- Email: [email protected]
Asymptotic fractals
Chaos, Solitons and Fractals 23 (2005) 731–737 www.elsevier.com/locate/chaos
Asymptotic fractals Andrzej Oknin´ski
*
Physics Division, Politechnika S´wie˛tokrzyska, Al. 1000-lecia PP 7, 25-314 Kielce, Poland Accepted 7 June 2004
Abstract A new class of fractals which magnified behave like a rectifiable curve and then, magnified further, disclose their inner structure, is defined analytically and investigated. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Fractal geometry is a fascinating field of research. Besides of being beautiful and intriguing, fractals have found applications in many fields of science. Fractal geometry provides tools to model natural phenomena, such as plants, clouds, lightnings or geological formations. Fractal theory has also contributed to such diverse fields as linguistics, psychology, image compression technology, superconductivity, finances, and computer science, signal and image processing including [1–3]. There are several general methods to generate fractals, for example initiator/generator scheme, Iterated Function Systems, L-systems [3]. We shall concentrate in the present paper on yet another method––the method of infinite series (a generalization of this method was described in [4]). For example, the Weierstrass function [5,7]: wðxÞ ¼
1 X
rk cosð2pbk xÞ;
ð0 < r < 1; br P 1Þ;
ð1Þ
k¼0
or the Weierstrass–Mandelbrot function [1]: W ðxÞ ¼
1 X
cðD2Þn ð1 expðicn xÞÞei/n ;
ð1 < D < 2; c > 1; /n arbitraryÞ;
ð2Þ
n¼1
are continuous and nowhere differentiable [6,8] and belong to this class. The Weierstrass–Mandelbrot function for / n = ln has nice scaling property, W(cx) = eilc 2DW(x), so that magnifications of the graph of the Weierstrass–Mandelbrot function reveal self-similar structure. The box-counting dimension D of the graph of the Weierstrass function log r (1) is D ¼ 2 þ log and is equal to the Hausdorff dimension with probability one if random phases are added [8,9]. b
*
Fax: +48 41 48698. E-mail addresses: [email protected], fi[email protected]
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.027
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
732
Weierstrass-type fractals defined in the form of Fourier series are particularly interesting since they are closely related to the fractal wavefunctions satisfying the Schro¨dinger equation for the simple potentials [10,11]. The general solu2 tions of the Schro¨dinger equation for a particle in one-dimensional infinite potential well, i oto Wðx; tÞ ¼ oxo 2 Wðx; tÞ, satisfying the boundary conditions W(0, t) = W(p, t) = 0, have the form [11,12]: 1 X 2 an sinðnxÞein t : ð3Þ Wðx; tÞ ¼ n¼0
It follows that for the appropriate choice of the coefficients an the following solutions can be obtained: WM ðx; tÞ ¼ N M
M X
rm sinðf ðmÞxÞeif
2 ðmÞt
ð4Þ
;
m¼0
where NM is the normalization constant, M is arbitrary but finite, r 2 (0, 1) and f(m) is a strictly growing integer valued function. For a fixed time and large M the function WM(x, t) approximates well the Weierstrass-type fractals if we P m m choose f(m) = bm in Eq. (4). Let us also notice that the truncated fractals, e.g. UM ðx; tÞ ¼ M m¼0 r sinðb ðx vtÞÞ, fulfill 2 2 the wave equation v12 oo2 t UM ðx; tÞ ¼ oxo 2 UM ðx; tÞ. In their path formulation of quantum mechanics Feynman and Hibbs noted that a quantum mechanical path is continuous but nowhere differentiable [13]. This is because, according to HeisenbergÕs principle, when a quantum particle is more and more precisely localized in space, its trajectory becomes more and more erratic. Abbott and Wise described classical to quantum transition considering the average distance hDli which the particle travels in a time Dt. They have demonstrated that the Hausdorff dimension D of so defined quantum-mechanical path is D = 1 when the distances being resolved by a detecting apparatus are much larger then the particleÕs wavelength and D = 2 when the distances being resolved are much smaller then the particleÕs wavelength [14]. This result was