In this paper we give some rigorous conditions for the occurrence of super chaotic attractors in a general piecewise smooth map of the plane

Elhadj Zeraoulia

Elhafsi Boukhalfa

Interdisciplinary Description of Complex Systems

English

ABSTRACT In this paper we give some rigorous conditions for the occurrence of super chaotic attractors in a general piecewise smooth map of the plane

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EXISTENCE OF SUPER CHAOTIC ATTRACTORS IN A GENERAL PIECEWISE SMOOTH MAP OF THE PLANE

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Interdisciplinary Description of Complex Systems 12(1), 92-98, 2014  *Corresponding author, : zeraoulia@mail.univ-tebessa.dz; +213 66 946 9880; *Université de Tébessa, 12002 Algeria * EXISTENCE OF SUPER CHAOTIC ATTRACTORS IN A GENERAL PIECEWISE SMOOTH MAP OF THE PLANE Elhafsi Boukhalfa and Elhadj Zeraoulia* Department of Mathematics, University of Tébessa Tébessa, Algeria DOI: 10.7906/indecs.12.1.6 Regular article Received: 30. September 2013. Accepted: 22. January 2014. ABSTRACT In this paper we give some rigorous conditions for the occurrence of super chaotic attractors in a general piecewise smooth map of the plane. KEY WORDS piecewise smooth map, super chaotic attractor, rigorous proof of chaos, Lyapunov exponents CLASSIFICATION JEL: Y91, Z00 PACS: 05.45.-a, 05.45.GgExistence of Super Chaotic Attractors in a General Piecewise Smooth Map of the Plane 93  INTRODUCTION There are many works that focused on the topic of chaotic behaviours of a discrete mapping. For example it has been studied from a control and anticontrol (chaotification) schemes or from the use of Lyapunov exponents [1-4], or by the use of several modified versions of the Marotto theorem [5], to prove the existence of chaos in n-dimensional dynamical discrete system, where the results in some way are to making an originally non-chaotic dynamical system chaotic, or enhancing the existing chaos of a chaotic system. Robust chaos is defined by the absence of periodic windows and coexisting attractors in some neighbourhood of the parameter space, since the existence of these windows in some chaotic regions imply that with small changes of the parameters would destroy the chaotic behaviour. This effect implies the fragility of this type of chaos. Contrary to this situation, there are many practical applications as in communication and spreading the spectrum of switch-mode power supplies to avoid electromagnetic interference [2-9], where it is necessary to obtain reliable operation in the chaotic mode where the robustness of chaos is required. A practical example can be found from electrical engineering to demonstrate robust chaos as shown in [6]. If all Lyapunov exponents are positive throughout the range, then the resulting attractors are called super-chaotic attractors. The importance of these attractors is that are more non-regular, and the iteration points are seemingly “almost” full of the considered space, which explains one of applications of chaos in fluid mixing, for example, refer to [7, 8]. A super-chaotification (or hyper-chaotification) scheme by making all Lyapunov exponents of a controlled dynamical system positive via the controller of some simple triangular function is given in [2]. In this paper, we shall determine rigorously a range in the parameters space for the existence of super chaotic attractors in a general 2-dimensional piecewise smooth discrete mapping, using an equivalence relation between the two matrices of the corresponding normal form of the considered map, and hence we compute analytically and assure the positivity of all the Lyapunov exponents. Since many practical applications require piecewise smooth map under discrete modeling [1, 6, 10], we consider a general two-dimensional piecewise smooth map f(x, y; ρ), which depends on a single parameter ρ. Let Γρ, given by x = h(y, ρ) denotes a smooth curve that divides the phase plane into two regions R1 and R2. The map is given by:  ., if   ),;,(,, if   ),;,();,(2211RyxyxfRyxyxfyxf  (1) For obtaining the general conditions of occurrence of super-chaotic attractors, it is assumed that the functions f1 and f2 are both continuous and have continuous derivatives. The map f is continuous but its derivative is discontinuous at the line Γρ, called the “border”. It is further assumed that the one-sided partial derivatives at the border are finite, and in each subregions R1 and R2 the map (1) has one fixed point P1 and P2, for certain value ρ of the parameter ρ. A normal form theory for border-collision bifurcations of two-dimensional piecewise smooth maps has been developed in [10]. Obviously, it has been shown that it is possible to choose an appropriate coordinate transformation so that the choice of axis is independent of the parameter. In so doing, the normal form of the map (1) is given by:  .0 if   ,1001,0 if   ,1001),(2211xyxxyxyxN (2) E. Boukhalfa and E. Zeraoulia 94  where μ is a parameter and τi, δi, i = 1, 2 are the traces and determinants of the corresponding matrices of the linearised map in the two subregions R1 and R2 evaluated at P1 and P2, respectively. Assume that:  τ1 > 1 + δ1, and τ2 < − (1 + δ2). (3) Then there are no fixed points for the map (1) when μ < 0, and there are altogether two fixed points, one in R1 and the other in R2, for μ > 0, given by:  222221111111,1,1,1 P . (4) Since as the parameter μ is varied through zero, the local bifurcation of the map (1) depends only on the values of τi and δi, (i = 1, 2), then it suffices to study the bifurcations in the normal form (2), and it is shown in [6] that the map (1) has a robust chaotic attractor for some parameter space region, when in first the condition (3) is verified. On the other hand, it is also shown in [6] that there is periodic attractor in the piecewise smooth map of the form (2) when τ1 > 1 + δ1, and −(1 + δ2) < τ2 < (1 + δ2), thus, we exclude this region from our analysis when looking for super chaotic behaviour. MAIN RESULT In this section, a rigorous proof for the occurrence of super chaotic attractors in the piecewise smooth map (1) is given, using an equivalence relation between the two matrices of the corresponding normal form (2), and hence we compute analytically all the Lyapunov exponents. The existence of fixed points P1 and P2 given in (4) imply that it is possible to write the map (1) under the normal form (2). Therefore, assume that  4 and 4222211  . (5) Then 21  −41 >0 and 22  −42 > 0 and these inequalities imply that the eigenvalues in R1 are 2/)4( 121111   , 2/)4( 121112   , while in R2 the eigenvalues are 2/)4( 222221   , 2/)4( 222212   . On the other hand, there are many ways for realizing an equivalence between matrices A1 and A2, with  01   111A ,   01   222A .  A simple way is to assume that their eigenvalues are equal. Suppose for example that 11 = 21 and 12 = 22, thus we obtain the following condition  2221212121 44)(2   . (6) Hence equation (6) has a solution if 2(1+2) – 12 < 0, or  22 1212 , (7) so that equation (6) becomes  0)2( 1212112222112122   , (8) Existence of Super Chaotic Attractors in a General Piecewise Smooth Map of the Plane 95  with the discriminant  = (2 – 1)2(12 – 41) > 0. Since (12 – 41) > 0, solutions of (8) with respect to 2 are  .2)4()(2,2)4()(212121221121)2(212121221121)1(2 (9) For brevity, we consider only the case 2 = 2(1), the case 2 = 2(2) being quite similar. Condition (7) with 2 = 2(1) gives the following inequality  242 22211 , (10) while condition (5) with 2 = 2(1) leads to  22)4()(2221211121212 , (11) with the additional condition  222221211 , (12) so that solution of the inequality (11) is possible. It is easy to verify that the inequality (11) still holds for all values of τ1, τ2, and δ1, thus we consider only the condition (12) for this case. Condition (3) with 2 = 2(1) gives  222 221211 . (13) Thus inequalities (3), (10), (12) and (13) imply that   1,222,22,42min 12212122212122211  . (14) We remark that  02)(2242 2122221212221  .  