Generation of fractals from complex logistic map

Chaos, Solitons and Fractals 42 (2009) 447–452

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos

Generation of fractals from complex logistic map Mamta Rani a,*, Rashi Agarwal b a b

Galgotias College of Engg. & Technology, Greater Noida, India IEC College of Engg. & Tech., Greater Noida, India

a r t i c l e

i n f o

Article history: Accepted 14 January 2009

a b s t r a c t Remarkably benign looking logistic transformations xn+1 = r xn(1  xn) for choosing x0 between 0 and 1 and 0 < r 6 4 have found a celebrated place in chaos, fractals and discrete dynamics. The strong physical meaning of Mandelbrot and Julia sets is broadly accepted and nicely connected by Christian Beck [Beck C. Physical meaning for Mandelbrot and Julia sets. Physica D 1999;125(3–4):171–182. Zbl0988.37060] to the complex logistic maps, in the former case, and to the inverse complex logistic map, in the latter case. The purpose of this paper is to study the bounded behavior of the complex logistic map using superior iterates and generate fractals from the same. The analysis in this paper shows that many beautiful properties of the logistic map are extendable for a larger value of r.  2009 Published by Elsevier Ltd.

1. Introduction Dynamical system is the study of iteration of functions from a space to itself in discrete repetitions or in a continuous flow of time [7]. The logistic map is a non-linear dynamical equation. The logistic model was originally introduced as a demographic model by Pierre François Verhulst [18]. Verhulst logistic map is given by a quadratic polynomial

pnþ1 ¼ rpn ð1  pn Þ;

n ¼ 0; 1; 2; 3; . . . ;

where pn denotes the population size at time n and r > 0 is the growth coefficient [1]. The relative simplicity of the logistic map makes it an excellent point of entry into a consideration of the concept of chaos. The main characteristic of the chaotic system is its extreme susceptibility to a change in initial conditions. Once the system becomes chaotic, there is no regularity and stability any more. The Verhulst’s logistic map is the basis of modern chaos theory. The occurrence of the logistic map in the large financial crashes had already been studied [1]. In recent years, chaotic dynamics has been used in cryptography using logistic map [10,15,23]. Also, for optimization of global searching capacity, chaos optimization has been used. For details, one may refer to [24] and several references thereof. For various applications and complexity of chaotic logistic map, one may refer to Beardon [2], Bunde and Havlin [4], Crownover [5], Devaney [6], Holmgren [7], Peitgen et al. [16,17] and M.S. El Naschie [11–14]. Comparative study of logistic map with Picard orbit and that of Nörlund orbit is given by Kumar and Rani [8], see also [7,9]. To control the chaos, different nonlinear control techniques have been used, e.g., feedback linearization, variable structure controllers, fuzzy methods and neural networks [22,24]. Picard iteration, also known as function iteration, is the most common method for solving the nonlinear equations. Although, the coding for Picard iterations is simple and of less complexity, but the rate of convergence is slow and for lower values of r. The logistic map converges to a fixed point for r lying between 0 and 2.75. Also, the Picard orbit shows a stable and cyclic behavior for 3.2 P r > 2.75 [6,16,20].

* Corresponding author. E-mail addresses: [email protected] (M. Rani), [email protected] (R. Agarwal). 0960-0779/$ - see front matter  2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2009.01.011

448

M. Rani, R. Agarwal / Chaos, Solitons and Fractals 42 (2009) 447–452

In 2002, logistic map had been studied using Mann iterative procedures [18–20]. This was the new beginning in the chaotic dynamics of the logistic map. It showed that convergence of the map is extendable to a value of r as large as 21. Recently, Rani and Agarwal studied the stability of logistic map using Mann iterative procedure and showed that the chaotic behavior of logistic map disappears in some cases [21]. Usually, talking about fractals, we think about self-similarity. Self-similarity seems to be a concept, which can be easily understood. But fractals adds a new dimension to the problems dealing with limits. It gives a refreshingly new perspective to understand the concept of limits. The r-values for which the sequence remains bounded, forms a set known as fractal [1]. In this, our main tool is superior iterations to study the bounded behavior of complex logistic map and generate the fractals from that. In Section 2, we have given preliminaries that have been taken into account during our study and Section 3 incorporates the description of our approach and main results followed by conclusion. 2. Preliminaries Following the two given quadratic laws:

pnþ1 ¼ pn þ apn ð1  pn Þ;

n ¼ 0; 1; 2; 3; . . .

