unit, Φ a fractal   (XG,YG) = Pj(U3,V3). processes system of E in E such as: E → E tangent as a bounded function on the first variable Φ:(f1, f2) → (XG,YG) X. as mentioned in the equation bellow:  Let E be the complete metric unit, Φ a fractal (U1,V1) = Pj((y1 + β1), (y2 + β2)).  processes system of E in E such as: (U2,V2) = Pj((y1 + β1), (y2 − β2)). (20)  (X ,Y ) = P ((U − U ), (V − V )). G G j 1 2 1 2 E → E

Φ:(f1, f2) → (XG,YG)

  (U1,V1) = Pj((y1 + β1), (y2 + β2). (U2,V2) = Pj((y1 + β1), (y2 − β2).  (XG,YG) = Pj((U1 − U2), tanh(V1 − V2). (22)

Fig. 7. CHUA attractor with 8 scrolls

B. Application of Bounded Function

In this section, we apply the hyperbolic tangent function in the last equation and we obtain the figures below.

1) Bounded function applied on the first variable: In the first example we apply the hyperbolic tangent as a bounded function on the first variable X. as mentioned in the equation bellow: Let E be the complete metric unit, Φ a fractal processes system of E in E such as: Fig. 9. chaotic attractor with bounded behavior applied on the second variable E → E 3) Julia Process applied on the chaotic attractor with Φ:(f1, f2) → (XG,YG) bounded behavior applied on the first and second variable :   (U1,V1) = Pj((y1 + β1), (y2 + β2). (U2,V2) = Pj((y1 + β1), (y2 − β2). in this attractor we have used like before the tangent  (XG,YG) = Pj(tanh(U1 − U2), (V1 − V2). hyperbolic function but we cascaded two times (21) the julia process and the figure below shows the multiplication of the number of scrolls thanks to the julia process. Let E be the complete metric unit, Φ a fractal processes system of E in E such as:

E → E

Φ:(f1, f2) → (XG,YG)

 (U1,V1) = Pj((y1 + β1), (y2 + β2). Fig. 8. Another behavior of bounded chaotic attractor applied on the first  variable  (U2,V2) = Pj((y1 + β1), (y2 − β2). (XG,YG) = Pj(tanh(U1 − U2), (V1 − V2).  2) Bounded function applied on the second variable:  (XG1,YG1) = Pj(XG,YG). In the following attractor we apply the hyperbolic (23) [2] E.O Rossler,¨ Phys.Lett A 71(1979)155. [3] Cafagna G. Grassi Int J Biffurcation Chaos13(2003)2889. [4] EE. Mohmoud Dynamics and synchronization of a new hyperchaotic complex , Math Computation Model 55(2012)1951 − 1962. [5] K.Bouallegue,chaotic with separated scrolls & AIP CHAOS an interdisciplinary journal of nonlinear science 2015; [6] K.Bouallegue, A. Chaari, A. Toumi, Multi-scroll and multi-wing chaotic attractor generated with Julia process fractal,Chaos, Solitons & 2011; 44 : 79 − 85 [7] Stephen.Lynuch, Dynamical Systems with Applications using Mathe- matica, Birkhauser Boston, 2007,ISBN − 13 : 978?0?8176?4482?6. [8] F.jianwen, Chen.X, jianliang.T. Controlling Chen’s chaotic attractor Fig. 10. Chaotic attractor with multi-scrolls bounded on the first variable using two different techniques based on parameter identification. Chaos, Solitons & Fractals 32(2007); 1413 − 1418 [9] chyun-chau.F,Hsun-Heng.T. Control of discrete-time chaotic systems via feedback linearization. Chaos, Solitons & Fractals 13(2002)285 − 294 [10] E.yalin.M, A.K. Suykens.J. Multi-Scroll and Hypercube Attractors from Let E be the complete metric unit, Φ a fractal Josephson Junctions. IEEE International Symposium on Circuits and Systems, 2006.ISCAS2006. processes system of E in E such as: [11] http : //www.scholarpedia.org/article/Chuacircuit [12] Bouallegue.K. A new class of control using fractal processes. Inter- E → E national Conference on Control, Engineering Information Technology (CEIT 013) Proceedings Engineering Technology - V ol.3, pp.91 − Φ:(f1, f2) → (XG,YG) 99, 2013. [13] Lorenz E.N. Deterministic nonperiodic flow // J. Atmos. Sci.1963.V.20.130 − 141  (U ,V ) = P ((y + β ), (y + β ). [14] Rossler O.E. An Equation for Continuous Chaos // Physics Letters.  1 1 j 1 1 2 2 1976.V.57A.N5.397 − 398  (U2,V2) = Pj((y1 + β1), (y2 − β2). [15] Chua L.O., Lin G.N. Canonical Realization of Chuas Circuit Family // IEEE Transactions on Circuits and Systems. 1990.V.37.N4.885 − 902  (XG,YG) = Pj((U1 − U2), tanh(V1 − V2). [16] Chen, G. Ueta, T. Yet another chaotic attractor, // Int. J. Bifurcation and  Chaos.1999.N9, 14651466.  (XG1,YG1) = Pj(XG,YG). [17] Qi Guoyuan, Chen Guanrong, van Wyk Michal Antonie, van Wyk (24) Barend Jacobus,Zhang Yuhui. A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system. Chaos Soliton Fract 2008; 38 : 70521. [18] Wang Xingyan, LiangQingyong. Reverse bifurcation and fractal of the compound . Communication in Nonlinear Science and Numerical Simulation2008; 13 : 913 − 27. [19] Stephen.Lynuch, Dynamical Systems with Applications using Mathe- matica, Birkhauser Boston, 2007,ISBN − 13 : 978 − 0 − 8176 − 4482 − 6.

Fig. 11. Chaotic attractor with multi-scrolls bounded on the second variable

IV. CONCLUSION In this paper, we have proposed a new approach to generate a new behavior of bounded chaotic attrac- tor combined with Julia Process. the Julia Process increase the number of scrolls and the bounded function chosen here is hyperbolic tangent which limit the chaotic attractor scheme. Numerical sim- ulations, demonstrate the validity and feasibility of proposed method. The procedure mentioned in this paper has practical application in many disciplines specially in encryption.

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