extended to relativistic particles [15–17] and for quantum strings [19]. Several authors considered the possibility that space–time itself forms fractal structure at PlanckÕs scale level [18,20–22] following WheelerÕs idea of quantum foam [23]. We shall undertake a general study of curves, defined analytically in terms of Fourier series, which magnified behave like a rectifiable curve and then, magnified further, disclose their inner structure, in view of interesting mathematical properties of such curves and their possible applications to quantum mechanics. We shall refer to such curves as fractal since their Hausdorff dimension exceeds unity. The paper is organized as follows. In Section 2 a class of nowhere differentiable functions, referred to as asymptotic fractals, is defined such that their graphs upon magnification look like as a straight line at first and next a nondifferentiable structure appears. We shall refer to such curves as asymptotic fractals. In Section 3 properties of such curves are studied numerically and their fractal nature such as affine similarity is revealed. In the last section we briefly discuss the properties of asymptotic fractals. 2. Design of asymptotic fractals We start with a simple but instructive example. Let us consider a modified Weierstrass function: gðxÞ ¼ f ðxÞ þ
1 X
rk cosð2pbk xÞ;
ð0 6 r < 1; br P 1Þ;
ð5Þ
k¼L
where f(x) is a differentiable function. When the graph of the function g(x) is magnified the contribution from f(x) will be rectified before we shall see the contribution from rL cos(2pbLx) provided that the integer parameter L is large enough. P1 k Underk larger magnifications contributions from the nondifferentiable truncated Weierstrass function k¼L r cosð2pb xÞ will be eventually seen. To construct a larger class of such curves which magnified behave like a rectifiable curve and then, magnified further, disclose their inner structure we shall use a theorem due to Belov [24] which generalizes the result obtained by Hardy [6]: Theorem 1 (BelovÕs theorem). The function f ðxÞ ¼
1 X k¼1
ak cosðbk x þ uk Þ;
ak P 0; bk > 0;
! 1 X bkþ1 P k > 1; ak < 1 ; bk k¼1
where the sequence uk is arbitrary, is continuous and nowhere differentiable if the sequence akbk does not tend to zero. We shall impose additional conditions on the parameters ak, bk to obtain asymptotic fractals.
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
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Definition 1 (Asymptotic fractals). Let us consider real functions: gðxÞ ¼
1 X
ak cosð2pbk x þ uk Þ;
ak P 0; bk > 0;
k¼1
where uk are arbitrary and the series sN ¼ ak decrease at least geometrically and
PN
k¼1 ak bk
! 1 X bkþ1 P k > 1; ak < 1 ; bk k¼1
ð6Þ
diverges for N ! 1. Moreover, we demand that the coefficients
1. (a) ak = 0 for 1 < N0 6 k 6 N1 (bk arbitrary), or bkþ1 (b) akþ1 < q1 < 1 ðak 6¼ 0Þ for k 6 N1, ak bk 2.
akþ1 bkþ1 ak bk
> q2 > 1 for N1 + 1 N2 6 k.
It follows directly P from the BelovÕs theorem that the functions g(x) are continuous and nowhere differentiable since the sequence sN ¼ Nk¼1 ak bk tends to infinity for N ! 1 and hence akbk cannot converge to zero. Let us now consider magnifications of the graph of g(x). Since the coefficients ak are assumed to decrease geometrically or faster then at magnifications which are not too large one perceives contributions to g(x) with k 6 N1 only gN ðxÞ ¼
N X
ak cosð2pbk xÞ;
ðN 6 N 1 Þ:
ð7Þ
k¼1
Derivative of the finite sum (7) exists and is equal: N X dgN ðxÞ ak bk sinð2pbk xÞ; ¼ 2p dx k¼1
ðN 6 N 1 Þ:
ð8Þ