ð1Þ

and

znþ1 ¼ z2n þ c;

n ¼ 0; 1; 2; 3; . . .

ð2Þ

2

are identical as we set c = (1  jaj )/4 and zn = ((1 + jaj)/2)  apn (see [16]). The complex logistic equation

xnþ1 ¼ rxn ð1  xn Þ;

n ¼ 0; 1; 2; 3; . . .

ð3Þ

and

pnþ1 ¼ pn þ apn ð1  pn Þ;

n ¼ 0; 1; 2; 3; . . .

ð1Þ

are identical as xn = (a  pn)/(1 + a) and r = a + 1 (see [16]). The iteration of any quadratic polynomial is equivalent to the iteration of logistic map. For each value of r in logistic map, we find the orbit, which is either bounded or diverges to infinity [1]. Peano–Picard iteration method is based on one-step machine, computed by the formula xn+1 = f(xn), where f is any function of x. It requires one number as input and returns a new number. Superior iteration is described as a two-step feedback machine and is computed by the formula xn+1 = g(xn, xn1). It requires two numbers xn and xn1 as inputs and returns a new number xn+1 and is computed as

xnþ1 ¼ gðf ðxn Þ; xn Þ ¼ bn f ðxn Þ þ ð1  bn Þxn : In this iterative procedure, the parameter bn lies between 0 and 1 and converges away from 0. In our study, we take bn = b. At b = 0, there is no change in the input and at b = 1, two-step machine works as a one-step machine [20,21]. The sequence {xn} constructed above will be called superior sequence of iterates and is denoted by SO(f, x0, b). 3. Complex logistic map and mann iterative procedure Verhulst logistic map is one of the main complex dynamical systems given by a quadratic polynomial

pnþ1 ¼ rpn ð1  pn Þ;

n ¼ 0; 1; 2; 3; . . . ;

ð4Þ

where p and r are to be of complex type "jpj 2 [0, 1] (refer [1]). Let p = px + ipy and r = rx + iry, then after nth iterations using superior iterates, pxn+1 and pyn+1 are given by the equations

pxnþ1 ¼ bðr xn ðpxn  p2xn þ p2yn Þ  r yn ðpyn  2pxn pyn ÞÞ þ ð1  bÞ  px ; pynþ1 ¼ bðr xn ðpyn  2pxn pyn Þ þ r yn ðpxn 

p2xn

þ

p2yn ÞÞ

þ ð1  bÞ  py :

ð5Þ ð6Þ

Eq. (4) converges to a fixed point if and only if Eqs. (5) and (6) converges to a fixed point " jpj 2 [0, 1]. In the present experimental study, we choose b in its prescribed range 0 < b 6 1. Considering jp0j in [0, 1], we attempt to find the maximum value of r correctable up to two decimal places, for which logistic map shows stable behavior. To find the optimum range of jrj, first we substitute ry = 0 in Eqs. (5) and (6) and find the maximum value of rx for which jpj remains bounded. After that, to find the maximum value of ry for which jpj remains bounded, we substitute rx = 0 in Eqs. (5) and (6). We have obtained optimum value of jrj for specific choices of b = 1, 0.7, 0.4 and 0.1 "jp0j 2 [0, 1]. From Fig. 1(a) and (b), we conclude that at b = 1, the Eqs. (5) and (6), remains bounded for 0 6 rx 6 3.57 and 0 6 r y 6 1.14 "jpj 2 [0, 1], respectively. Similarly, at b = 0.7 the bounded range of complex logistic map is 0 6 r x 6 4.69 and 0 6 ry 6 1.30, at b = 0.4 it is 0 6 rx 6 6.60 and 0 6 ry 6 1.95, and at b = 0.1 it is 0 6 rx 6 17.05 and 0 6 ry 6 4.01, "jpj 2 [0, 1]. Fig. 1(a)–(h) shows the time series for maximum value of rx and ry for which Verhulst logistic map is bounded at different values of b.

