Int. Journal of Math. Analysis, Vol. 7, 2013, no. 29, 1433 - 1449 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.3361 Some Dynamical Properties of the Family of Tent Maps* Wadia Faid Hassan Al-shameri 1 and Mohammed Abdulkawi Mahiub 2 1 Department of Mathematics, Faculty of Science and Arts, Najran University [email protected] 2 Department of Mathematics, Faculty of Science and Arts, Najran University [email protected] Copyright © 2013 Wadia Faid Hassan Al-shameri and Mohammed Abdulkawi Mahiub. This is an open access article 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 In this paper, we investigated some dynamical behavior of family of tent maps. Such dynamical properties include fixed points and their stability, period orbits, visualize the iterations using a kind of plot called a cobweb plot and demonstrate bifurcation diagram forTr . Furthermore, detecting the presence of chaos in the discrete dynamical systemTr is investigated. Finally, we develop Matlab computer programs that reflect the results interpreting such dynamical behavior. Mathematics Subject Classification: 37C25, 37E15, 34C23 Keywords: Fixed points, periodic points, periodic orbits, bifurcation 1. INTRODUCTION Consider the parameterized tent map, which can be described piecewise by ⎧2,ifrx 0≤≤ x 1 Tx()= 2 r ⎨ 1 ⎩2(1rx− ),if 2 <≤ x1 1434 Wadia Faid Hassan Al-shameri and Mohammed Abdulkawi Mahiub where 02<≤r . This map is continuous, linear on each of the intervals (−1, 1 / 2] and [1/2, 1) (with respect to slopes r and −r ) and has the points (0,0) and (1, 0) in its graph. This map is known as tent maps and often introduced as one of the first examples of chaotic maps literature for nonlinear discrete dynamical systems. Its dynamics exhibit various features are commonly used to identify chaotic systems (see for instance [5]). Chaos theory describes complex motion and the dynamics of sensitive systems. Chaotic systems are mathematically deterministic but nearly impossible to be predicted. Chaos is more evident in long-term systems than in short-term systems. Behavior in chaotic systems is a periodic, in other words, no variable describes the state of the system that undergoes a regular repetition of values. A chaotic system can actually evolve in a way that appears to be smooth and ordered; however, chaos refers to the issue of whether or not it is possible to make accurate long-term predictions of any system if the initial conditions are known to an accurate degree [8]. In addition to the sensitivity condition, there is a mixing condition called transitivity and a regularity condition called density of periodic points[2]. Deterministic chaos can be described by deterministic map, such as tent map Tr as follows: starting from any point between 0 and 1, the trace of the map would still lie in the range [0, 1] of the tent map, which is a absorbing set. Also, each point inside [0, 1] will be often visited arbitrarily closely and infinitely by any ’typical’ solution, hence it is also an attractor [2]. The purpose of this paper is to investigate the dynamical behavior of the family of tent maps, we will limit ourselves to the case 02< r ≤ [3]. For certain parameter values, the mapping undergoes stretching and folding transformations and displays sensitivity to initial conditions and periodicity (see [1] and [6]). The tent map is studied in the mathematics of dynamical systems. Because of its simple shape, the tent map shape under iteration is very well understood. And despite its simple shape, it has several interesting properties [10]. The paper is organized as follows; The first section is the introduction and the second section describes the dynamical behavior for this discrete dynamical systemTr such as fixed points and their stability and period orbits followed by section three, which presents cobweb plots. Section four demonstrates period doubling and obtains the bifurcation diagram ofTr , for the parameter 02.