Fractal sets generated by chemical reactions discrete chaotic dynamics

Chaos, Solitons and Fractals 32 (2007) 496–502 www.elsevier.com/locate/chaos

Fractal sets generated by chemical reactions discrete chaotic dynamics V. Gontar *, O. Grechko International Group for Chaos Studies, Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel Accepted 29 June 2006

Communicated by Professor Mohamed Saladin Elnaschie

Abstract Fractal sets composed by the parameters values of difference equations derived from chemical reactions discrete chaotic dynamics (DCD) and corresponding to the sequences of symmetrical patterns were obtained in this work. Examples of fractal sets with the corresponding symmetrical patterns have been presented.  2006 Elsevier Ltd. All rights reserved.

1. Introduction R&D conducted over the past few decades has culminated in the widespread use of iterated functions, cellular automata and systems of difference equations for the mathematical modeling of complex systems dynamics. Solutions of difference equations, cellular automata rules, the ‘‘game of life’’, the prisoner’s dilemma have given different types of patterns [1–4]. Complex chaotic systems dynamics simulated by difference equations may be identified and presented as fractal sets in the form of strange attractors of a time series or in the form of specific patterns in the parameter space (Julia and Mandelbrot sets [1,2]). The development of a special type of difference equations that can be related to basic physicochemical principles and to the ‘‘laws of nature’’ with the corresponding fractal sets will enrich our understanding of the dynamics of complex systems. In this paper we present a number of symmetrical patterns and their fractal sets composed by corresponded parameters values of the specific type of difference equations. These equations have been derived from the discrete chaotic dynamics (DCD) of physicochemical reactions [5]. The DCD theory describes chemical reaction dynamics of multicomponent systems in discrete time and space. According to this theory, the interaction between the constituents of a chemical system—in addition to undergoing chemical transformations—also enter into ‘‘information exchange’’ as a particular type of feedback. It has been shown that when these equations written for a particular hypothesis about the mechanism of interaction of constituents are solved in discrete time and space (on a 2D discrete lattice), a variety of patterns, including symmetrical patterns are obtained [5–8]. Selection of the symmetrical patterns for the analysis *

Corresponding author. Tel.: +972 7 646 1935; fax: +972 7 647 2969. E-mail address: [email protected] (V. Gontar).

0960-0779/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.092

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presented here relates to their possible application for the simulation of human brain creativity in the form of mandalas and ornaments [9]. It is important to evaluate and characterize the regions (sets) of multidimensional parameter space of the DCD difference equations in which the desired patterns exist. As will be demonstrated on 2D and 3D cross-sections of a 9D parameter space, the sets of parameters that result in symmetrical patterns are composing fractal set with non-integer box-counting dimension [2]. 2. Background According to DCD any mechanism of system constituents (Xi, i = 1, 2, . . . , N) transformations could be presented in discrete time tq and discrete 2D space designated by the integer coordinates 1 < Rp, Rs < 1, where p denoting the rows and s denoting the columns of the lattice: for example X1(25, 30) designates concentration value of constituent X1 in the cell situated on 25 row and 30 column. For practical reasons we limited our consideration with the discrete square lattice of final size R · R with coordinates Rp, Rs = 1, 2, . . . , R. Here, we intend to investigate the following mechanism of transformation for three constituents X1, X2 and X3:

ð1Þ

where the solid arrows ( ) denote the chemical transformations of the constituents, the broken-line arrows ( ) denote ‘‘information exchange’’ between the constituents inside each cell of the lattice, and finely dotted-line arrows ( ) denote ‘‘information exchange’’ between the constituents in the cell and the constituents in the closest neighboring cells. The molecular matrix (kaijk) and matrix of stoichiometric coefficients (kmlik) corresponding to the above-described mechanism (1) should be written as   1 i ¼ 1; 2; 3      1 1 0    ; j ¼ 1 kaij k ¼  1 ; kmli k ¼  ð2Þ   1 0 1  1 l ¼ 1; 2 According to the DCD basic equations [6] we obtain the following system of equations: t