449

M. Rani, R. Agarwal / Chaos, Solitons and Fractals 42 (2009) 447–452 Behavior of logistic map

Behavior of logistic map 1.00 0.80

|p |

|p|

0.60 0.40 0.20 0.00

0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 1

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

5

(b) ( rx = 0, ry = 1.14 , β = 1)

(a) (rx = 3.57, ry = 0, β = 1)

Behavior of logistic map 0.01

1.00

0.01

0.80

0.01

0.60

0.01

|p|

|p|

Behavior of logistic map 1.20

0.40

0.00

0.20

0.00

0.00

1

5

0.00

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93

1

5

No. of iterations

(d) (rx = 0, ry = 1.30, β = 0.7) Behavior of logistic map

Behavior of logistic map 1.20

0.01

1.00

0.01

0.80

0.01

0.60

0.01

|p|

|p|

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 No. of iterations

(c) (rx = 4.69, ry = 0, β = 0.7)

0.00

0.40

0.00

0.20

0.00

0.00 1

5

1

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93

5

(f) (rx = 0, ry = 1.95, β = 0.4)

(e) (rx = 6.60, ry = 0, β = 0.4) Behavior of logistic map

Behavior of logistic map 0.01

1.00

0.01

0.80

0.01

0.60

0.01

|p|

1.20

0.40

0.00

0.20

0.00

0.00 1

5

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 No. of iterations

(g) (rx = 17.05, ry = 0, β = 0.1)

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 No. of iterations

No. of iterations

|p|

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 No. of iterations

No. of iterations

0.00

1

5

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 No. of iterations

(h) (rx = 0, ry = 4.01, β = 0.1)

Fig. 1. Time series of Verhulst logistic map using superior iterations at different values of b.

In superior orbit, the logistic map shows stable behavior for larger values of jrj for all jpj 2 [0, 1]. The maximum value of r depends on the value of parameter b for stable behavior. We notice, there is a remarkable increase in the value of jrj "j pj 2 [0, 1]. Eq. (4) depends on both rx and ry. The r-value for which the sequence stays bounded forms a set, which is known as Verhulst fractal [1] (see also [3]). These fractals are superior Julia sets and can easily be generated on computer using Eq. (4). Here, we have given some of the superior fractals generated from complex logistic map using superior iterations, in C programming language. See Figs. 2–7. The same fractals can also be generated by znþ1 ¼ z2n þ c, where n = 0, 1, 2, 3,. . . As we find that Eqs. (2) and (3) become identical as we set

c ¼ r  ð2  rÞ=4 and pn ¼ ðr=2Þ  r  xn :

450

M. Rani, R. Agarwal / Chaos, Solitons and Fractals 42 (2009) 447–452

Fig. 2. (rx, ry, b) = (3.57, 0, 1).

Fig. 3. (rx, ry, b) = (0, 1.14, 1).

Fig. 4. (rx, ry, b) = (3.80, 0.01, 0.9).

Fig. 5. (rx, ry, b) = (4.236, 0.1, 0.7).

By taking c = cx + icy and r = rx + iry, we get

cx ¼ ð2r x  r x  rx þ ry  ry Þ=4 and cy ¼ ðr y  r x  r y Þ=4: jcj is always less than or equal to 2/b, "jpj 2 [0, 1].

M. Rani, R. Agarwal / Chaos, Solitons and Fractals 42 (2009) 447–452

451

Fig. 6. (rx, ry, b) = (5.07, 0.01, 0.63).

