A Trivial Dynamics in 2-D Square Root Discrete Mapping

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 8, HIKARI Ltd, A Trivial Dynamics in 2-D Square Root Discrete Mapping M. Mammeri Department of Mathematics University of Kasdi Merbah, Ouargla Algeria Copyright c 2018 M. Mammeri. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We investigate in this paper a new simple two-dimensional square root discrete mapping with only one nonlinearity and we will give a rigorous analysis proof of trivial dynamics on long-term of the 2-D square root discrete mapping. Keywords: 2-D square root discrete mapping, fixed point, trivial dynamics 1 Introduction It is well known the complex (chaotic) behavior of dynamical system is characterized by its sensitive dependence on initial conditions (is popularly as the butterfly effect). Small changes in initial conditions can possibly lead to immense changes in subsequent on large time intervals and it is difficult to predict precisely the future behaviors of any chaotic system see for instance the work done in [16,17,18,19,20,21]. In recent years many 2-D and 3-D chaotic maps and systems have been studied in the chaos literatures such as [1,2,3,5,6,7,8,9,10,11,12,13,14,15]. Doubtless, the study of 2-D discrete mapping such as with square root is interesting contribution to the development of the theory of dynamical systems. The square root map play an important role in mathematics due to its properties in addition to mathematics, the square root map occur in other fields of study such as sciences, physics, engineering and economics systems.

2 364 M. Mammeri In this paper we investigate a new simple two-dimensional square root discrete mapping with only one nonlinearity and with determinant of Jacobi matrix equal to zero at all its fixed points. We will offers a rigorous analysis proof of trivial dynamics of the 2-D square root discrete mapping given by equation (1) in the sense that all its dynamics on long-term are trivial and converges to a fixed point. 2 A 2-D square root discrete mapping We consider the two-dimensional discrete mapping is given as follows: ( ) xy f(x, y) = 1(x + y) 2 (1) Where (x, y) R 2 is the state variable, (x 0, y 0 ) = (a, b) R 2 is the initial state where 0 < a < b. The fixed points of the mapping (1) are (x = x, y = x). Possibly, the mapping (1) is the first simple 2-D square root discrete mapping studied whose determinant of Jacobi matrix detj(x = x, y = x) = 0 for all its fixed points (x = x, y = x) R 2. We can deduce that the Jacobi matrix of the mapping (1) calculated at the fixed points has at least one zero eigenvalue, by using the standard definition of the largest Lyapunov exponent of discrete time systems [22] (Lyapunov exponents as the usual test for chaos and hyperchaos), which means that the mapping (1) has a minus infinity Lyapunouv exponent. 3 A rigorous proof of trivial dynamics This section includes some propositions that determine rigorously the proof of trivial dynamics on large time intervals in the 2-D square root discrete mapping (1). We consider the discrete time n = 0, 1, 2,... Proposition 1 For all positive integral values of n one has: x 0 x n < x n+1 < y n+1 < y n y 0 (2) Proof. It is clear that all terms of the two sequences are strictly positive real numbers. We will prove that the inequalities (2) proposed by recurrence on the integer n. For n = 0, it is clear that x 0 < x 0 y 0 and 1(x y 0 ) < y 0 we prove that ab < 1 (a + b), by squaring the two members of the inequality, we get 2 4ab < (a+b) 2, i.e., 0 < a 2 2ab+b 2, then one has x 0 x 0 < x 1 < y 1 < y 0 y 0. Suppose the inequalities are true for n 1, then we have x 0 x n 1 < x n < y n < y n 1 y 0. But as a precedent x n < x n y n, 1(x + y n ) < y n, we got