X 2q ðRp ; Rs Þ ¼ p1 ðtq1 ; rÞ t X 1q ðRp ; Rs Þ

ð3Þ

t

X 3q ðRp ; Rs Þ ¼ p2 ðtq1 ; rÞ t X 2q ðRp ; Rs Þ t

ð4Þ

t

t

X 1q ðRp ; Rs Þ þ X 2q ðRp ; Rs Þ þ X 3q ðRp ; Rs Þ ¼ b ( " #) 3 3 X X tq1 tq1 r p1 ðtq1 ; rÞ ¼ k 1 exp  a1i X i ðRp ; Rs Þ þ b1i X i ðrÞ ( " p2 ðtq1 ; rÞ ¼ k 2 exp 

i¼1 3 X i¼1 t

i¼1 t a2i X iq1 ðRp ; Rs Þ

þ

3 X

ð5Þ ð6Þ

#) t br2i X iq1 ðrÞ

ð7Þ

i¼1

where i = 1, 2, 3; j = 1; l = 1, 2, X iq ðRp ; Rs Þ is the concentration of the ith constituent calculated in each cell of the lattice with coordinates (Rp, Rs) at the discrete time tq (q = 1, 2, . . . , Q). Here kaijk is molecular matrix defining the number of main constituents of type ‘‘j’’ in the ith constituent, kmlik is a matrix of stoichiometric coefficients reflecting the mechanism of constituents chemical transformations, pl(tq1, r) is the function of system’s constituents concentrations calculated at t discrete time tq1 and neighboring concentrations X iq1 ðrÞ of the lth reaction, b is the total concentration of the main constituent, kl-rate constant of lth reaction, ali are empirical parameters characterizing local ‘‘information exchange’’ between the constituents inside the considered cell, brli are empirical parameters characterizing the ‘‘information exchange’’ emanating from the constituents in eight closest neighboring cells including considered cell (r = 1, 2, . . . , 9). Therefore each vector r contains nine discrete coordinates: r  = [(Rp  1, Rs  1), (Rp  1, Rs), (Rp  1, Rs + 1), (Rp, Rs  1), (Rp, Rs), (Rp, Rs + 1), (Rp + 1, Rs  1), (Rp + 1, Rs), (Rp + 1, Rs + 1)], where Rp, Rs = 1, 2, . . . , R.

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For this particular mechanism of transformation of the constituents, Eqs. (3)–(5) may be solved analytically by subt t stitution of X 2q ðRp ; Rs Þ and X 3q ðRp ; Rs Þ taken from Eqs. (3) and (4) into Eq. (5) t

t

t

X 1q ðRp ; Rs Þ þ X 1q ðRp ; Rs Þp1 ðtq1 ; rÞ þ X 1q ðRp ; Rs Þp1 ðtq1 ; rÞp2 ðtq1 ; rÞ ¼ b Therefore, by taking into account p1(tq1, r) and p2(tq1, r) it is easy to obtain solutions for each following form: b 1 þ p1 ðtq1 ; rÞ þ p1 ðtq1 ; rÞp2 ðtq1 ; rÞ bp1 ðtq1 ; rÞ t X 2q ðRp ; Rs Þ ¼ 1 þ p1 ðtq1 ; rÞ þ p1 ðtq1 ; rÞp2 ðtq1 ; rÞ bp1 ðtq1 ; rÞp2 ðtq1 ; rÞ t X 3q ðRp ; Rs Þ ¼ 1 þ p1 ðtq1 ; rÞ þ p1 ðtq1 ; rÞp2 ðtq1 ; rÞ t

X 1q ðRp ; Rs Þ ¼

t

ð8Þ t X iq ðRp ; Rs Þ

in the

ð9Þ ð10Þ ð11Þ t

t

which result in the evolution of the concentrations: X 1q ðRp ; Rs Þ, X 2q ðRp ; Rs Þ, X 3q ðRp ; Rs Þ in discrete time tq (q = 1, 2, . . . , Q) on three discrete square lattices, starting with the following initial conditions: X t10 ðRp ; Rs Þ ¼ b;