Note that since jakbk sin(2pbkx)j 6 akbk absolute values of subsequent contributions to the derivative (8) are rapidly decreasing (geometrically or faster) and the series (8) converges uniformly as long as N 6 N1. Hence the graph of the function (6) will look like a rectifiable curve under such magnifications. However, at larger magnification contributions to gN(x) with k P N2 will be perceived. Since the function g(x) is nowhere differentiable due to the BelovÕs theorem, Theorem 1, its graph will eventually display its inner structure. The transition from smooth to nondifferentiable behaviour shoud be sharp due to fast divergence of the series sN for N P N2. The series sN belongs to the class of asymptotic series [25] and this is why we call graphs of functions constructed according to the Definition 1 asymptotic fractals. 3. Numerical examples We consider three examples of continuous nowhere differentiable functions constructed along lines described in the previous section, cf. Definition 1. All functions considered in the present section have periods T = 1 and hence their graphs are drawn in [0, 1] interval. All graphs shown below look initially, upon small magnifications, like cos(2px) and we shall magnify all the graphs around the point x ¼ 12. Let us also note that to see one period of the term akcos(2pbkx + uk) in a Fourier series (6) we have to magnify the graph bk times in the x variable to perceive intervals 1 of length b1 k and we have to magnify this graph ak times on the g axis to make this contribution of order of unity––we shall denote such nonisotropic magnifications as ðbk ; a1 k Þ. 3.1. Example 1 As a first example we consider a modified Weierstrass function (5) with f(x) = cos(2px): 1 X rk cosð2pbk xÞ; ð0 6 r < 1; br P 1Þ; g1 ðxÞ ¼ cosð2pxÞ þ
ð9Þ
k¼L
where r = 0.4, b = 3, L = 15, and summation over k was truncated at N = 50. The function (10) corresponds to Definition 1, case 1a. The graph of the function (9) looks for 0 6 x 6 1 like cos(2px). When the graph of the function g1(x) is magnified (bK, rK) times, i.e. bK times on the x axis and rK times on the g1 axis, 1KL, the contribution from f(x) is rectified before we can see contribution from rL cos(2pbLx) provided that the parameter L is large enough (isotropic magnifications of order rK (rK < bK) also show nearly a straight line). Under further magnifications we get the
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
734
0.8
0.6
0.4
0.2
0
0.4996
0.4998
0.5
0.5002
0.5004
Fig. 1. Function (g1(x) g1(0.5))l, r = 0.4, b = 3, L = 15, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (103, 105).
3 2.5 2 1.5 1 0.5 0 0.49999998
0.5
0.50000002
Fig. 2. Function (g1(x) g1(0.5))l, r = 0.4, b = 3, L = 15, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (315, 0.415).
nonsmooth curve, Fig. 1: which magnified and magnified again displays a self-affine structure at magnification bL = 315 on the x axis and rL = 0.415 on the g1 axis which can be associated with the truncated Weierstrass function P k k k¼15 0:4 cosð2p3 xÞ, Fig. 2. 0:4 The box-counting dimension of this graph is D ¼ 2 þ log 1:166 [8,9]. log 3 3.2. Example 2 As a second example we consider: 1 X rk cosð2pk!xÞ; ð0 6 r < 1Þ; g2 ðxÞ ¼
ð10Þ
k¼0
where r = 0.1 and summation over k was truncated at N = 50. The function (10) corresponds to Definition 1, case 1b, with N2 = 11. The graph of the function (10) looks for 0 6 x 6 1 like cos(2px). In the present case the value r = 0.1 is not very small and the inner structure of g2(x) is quickly seen upon isotropic magnifications. We shall however magnify the graph of the function (10) nonisotropically: n! times on the x axis and rn times on the g2 axis. When the initial graph of g2(x) is magnified (8! = 40, 320, 0.18 = 108) times the contribution from 0.18 cos 2p8!x is seen and due to a rather large value of r a contribution from higher oscillating term, 0.19 cos 2p9!x, can be also perceived, Fig. 3. When the initial graph of g2(x) is magnified (10! = 3, 628, 800, 0.110 = 1010) times the contribution from (0.110 cos 2p10!x is seen and a contribution from higher oscillating term, 0.111 cos 2p11!x, can be also perceived, Fig. 4: 3.3. Example 3 As the third example we consider series containing terms with rapidly growing phases: 1 X 2 rk cosð2pbk xÞ; ð0 6 r < 1; b > 1Þ; g3 ðxÞ ¼ k¼0
ð11Þ
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
735
3 2.5 2 1.5 1 0.5
0.49999
0.499995
0.5
0.500005
0.50001
Fig. 3. Function (g2(x) g2(0.5))l + 3, r = 0.1, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (8!, 0.18).