3 A trivial dynamics in 2-D square root discrete mapping 365 xn y n < 1 2 (x n + y n ). Then we have x 0 x n 1 < x n < x n+1 < y n+1 < y n < y n 1 y 0. Hence the inequalities x 0 x n < x n+1 < y n+1 < y n y 0 is true for all positive integers n. Proposition 2 For all positive integral values of n one has: y n x n 1 (b a) (3) 2n Proof. We will prove the inequalities (3) proposed by recurrence on the integer n, It is clear that y 0 x 0 1 (x y 0 ). Suppose that the inequalities is true for all positive integers n, then we obtained y n x n 1 (x 0 + y 0 ), then we have 1(x + y n ) x n y n 1(y x n ) after simplification we have x n y n y n. Hence y n+1 x n+1 1(y x n ) 1 (y +1 0 x 0 ) this implies y n x n 1 (y 0 x 0 ) i.e., y n x n 1 (b a) is true for all positive integers n. Proposition 3 For all positive integral values of n one has: lim (y n x n ) = 0 (4) 1 Proof. It is well known that lim = 0, but 0 y n x n 1 (b a) then one has 0 lim (y 1 n x n ) lim (b a). Finally, one has lim (y n x n ) = 0. Proposition 4 For all positive integral values of n one has: lim f(x n, y n ) = (l, l), where l R (5) Proof. It s clear that the sequence (x n ) n is increasing and the sequence (y n ) n is decreasing, the two sequences both satisfy x n < y n for all positive integers n. The sequences (x n ) n and (y n ) n are Adjacent, then they are convergent and converge to the same limit l where l R then one has lim (x n, y n ) = (l, l) thus lim (x n+1, y n+1 ) = (l, l). This implies lim f(x n, y n ) = (l, l). For example, by numerical simulation we need to find approximate value to 10 3 for the limit l. First we chose a = 1 and b = 4 then one has y n x n 3. Hence 3 < 10 3, thus one has > We know that 2 11 = 2948 and 2 12 = 4096 then one has n 12, then we have the table 1: n x n y n

4 366 M. Mammeri Secondly if we chose a = and b = then we have the table 2: n x n y n From the tables 1 and 2 one has l = 2.243, i.e., lim f(x n, y n ) = (2.243, 2.243) in the sense that the mapping (1) converges to a fixed point (2.243, 2.243). 4 Conclusion We have described a square root discrete mapping of the plane with trivial dynamics on long-term that the results have been confirmed by a rigorous analysis proof. References [1] M. Hénon, A Two-Dimensional Mapping with a Strange Attractor, Comm. Math. Phys., 50 (1976), [2] R. Lozi, Un attracteur étrange du type attracteur de Hénon, Journal de Physique, Colloque, 39 (1978), no. C5, [3] E. Zeraoulia, J.C Sprott, 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach, World Scientific Series on Nonlinear Science Series A, Vol. 73, [4] M. Benedicks and L. Carleson, The dynamics of the Hénon maps, Ann. Math., 133 (1991), [5] Y. Cao and Z. Liu, Strange Attractors in the Orientation-preserving Lozi map, Chaos, Solitons and Fractals, 9 (1998), no. 11, [6] M. A. Aziz Alaoui, Carl Robert and C. Grebogi, Dynamics of a Hénon- Lozi map, Chaos, Solitons and Fractals, 12 (2001), no. 12, [7] D. A. Miller & G. Grassi, A discrete generalized hyperchaotic Hénon map circuit, Proc. 44th IEEE 2001 Midwest Symp. on Circuits and Systems. MWSCAS 2001, (2001),

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6 368 M. Mammeri [19] K.J. Arrow, R. Forsythe, M. Gorham, R. Hahn, R. Hanson, J. O. Ledyard, S. Levmore, R. Litan, P. Milgrom, F. D. Nelson, G. R. Neumann et al., The promise of prediction markets, Science, 320 (2008), [20] C. Boettiger & A. Hastings, Tipping points: From patterns to predictions, Nature, 493 (2013), [21] J. Dauwels, F. Vialatte, C. Latchoumane, J. Jeong & A. Cichocki, EEG synchrony analysis for early diagnosis of Alzheimer s disease: a study with several synchrony measures and EEG data sets, 2009 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, US-New York: IEEE, 2009, [22] Zeraoulia Elhadj, J. C. Sprott, Chaotifying 2-D piecewise-linear maps via a piecewise linear controller function, Nonlinear Oscillations, 13 (2010), no. 3, Received: February 11, 2018; Published: March 6, 2018