<≤r The dramatic bifurcation that occurs at r = 1 will be explained. The results will be obtained using Matlab computer programs. One of these programs is to visualize the iterations using a kind of plot called a cobweb plot. Sketching the bifurcation diagram ofTr , for the parameter 01< r ≤ is by using another Matlab program. Fixed points will show up as a single point, a periodic orbit as several points, and a chaotic orbit as a band or several bands of points. Section five presents the results that are detecting the presence of chaos in the tent mapsTr and examining sensitivity to initial conditions; it is one of the dynamical Some dynamical properties of the family of tent maps 1435 properties of strange attractors. Results and discussions will be displayed. Finally, in the last section some concluding remarks will be highlighted. 2. FIXED POINTS AND PERIODIC ORBITS Consider an iteration x = fx() (2.1) nn+1 A fixed point, or point of period one, of discrete dynamical system (2.1) is a point at which xnnn+1 ==fx() x for all n[7] For the tent map, this implies thatTxrn()= x n, for all n. Graphically, the fixed points can be found by identifying intersections of the map Txr ()with the line y = x , see [3]. Figures 2.1(a) through 2.1(c) displayTr for r = 2/7,1/2 and 5/6by Matlab program 1 in the Appendix. As r increases, the height of the graph ofTr rises, because of the factor r in the formula forTr . From this observation and the three graphs in Figure 2.1; we deduce that if 01/2<<r , thenTr intersects the line y = x once (at 0), whereas if 1/2<<r 1, then there are two points of intersection. We are led to analyze separately the members of{ Tr } for which 01/2<<r , r = 1/2 and 1/2< r < 1. Finally, we will studyT1 , which is the original tent mapT and which has some very interesting features, we shall follow [3]. Graphical iteration of tent map:T (x) r Graphical iteration of tent map:T (x) 1 r 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 r =0.5. r =0.2857. r 0.4 (x), is in (0, 2] r 0.4 (x), is in (0, 2] r T 0.3 T 0.3 0.2 0.2 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x Figure 2.1 (a) Figure 2.1 (b) 1436 Wadia Faid Hassan Al-shameri and Mohammed Abdulkawi Mahiub Graphical iteration of tent map:T (x) r 1 0.9 0.8 0.7 0.6 0.5 r =0.8333. r (x), isin (0, 2] 0.4 r T 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 2.1(c) Case 1. 0<<r 1/ 2 The graph in Figure 2.1(a) shows that 0 is the only fixed point of Tr . Since 01/2<<r , it follows from the definition of Tr that if 0 < x ≤ 1/ 2 , then 02≤ Txr ( ) =< rxx and if 1/ 2 < x ≤ 1, then −1 02(1)1≤=−<−<<Tx() r x x x r 2 ∞ Consequently for any x ∈[0,1], the sequence T{n} x is bounded and decreasing and by { μ ()}n=0 the continuity of Tr , the sequence converges to the fixed point 0. Therefore, 0 is an attracting fixed point whose basin of attraction is [0,1] . Case 2. r = 1/2 First we notice that if 0 ≤ x ≤ 1/ 2 , then T1/ 2 (x) = 2(1/ 2)x = x , so that x is a fixed point of T1/ 2 (Figure 2.1(b)). Next, we calculate that 1/ 2 < x ≤ 1, then 1 02(1/2)(1)1≤=Tx() −=−≤ x x r 2 So that Txx1/2 ()= is a fixed point of T1/ 2 . Consequently, every point in [0,1] either is a fixed point of T1/ 2 or has a fixed point its first iterate. Case 3. 1/2<<r 1 In addition to the fixed point 0, there is a second fixed point p that lies in (1 / 2,1] , as one can see in Figure 2.1(c). To evaluate p we solve the equation p = Tpr ( ) =−2(1 r p ) which yields Some dynamical properties of the family of tent maps 1437 2r p = 12+ r / As r increases from ½ toward 1, p increases from 1/2 toward 2/3. Because Txr ()=>21 r on [0,1] except at 1/2, both 0 and p are repelling fixed points. The period-2 points of Tr [2] are the fixed points of Tr , which is given by ⎧401/4rx2 for≤≤ x r ⎪ [2] ⎪2(12rrxforrx−<≤ ) 1/4 1/2 Txr ()= ⎨ ⎪2rrrxforx (1−+ 2 2 ) 1/2 <≤− 1 1/4 r ⎪ 2 ⎩4(1)rxforrx−−≤≤ 11/4 1 [2] The graph of T8 appears in Figure 2.2 plotted by Matlab program 2 in the Appendix, [2] and suggests that for 1/2<<r 1, Tr has four fixed points that can be found by solving [2] [2] the four equations x = Txr () arising from the definition of Tr . We find that the fixed 22rr 4 r2 points are0, , , and 14++rr22 12 14 + r Graphical iteration of tent map, r=0.8333 1 0.9 0.8 0.7 0.6 0.5 (x), n=2 (x), [n] T 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 2.2 ⎧ 24rr2 ⎫ The first and third are the two fixed points of Tr , so it follows that ⎨ 22, ⎬ ⎩⎭14++rr 14 [2]′ 2 is a 2-cycle forTr .