3 2.5 2 1.5 1 0.5 0.4999999
0.5
0.5000001
Fig. 4. Function (g2(x) g2(0.5))l + 3, r = 0.1, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (10!, 0.110).
where r = 0.01, b = 3 and summation over k was truncated at N = 20. The function (11) corresponds to the Definition 1, case 1b, with N2 = 3. Again the graph of the function (11) looks for 0 6 x 6 1 like cos(2px). Magnifying isotropically the graph of the function g3(x) 1000 times we get a very flat, nearly straight line. At larger magnifications the inner structure of g3(x) can be seen (the smaller is the value of r the larger is magnification for which the transition occurs). 2 We shall magnify the graph of the function (11) 3n times on the x axis and rn times on the g3 axis. To see contri4 16 bution from a higher order term, say 0.01 cos 2p3 x, we need magnification of order (316 = 43, 046, 721, 0.014 = 108). At such magnification contributions from all lower order terms, for example r3 cos 2p39x, are rectified and contributions from higher order terms are barely visible due to small value of r––therefore we shall only see a slightly perturbed term 0.014 cos 2p316x, see Fig. 5:
1 0.8 0.6 0.4 0.2 0 -0.2
0.5
0.50000001
-0.4 -0.6 -0.8 -1 2
Fig. 5. Function (g3(x) g3(0.5))l1, r = 0.01, b = 3, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (34 , 0.014).
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
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-0.5 -0.6 -0.7 -0.8 -0.9 -1 0.499999998
0.5
0.500000002 1 2
Fig. 6. Function (g3(x) g3(0.5))l1, r = 0.01, b = 3, range
1 2k
6 x 6 12 þ 2k1 , (k, l) = (317.35, 0.014.15).
For larger magnification contributions from higher order terms can be seen, see Fig. 6. For magnification (325 = 847, 288, 609, 443, 0.015 = 1010) we can see contribution from 0.015 cos 2p325x––this picture is very similar to Fig. 5. At still larger magnifications we can also see picture similar to Fig. 6. It can be thus concluded that the fractal (11) is approximately self-affine.
4. Closing remarks In all examples presented in the last section the transition from rectifiable to nondifferential behaviour was observed (strictly speaking the transition takes place in N ! 1 limit but it can be anticipated after examination of the first N terms in expansion (6)). The resolution at which the transition occurs can be controlled by the parameter L in Example 1, by parameter r in Example 2, and by parameters r, b in Example 3. The transitions occur at larger magnifications for increasing value of L and for decreasing value of r. It follows that the wave functions W 1(x, 0) defined in (4) will have 2 the same properties for f(m) = bk, k!, b k as fractals discussed in Examples 1–3. Time evolution of such wavefunctions W1(x, t) was investigated in Ref. [11,12] where it was shown that the probability density of Weierstrass-type wave functions exhibited fractal nature (this result applies also to our Example 1). Computing the probability density 2 P(x, t) = jW1(x, t) j2 of the wave function (4) for f(m) = k!, bk and isolating the time independent part Px(x) as in [12] it is possible to show using the BelovÕs theorem, Theorem 1, that Px(x) is nowhere differentiable. This transition is similar to classical to quantum transition for quantum mechanical path. Important difference consists in transition occurring for the wave function rather than the average distance hDli.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Mandelbrot B. The fractal geometry of nature. New York: Freeman; 1982. Mandelbrot B. Fractals and scaling in finance. New York: Springer Verlag; 1997. Peitgen H-O, Jurgens H, Saupe D. Chaos and fractals, new frontiers of science. New York: Springer-Verlag; 1992. Oknin´ski A. Chaos in discrete maps, deterministic scattering, and nondifferentiable functions. J Stat Phys 1988;52:577–94. ¨ ber kontinuerliche funktionen eines reellen arguments, die fu¨r keinen wert des letzteren einen bestimmten Weierstrass K. U differential quotienten besitzen. In: Mathematische Werke II. Berlin: Mayer & Muller; 1895. Hardy GH. WeierstrassÕs nondifferentiable function. Trans Am Math Soc 1916;17:301–25. Edgar GA, editor. Classics on fractals. Reading, Massachusetts: Addison-Wesley; 1993. Berry MV, Lewis ZV. On the Weierstrass–Mandelbrot fractal function. Proc R Soc London, A 1980;370:459–84. Hunt BR. The Hausdorff dimension of graphs of Weierstrass functions. Proc Am Math Soc 1988;126:791–800. Berry MV. Quantum fractals in boxes. J Phys A: Math Gen 1996;29:6617–29. _ Wo´jcik D, Białynicki-Birula I, Zyczkowski K. Time evolution of quantum fractals. Phys Rev Lett 2000;85(24):5022–5. _ Wo´jcik D, Zyczkowski K. Fractality of certain quantum states. arXiv:math-ph/0107030. Feynman RP, Hibbs AR. Quantum mechanics and path integrals. New York: McGraw-Hill; 1965. Abbott LF, Wise MB. Dimension of a quantum-mechanical path. Am J Phys 1981;49:37–9. Ord GN. Fractal space–time: a geometric analogue of relativistic quantum mechanics. J Phys A: Math Gen 1983;16:1869–84. Cannata F, Ferrari L. Dimension of relativistic quantum mechanical paths. Am J Phys 1988;56:721–5. Nottale L. Fractals and the quantum theory of space–time. Int J Mod Phys A 1989;4:5047–117.