because of what the condition (14) transforms into   1,222,42min 12212122211  . (15) Note that λ11 = λ21 > 1, if (3) holds. i.e., δ1 < τ1 − 1, and λ12 = λ22 < −1, if and only if:  )1( 11   . (16) Lyapunov exponents of the map (1) in the region R1 are 1(X0) = ln|λ11|, and 2(X0) = ln|λ12|, and in the region R2 the Lyapunov exponents are 1(X0) = ln|λ21| and 2(X0) = ln|λ22|, for all X0 ∈  R2, thus, according to (3) and (16) we obtain: E. Boukhalfa and E. Zeraoulia 96   1(X0) = 1(X0) = ln|λ11| > 0 and 2(X0) = 2(X0) = ln|λ12| > 0 (17) Finally, the Lyapunov exponents are identical in both regions R1 and R2, then the map (1) has a super chaotic attractor when conditions (9), (15) and (16) are verified. The main result of this article is now given. Theorem 1. Consider the piecewise smooth map (1) written in the normal form (2), and assume the following:  (1)  1  ,1  ,222  ,42min 112212122211  .  (2) 2)4()(2 121212211212 . Then the map (1) converges to a super chaotic attractor. ELEMENTARY EXAMPLE Let us consider the following piecewise linear map of the plane:   .0 f,2,000   1,0 f,2,0001   ),(xiyxbaxiyxbayxf  (18) It is easy to verify that the map (18) is a special case of the map (1) with all its assumptions given in this paper. Indeed, using the main theorem of this article we find that if a = 1,2 then a portion of the range for the occurrence of super chaotic attractor is b < −1,6667, and in this case the Lyapunov exponents are  0218,4121ln)( 01  bX , and 08,412121ln)( 02  bX .  On the one hand, the super chaotic attractor and the bifurcation diagram of the map (18) are shown in Fig. 1a) and Fig. 1b), respectively. On the other hand, we remark that the given attractor is more non-regular, and the iteration points are seemingly “almost” full of the considered space, which explains an application of chaos in fluid mixing, for example, see [7, 8].  Figure 1. a) The super chaotic attractor obtained for a = 1,2, b = −1,6668, and the initial condition x = y = 0,001. b) The border collision bifurcation for the map (18) for −1,66685 < b < 0 and a = 1,2. a)      b) Existence of Super Chaotic Attractors in a General Piecewise Smooth Map of the Plane 97  CONCLUSION We have reported some analytical results on the existence of super chaotic attractors in a general piecewise smooth map of the plane. An elementary example is also given and discussed. REFERENCES [1] Zeraoulia, E.: Robust chaos in piecewise nonsmooth map of the plane. Journal of Mathematical Sciences 177(3), 366-372, 2011, http://dx.doi.org/10.1007/s10958-011-0462-z, [2] Li, C.: On super-chaotifying discrete dynamical systems. Chaos, Solitons and Fractals 21(4), 855-861, 2004, http://dx.doi.org/10.1016/j.chaos.2003.12.023, [3] Wang, X.F. and Chen, G.: On feedback anticontrol of discrete chaos. 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Science 257(5071), 754-760, 1992, http://dx.doi.org/10.1126/science.257.5071.754, [9] Barreto, E.; Hunt, E.B.; Grebogi, C. and Yorke, J.A.: From High Dimensional Chaos to Stable Periodic Orbits: The Structure of Parameter Space. Physical Review Letters 78(24), 4561-4564, 1997, http://dx.doi.org/10.1103/PhysRevLett.78.4561, [10] Nusse, H.E. and Yorke, J.A.: Border Collision Bifurcations Including “period two to period three” for Piecewise Smooth Maps. Physica D: Nonlinear Phenomena 57(1-2), 39-57, 1992, http://dx.doi.org/10.1016/0167-2789(92)90087-4.    E. Boukhalfa and E. Zeraoulia 98  EGZISTENCIJA SUPERKAOTIČNIH ATRAKTORA U PO DIJELOVIMA GLATKOJ RAVNINI E. Boukhalfa i E. Zeraoulia Odsjek za matematiku – Sveučilište u Tebessi Tebessa, Alžir SAŽETAK U radu su dani rigorozni uvjeti za pojavu superkaotičnih atraktora u po dijelovima glatkoj mapi ravnine. KLJUČNE RIJEČI po dijelovima glatka mapa, superkaotični atraktor, rigorozni dokaz kaosa, eksponent Lyapunova 

Existence of Super Chaotic Attractors in a General Piecewise Smooth Map of the Plane

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Existence of Super Chaotic Attractors in a General Piecewise Smooth Map of the Plane

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