Fig. 7. (rx, ry, b) = (6.3, 0.3, 0.4).

4. Conclusion We conclude following results from our paper: 1. The range of jrj for which complex logistic map is bounded depends on the value of b in superior orbit. The bounded range of jrj increases with the decrease in b. 2. We have generated fractals from complex logistic map and supported the results by the fact that the same fractals can be generated from znþ1 ¼ z2n þ c; n ¼ 0; 1; 2; 3 . . . by substituting the corresponding c value, where c = r(2  r)/4.

Acknowledgement Authors thank to Prof. S.L. Singh for his very auspicious suggestions and remarks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

Ausloos M, Dirickx M. The logistic map and the route to chaos. Berlin: Springer; 2006. MR2202732. Beardon AF. Iteration of rational functions. Berlin: Springer; 1991. MR1128089. Beck C. Physical meaning for Mandelbrot and Julia sets. Physica D 1999(3–4):171–82. Zbl 0988.37060. Bunde A, Havlin S, editors. Fractals in science. Berlin: Springer; 1994. Crownover RM. Introduction to fractals and chaos. Boston: Jones & Barlett; 1995. Devaney RL. A first course in chaotic dynamical systems. Theory and experiment. Reading (MA): Addison-Wesley; 1992. MR1202237. Holmgren RA. A first course in discrete dynamical systems. Berlin: Springer; 1996. MR1410752. Kumar M. Fractal modeling in graphics. Ph.D. thesis, Gurukala Kangri Vishwavidyalaya, Hardwar, India, 2006.. Kumar M, Rani M. An experiment with summability methods in the dynamics of the logistic model. Indian J Math 2005(471):77–89. MR2155130. Mooney A, Keating JG, Heffernan DM. A detailed study of the generation of optically detectable watermarks using the logistic map. Chaos, Solitons and Fractals 2006(305):1088–97. MR2249218. El Naschie MS. On the universality class of all universality classes and E-infinity spacetime physics. Chaos, Solitons and Fractals 2007(323):927–36. El Naschie MS. From E-eight to E-Infinity. Chaos, Solitons and Fractals 2008(352):285–90. El Naschie MS. Quantum golden field theory – ten theorems and various conjectures. Chaos, Solitons and Fractals 2008(365):1121–5. El Naschie MS. New elementary particles as a possible product of a disintegrating symplictic vacuum. Chaos, Solitons and Fractals 2004(204):905–13. de Oliveira LPL, Sobottka M. Cryptography with chaotic mixing. Chaos, Solitons and Fractals 2008(353):466–71. Peitgen HO, Jurgens H, Saupe D. Chaos and fractals. New York: Springer; 2006. Peitgen H, Saupe D, editors. The science of fractal images. Berlin: Springer; 1988. Rani M, Kumar V. A new experiment with the logistic function. J Indian Acad Math 2005(271):143–56. MR2224669.

452 [19] [20] [21] [22] [23]

M. Rani, R. Agarwal / Chaos, Solitons and Fractals 42 (2009) 447–452

Rani M, Kumar V. Superior Julia set. J Korea Soc Math Edu Ser D: Res Math Edu 2004(84):261–77. Rani M, Kumar V. Superior Mandelbrot set. J Korea Soc Math Edu Ser D: Res Math Edu 2004(84):279–91. Rani M, Agarwal R. A new experimental approach to study the stability of logistic map. Chaos, Solitons and Fractals 2008, preprint. Salarieh H, Shahrokhi M. Indirect adaptive control of discrete chaotic systems. Chaos, Solitons and Fractals 2007(344):1188–201. MR2331274. Vázquez-Medina R, Díaz-Méndez A, Del Río-Correa JL. Design of chaotic analogue noise generators with logistic map and MOS QT circuits. Chaos, Solitons and Fractals 2007. [24] Yang D, Li G, Cheng G. On the efficiency of chaos optimization algorithms for global optimization. Chaos, Solitons and Fractals 2007(344):1366–75.