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[18] Nottale L. Fractal space–time and microphysics. Singapore: World Scientific; 1993. [19] Ansoldi S, Aurilia A, Spalluci E. Hausdorff dimension of a quantum string. Phys Rev D 1997;56:2352–61. [20] Ansoldi S, Aurilia A, Spalluci E. Loop quantum mechanics and the fractal structure of quantum space–time. Chaos, Solitons & Fractals 1999;10:197–212. [21] El Naschie MS. Jones invariant, Cantorian geometry and quantum space–time. Chaos, Solitons & Fractals 1999;10:1241–50. [22] Ord GN. Fractals and the quantum classical boundary. Chaos, Solitons & Fractals 1999;10:1281–94. [23] Wheeler JA. Geometrodynamics. New York: Academic Press; 1962. [24] Belov AS. A study of certain trigonometric series. Math Notes 1973;13:291–8 (translated from Russian, Mat Zametki 1973;13:481–92). [25] Erde´lyi A. Asymptotic expansions. New York: Dover; 1987.
Asymptotic fractals Andrzej Oknin´ski
*
Physics Division, Politechnika S´wie˛tokrzyska, Al. 1000-lecia PP 7, 25-314 Kielce, Poland Accepted 7 June 2004
Abstract A new class of fractals which magnified behave like a rectifiable curve and then, magnified further, disclose their inner structure, is defined analytically and investigated. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Fractal geometry is a fascinating field of research. Besides of being beautiful and intriguing, fractals have found applications in many fields of science. Fractal geometry provides tools to model natural phenomena, such as plants, clouds, lightnings or geological formations. Fractal theory has also contributed to such diverse fields as linguistics, psychology, image compression technology, superconductivity, finances, and computer science, signal and image processing including [1–3]. There are several general methods to generate fractals, for example initiator/generator scheme, Iterated Function Systems, L-systems [3]. We shall concentrate in the present paper on yet another method––the method of infinite series (a generalization of this method was described in [4]). For example, the Weierstrass function [5,7]: wðxÞ ¼
1 X
rk cosð2pbk xÞ;
ð0 < r < 1; br P 1Þ;
ð1Þ
k¼0
or the Weierstrass–Mandelbrot function [1]: W ðxÞ ¼
1 X
cðD2Þn ð1 expðicn xÞÞei/n ;
ð1 < D < 2; c > 1; /n arbitraryÞ;
ð2Þ
n¼1
are continuous and nowhere differentiable [6,8] and belong to this class. The Weierstrass–Mandelbrot function for / n = ln has nice scaling property, W(cx) = eilc 2DW(x), so that magnifications of the graph of the Weierstrass–Mandelbrot function reveal self-similar structure. The box-counting dimension D of the graph of the Weierstrass function log r (1) is D ¼ 2 þ log and is equal to the Hausdorff dimension with probability one if random phases are added [8,9]. b
*
Fax: +48 41 48698. E-mail addresses: [email protected], fi[email protected]
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.027
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
732
Weierstrass-type fractals defined in the form of Fourier series are particularly interesting since they are closely related to the fractal wavefunctions satisfying the Schro¨dinger equation for the simple potentials [10,11]. The general solu2 tions of the Schro¨dinger equation for a particle in one-dimensional infinite potential well, i oto Wðx; tÞ ¼ oxo 2 Wðx; tÞ, satisfying the boundary conditions W(0, t) = W(p, t) = 0, have the form [11,12]: 1 X 2 an sinðnxÞein t : ð3Þ Wðx; tÞ ¼ n¼0
It follows that for the appropriate choice of the coefficients an the following solutions can be obtained: WM ðx; tÞ ¼ N M
M X
rm sinðf ðmÞxÞeif