X t20 ðRp ; Rs Þ ¼ 0;

X t30 ðRp ; Rs Þ ¼ 0;

Rp ; Rs ¼ 1; 2; . . . ; R

The boundary conditions used here are ( t X iq ðRp ; Rs Þ; 1 6 Rp ; Rs 6 R ðinside the latticeÞ t X iq ðRp ; Rs Þ ¼ 0; Rp ; Rs < 1; Rp ; Rs > R ðoutside the latticeÞ

ð12Þ

ð13Þ

t

According to Eq. (5) all concentrations are within the interval 0 < X iq ðRp ; Rs Þ < b. Values of the concentrations calculated within the each cell of the considered lattice will be encoded in colors using colored palette with 256 arbitrary distributed colors, and with the image resolution 2 · 2 pixels. As a result we receive three different sequences t t t (i = 1, 2, 3) of patterns corresponded to X 1q ðRp ; Rs Þ, X 2q ðRp ; Rs Þ and X 3q ðRp ; Rs Þ calculated at the discrete time tq tq (q = 1, 2, . . . , Q). In this work the sequences of the patterns for each X i ðRp ; Rs Þ were generated on a lattice of size 100 · 100 (R = 100) cells. We expected that different patterns could be obtained by varying the model’s parameters. As pointed out in Section 1, we are interested only in the parameters that result in different sequences of symmetrical patterns. A pattern is cont sidered to be symmetrical if the concentrations X iq ðRp ; Rs Þ have equal values within their relationship to the two main diagonals of the square lattice. Eqs. (9)–(11) have 63 parameters with clear physicochemical sense and should be defined empirically according to t the concrete goal. In this paper we intend to obtain distributions of X iq ðRp ; Rs Þ in the form of symmetrical patterns and would like to start our computations with the minimal number of parameters. Here we are assuming that local ‘‘information exchange’’ between the constituents inside the considered cell (ali) can be equal for both reactions (l = 1, 2); the ‘‘information exchange’’ emanating from the constituents in neighboring cells (brli ) also can be equal for both reactions (l = 1, 2); and the ‘‘information exchange’’ emanating from the constituents in each of the neighboring cells including considered cell can be equal (r = 1, 2, . . . , 9). These assumptions lead to the following conditions: ali ¼ al0 i ¼ ai ;

l 6¼ l0 ; l; l0 ¼ 1; 2; i ¼ 1; 2; 3

brli

l 6¼ l0 ; l; l0 ¼ 1; 2; i ¼ 1; 2; 3

bri

¼

brl0 i r0

¼

bri ;

¼ bi ¼ bi ;

0

ð14Þ

0

r 6¼ r ; r; r ¼ 1; 2; 3; . . . ; 9; i ¼ 1; 2; 3

Hence, the number of parameters was reduced to nine b, k1, k2, a1, a2, a3, b1, b2, b3. Numerical investigation of Eqs. (9)–(11) indicated that not all the parameters values within given intervals will result to the desired symmetrical patterns [8]. The topological properties of the 9D parameters space resulted to the concrete type of patterns are not known a priori. In our case the reconstruction of 9D parameters space presents an extremely computer time consuming problem. Therefore we will constrain our consideration by the reconstruction of 2D and 3D cross-sections of 9D parameters space corresponded to the symmetrical patterns.