2 ðmÞt
ð4Þ
;
m¼0
where NM is the normalization constant, M is arbitrary but finite, r 2 (0, 1) and f(m) is a strictly growing integer valued function. For a fixed time and large M the function WM(x, t) approximates well the Weierstrass-type fractals if we P m m choose f(m) = bm in Eq. (4). Let us also notice that the truncated fractals, e.g. UM ðx; tÞ ¼ M m¼0 r sinðb ðx vtÞÞ, fulfill 2 2 the wave equation v12 oo2 t UM ðx; tÞ ¼ oxo 2 UM ðx; tÞ. In their path formulation of quantum mechanics Feynman and Hibbs noted that a quantum mechanical path is continuous but nowhere differentiable [13]. This is because, according to HeisenbergÕs principle, when a quantum particle is more and more precisely localized in space, its trajectory becomes more and more erratic. Abbott and Wise described classical to quantum transition considering the average distance hDli which the particle travels in a time Dt. They have demonstrated that the Hausdorff dimension D of so defined quantum-mechanical path is D = 1 when the distances being resolved by a detecting apparatus are much larger then the particleÕs wavelength and D = 2 when the distances being resolved are much smaller then the particleÕs wavelength [14]. This result was extended to relativistic particles [15–17] and for quantum strings [19]. Several authors considered the possibility that space–time itself forms fractal structure at PlanckÕs scale level [18,20–22] following WheelerÕs idea of quantum foam [23]. We shall undertake a general study of curves, defined analytically in terms of Fourier series, which magnified behave like a rectifiable curve and then, magnified further, disclose their inner structure, in view of interesting mathematical properties of such curves and their possible applications to quantum mechanics. We shall refer to such curves as fractal since their Hausdorff dimension exceeds unity. The paper is organized as follows. In Section 2 a class of nowhere differentiable functions, referred to as asymptotic fractals, is defined such that their graphs upon magnification look like as a straight line at first and next a nondifferentiable structure appears. We shall refer to such curves as asymptotic fractals. In Section 3 properties of such curves are studied numerically and their fractal nature such as affine similarity is revealed. In the last section we briefly discuss the properties of asymptotic fractals. 2. Design of asymptotic fractals We start with a simple but instructive example. Let us consider a modified Weierstrass function: gðxÞ ¼ f ðxÞ þ
1 X
rk cosð2pbk xÞ;
ð0 6 r < 1; br P 1Þ;
ð5Þ
k¼L
where f(x) is a differentiable function. When the graph of the function g(x) is magnified the contribution from f(x) will be rectified before we shall see the contribution from rL cos(2pbLx) provided that the integer parameter L is large enough. P1 k Underk larger magnifications contributions from the nondifferentiable truncated Weierstrass function k¼L r cosð2pb xÞ will be eventually seen. To construct a larger class of such curves which magnified behave like a rectifiable curve and then, magnified further, disclose their inner structure we shall use a theorem due to Belov [24] which generalizes the result obtained by Hardy [6]: Theorem 1 (BelovÕs theorem). The function f ðxÞ ¼
1 X k¼1
ak cosðbk x þ uk Þ;
ak P 0; bk > 0;
! 1 X bkþ1 P k > 1; ak < 1 ; bk k¼1
where the sequence uk is arbitrary, is continuous and nowhere differentiable if the sequence akbk does not tend to zero. We shall impose additional conditions on the parameters ak, bk to obtain asymptotic fractals.