3. Results To begin to elucidate the structure of the 9D parameter space (composed of the points corresponding to the symmetrical patterns) and to investigate its properties, we set out to calculate and investigate the 2D and 3D cross-sections

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of this space for two and three arbitrarily chosen parameters (here, b1, b2 and b1, b2, b3). To this end, chosen parameters were varied within some intervals around the point ðb ; k 1 ; k 2 ; a1 ; a2 ; a3 ; b1 ; b2 ; b3 Þ corresponding to a symmetrical pattern, while the remaining parameters remained fixed. The initial values of nine parameters that resulted in symmetrical patterns ðb ; k 1 ; k 2 ; a1 ; a2 ; a3 ; b1 ; b2 ; b3 Þ were obtained by using a real coded generic algorithm [10] (RCGA). We are supposing that the number of points within the considered cross-sections should depend on the step D of varying parameter. To quantitatively characterize the obtained cross-sections, we calculated their box-counting dimensions. Fig. 1a and b show two 2D cross-sections calculated with Db = 0.05 and Db = 0.01 within the intervals 4 6 b1 6 3 and 6 6 b2 6 1, resulting in 380 and 8591 points, respectively (the values of the fixed parameters are: b* = 0.79, k 1 ¼ 4:18, k 2 ¼ 1:11, a1 ¼ 1:85, a2 ¼ 2:06; a3 ¼ 1:76; b3 ¼ 9:36). The calculated box-counting dimension for 380 points and 8591 points are Df = 1.91 and, Df = 1.94, respectively. Fig. 2 presents 3D cross-section calculated with Db = 0.05 within the intervals 4 6 b1 6 3, 6 6 b2 6 1 and 3 6 b3 6 10, resulting in 68,441 points corresponding to symmetrical patterns (the values of the fixed parameters are: b* = 0.79, k 1 ¼ 4:18; k 2 ¼ 1:11; a1 ¼ 1:85; a2 ¼ 2:06; a3 ¼ 1:76). The calculated box-counting dimension for this cross-section is Df = 2.87. Fig. 3 shows 12 examples of symmetrical patterns corresponding to the points taken from the cross-sections presented in Fig. 1b.

Fig. 1. (a) Two parameters cross-section (4 6 b1 6 3, 6 6 b2 6 1) of the 9D parameter space with Db = 0.05 and fixed parameters b ¼ 0:79; k 1 ¼ 4:18; k 2 ¼ 1:11, a1 ¼ 1:85; a2 ¼ 2:06; a3 ¼ 1:76; b3 ¼ 9:30. (b) Two parameters cross-section (4 6 b1 6 3, 6 6 b2 6 1) of the 9D parameter space with Db = 0.01 and the same fixed parameters.

Fig. 2. Three parameters cross-section (4 6 b1 6 3, 6 6 b2 6 1, 3 6 b3 6 10) of the 9D parameter space with Db = 0.05 and fixed parameters b ¼ 0:79; k 1 ¼ 4:18; k 2 ¼ 1:11, a1 ¼ 1:85; a2 ¼ 2:06; a3 ¼ 1:76.

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Fig. 3. Twelve examples of symmetrical patterns corresponding to the parameters taken from the cross-section presented in Fig. 1b.

Fig. 4a and b present two cross-sections calculated with Db = 0.2 and Db = 0.1 within the intervals 50 6 b1 6 0 and 10 6 b2 6 40, resulting in 1500 and 6052 points, respectively (fixed parameters are: (b* = 0.74, k 1 ¼ 0:29; k 2 ¼ 1:81; a1 ¼ 2:12; a2 ¼ 7:41; a3 ¼ 7:8; b3 ¼ 1:19). The calculated box-counting dimension for 1500 points and 6052 points are Df = 1.94 and Df = 1.88, respectively. Fig. 5 shows 3D cross-section calculated with Db = 0.2 within the intervals 25 6 b1 6 0, 10 6 b2 6 15 and 25 6 b3 6 0, resulting in 6589 points corresponding to symmetrical patterns (the values of the fixed parameters are: b* = 0.74, k 1 ¼ 0:29; k 2 ¼ 1:81; a1 ¼ 2:12; a2 ¼ 7:41; a3 ¼ 7:8). The calculated box-counting dimension for 3D cross-section is Df = 2.76. Fig. 6 shows examples of symmetrical patterns corresponding to the points taken from the cross-sections presented in Figs. 4 and 5. The obtained cross-sections are characterized by infinitely complex confined structures within the considered parameters intervals. The calculated box-counting dimension indicates that the obtained cross-sections are fractal sets. Symmetrical patterns corresponding to the obtained cross-sections are extremely sensitive to the tiny changes in parameter’s