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
733
Definition 1 (Asymptotic fractals). Let us consider real functions: gðxÞ ¼
1 X
ak cosð2pbk x þ uk Þ;
ak P 0; bk > 0;
k¼1
where uk are arbitrary and the series sN ¼ ak decrease at least geometrically and
PN
k¼1 ak bk
! 1 X bkþ1 P k > 1; ak < 1 ; bk k¼1
ð6Þ
diverges for N ! 1. Moreover, we demand that the coefficients
1. (a) ak = 0 for 1 < N0 6 k 6 N1 (bk arbitrary), or bkþ1 (b) akþ1 < q1 < 1 ðak 6¼ 0Þ for k 6 N1, ak bk 2.
akþ1 bkþ1 ak bk
> q2 > 1 for N1 + 1 N2 6 k.
It follows directly P from the BelovÕs theorem that the functions g(x) are continuous and nowhere differentiable since the sequence sN ¼ Nk¼1 ak bk tends to infinity for N ! 1 and hence akbk cannot converge to zero. Let us now consider magnifications of the graph of g(x). Since the coefficients ak are assumed to decrease geometrically or faster then at magnifications which are not too large one perceives contributions to g(x) with k 6 N1 only gN ðxÞ ¼
N X
ak cosð2pbk xÞ;
ðN 6 N 1 Þ:
ð7Þ
k¼1
Derivative of the finite sum (7) exists and is equal: N X dgN ðxÞ ak bk sinð2pbk xÞ; ¼ 2p dx k¼1
ðN 6 N 1 Þ:
ð8Þ
Note that since jakbk sin(2pbkx)j 6 akbk absolute values of subsequent contributions to the derivative (8) are rapidly decreasing (geometrically or faster) and the series (8) converges uniformly as long as N 6 N1. Hence the graph of the function (6) will look like a rectifiable curve under such magnifications. However, at larger magnification contributions to gN(x) with k P N2 will be perceived. Since the function g(x) is nowhere differentiable due to the BelovÕs theorem, Theorem 1, its graph will eventually display its inner structure. The transition from smooth to nondifferentiable behaviour shoud be sharp due to fast divergence of the series sN for N P N2. The series sN belongs to the class of asymptotic series [25] and this is why we call graphs of functions constructed according to the Definition 1 asymptotic fractals. 3. Numerical examples We consider three examples of continuous nowhere differentiable functions constructed along lines described in the previous section, cf. Definition 1. All functions considered in the present section have periods T = 1 and hence their graphs are drawn in [0, 1] interval. All graphs shown below look initially, upon small magnifications, like cos(2px) and we shall magnify all the graphs around the point x ¼ 12. Let us also note that to see one period of the term akcos(2pbkx + uk) in a Fourier series (6) we have to magnify the graph bk times in the x variable to perceive intervals 1 of length b1 k and we have to magnify this graph ak times on the g axis to make this contribution of order of unity––we shall denote such nonisotropic magnifications as ðbk ; a1 k Þ. 3.1. Example 1 As a first example we consider a modified Weierstrass function (5) with f(x) = cos(2px): 1 X rk cosð2pbk xÞ; ð0 6 r < 1; br P 1Þ; g1 ðxÞ ¼ cosð2pxÞ þ
ð9Þ
k¼L
where r = 0.4, b = 3, L = 15, and summation over k was truncated at N = 50. The function (10) corresponds to Definition 1, case 1a. The graph of the function (9) looks for 0 6 x 6 1 like cos(2px). When the graph of the function g1(x) is magnified (bK, rK) times, i.e. bK times on the x axis and rK times on the g1 axis, 1KL, the contribution from f(x) is rectified before we can see contribution from rL cos(2pbLx) provided that the parameter L is large enough (isotropic magnifications of order rK (rK < bK) also show nearly a straight line). Under further magnifications we get the
A. Oknin´ski / Chaos, Solitons and Fractals 23 (2005) 731–737
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0.8
0.6
0.4
0.2
0
0.4996
0.4998
0.5
0.5002
0.5004
Fig. 1. Function (g1(x) g1(0.5))l, r = 0.4, b = 3, L = 15, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (103, 105).