Fig. 4. (a) Two parameters cross-section (50 6 b1 6 0, 10 6 b2 6 40) of the 9D parameter space with Db = 0.2 and fixed parameters: b ¼ 0:74; k 1 ¼ 0:29; k 2 ¼ 1:81; a1 ¼ 2:12; a2 ¼ 7:41; a3 ¼ 7:8; b3 ¼ 1:19. (b) Two parameters cross-section (50 6 b1 6 0, 10 6 b2 6 40) of the 9D parameter space with Db = 0.1 and the same fixed parameters.

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Fig. 5. Three parameters cross-section (25 6 b1 6 0, 10 6 b2 6 15 and 25 6 b3 6 0) of the 9D parameter space with Db = 0.2 and fixed parameters b ¼ 0:74; k 1 ¼ 0:29; k 2 ¼ 1:81; a1 ¼ 2:12; a2 ¼ 7:41; a3 ¼ 7:8.

values (for example, see Fig. 7). This implies fractal character of the obtained parameter’s space and reflected existence of embedded chaotic regimes within the cells of 2D lattice [11]. When D ! 0 for considered parameters, the number of points resulted to symmetrical patterns tends to infinity. In the same way we performed calculations of box-counting dimension for cross-sections with different combinations of two and three parameter’s (a1, a2; b, a1; b, b1; b, a1, b1; a1, a2, a3 and others) of the 9D parameters space, for different fixed parameters and with different varying intervals. These calculations demonstrated similar to the presented here results. This fact allows us to assume that the 9D space of parameters corresponding to symmetrical patterns has a fractal character.

Fig. 6. Twelve examples of symmetrical patterns corresponding to the parameters taken from the cross-section presented in Fig. 1b.

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Fig. 7. Example of the extreme sensitivity of symmetrical patterns (corresponding to cross-section presented on Fig. 1b) to the infinitely small changes in parameters (the fixed parameters are b ¼ 0:79; k 1 ¼ 4:18; k 2 ¼ 1:11; a1 ¼ 1:85; a2 ¼ 2:06; a3 ¼ 1:76; b1 ¼ 0:01; b3 ¼ 9:36).

4. Conclusions In addition to the known difference equations revealing chaotic behavior with corresponding fractal sets, a specific system of difference equations derived from the DCD of physicochemical reactions was investigated. As was demonstrated here, DCD difference equations can serve as a source for practically unlimited different symmetrical patterns. This was confirmed by the fractal character of the parameter space corresponding to the symmetrical patterns and their specific property – extreme sensitivity to infinitesimal changes in parameters. The results presented here can be related to the mathematical modeling of the living systems dynamics [8] and simulation of brain creativity in a form of symmetrical images [7].

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Mandelbrot BB. Fractals and chaos. New York: Springer; 2004. Peitgen H, Jurgens H, Saupe D. Chaos and fractals. New York: Springer-Verlag; 2004. Wolfram S. A new kind of science. Wolfram Media; 2002. Nowak MA, Bonhoeffer S, May RM. Int J Bifurcat Chaos 1994;4:33. Gontar V, Il’in AV. J Phys D 1991;52:528. Gontar V. In: Field RJ, Gyorgui L, editors. Chaos in chemistry and biochemistry. London: World Scientific; 1993. p. 225–47. Gontar V. Discr Dyn Nat Soc 2000;5(1):19. Gontar V. Chaos, Solitons & Fractals 2000;11:231. Jung CG. Mandala symbolism. Princeton University Press; 1973. Eshelman LJ, Schaffler JD. In: Whitley D, editor. Foundations of genetic algorithms II. San Mateo: Morgan Kaufman; 1993. p. 187–202. [11] Gontar V, Grechko O. In: Lesnic D, editor. Proceedings of the 5th international conference on inverse problems in engineering: theory and practice. Leeds University Press; 2005. p. G03.