3 2.5 2 1.5 1 0.5 0 0.49999998
0.5
0.50000002
Fig. 2. Function (g1(x) g1(0.5))l, r = 0.4, b = 3, L = 15, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (315, 0.415).
nonsmooth curve, Fig. 1: which magnified and magnified again displays a self-affine structure at magnification bL = 315 on the x axis and rL = 0.415 on the g1 axis which can be associated with the truncated Weierstrass function P k k k¼15 0:4 cosð2p3 xÞ, Fig. 2. 0:4 The box-counting dimension of this graph is D ¼ 2 þ log 1:166 [8,9]. log 3 3.2. Example 2 As a second example we consider: 1 X rk cosð2pk!xÞ; ð0 6 r < 1Þ; g2 ðxÞ ¼
ð10Þ
k¼0
where r = 0.1 and summation over k was truncated at N = 50. The function (10) corresponds to Definition 1, case 1b, with N2 = 11. The graph of the function (10) looks for 0 6 x 6 1 like cos(2px). In the present case the value r = 0.1 is not very small and the inner structure of g2(x) is quickly seen upon isotropic magnifications. We shall however magnify the graph of the function (10) nonisotropically: n! times on the x axis and rn times on the g2 axis. When the initial graph of g2(x) is magnified (8! = 40, 320, 0.18 = 108) times the contribution from 0.18 cos 2p8!x is seen and due to a rather large value of r a contribution from higher oscillating term, 0.19 cos 2p9!x, can be also perceived, Fig. 3. When the initial graph of g2(x) is magnified (10! = 3, 628, 800, 0.110 = 1010) times the contribution from (0.110 cos 2p10!x is seen and a contribution from higher oscillating term, 0.111 cos 2p11!x, can be also perceived, Fig. 4: 3.3. Example 3 As the third example we consider series containing terms with rapidly growing phases: 1 X 2 rk cosð2pbk xÞ; ð0 6 r < 1; b > 1Þ; g3 ðxÞ ¼ k¼0
ð11Þ
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735
3 2.5 2 1.5 1 0.5
0.49999
0.499995
0.5
0.500005
0.50001
Fig. 3. Function (g2(x) g2(0.5))l + 3, r = 0.1, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (8!, 0.18).
3 2.5 2 1.5 1 0.5 0.4999999
0.5
0.5000001
Fig. 4. Function (g2(x) g2(0.5))l + 3, r = 0.1, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (10!, 0.110).
where r = 0.01, b = 3 and summation over k was truncated at N = 20. The function (11) corresponds to the Definition 1, case 1b, with N2 = 3. Again the graph of the function (11) looks for 0 6 x 6 1 like cos(2px). Magnifying isotropically the graph of the function g3(x) 1000 times we get a very flat, nearly straight line. At larger magnifications the inner structure of g3(x) can be seen (the smaller is the value of r the larger is magnification for which the transition occurs). 2 We shall magnify the graph of the function (11) 3n times on the x axis and rn times on the g3 axis. To see contri4 16 bution from a higher order term, say 0.01 cos 2p3 x, we need magnification of order (316 = 43, 046, 721, 0.014 = 108). At such magnification contributions from all lower order terms, for example r3 cos 2p39x, are rectified and contributions from higher order terms are barely visible due to small value of r––therefore we shall only see a slightly perturbed term 0.014 cos 2p316x, see Fig. 5:
1 0.8 0.6 0.4 0.2 0 -0.2
0.5
0.50000001
-0.4 -0.6 -0.8 -1 2
Fig. 5. Function (g3(x) g3(0.5))l1, r = 0.01, b = 3, range 12 2k1 6 x 6 12 þ 2k1 , (k, l) = (34 , 0.014).
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-0.5 -0.6 -0.7 -0.8 -0.9 -1 0.499999998
0.5
0.500000002 1 2
Fig. 6. Function (g3(x) g3(0.5))l1, r = 0.01, b = 3, range
1 2k
6 x 6 12 þ 2k1 , (k, l) = (317.35, 0.014.15).
For larger magnification contributions from higher order terms can be seen, see Fig. 6. For magnification (325 = 847, 288, 609, 443, 0.015 = 1010) we can see contribution from 0.015 cos 2p325x––this picture is very similar to Fig. 5. At still larger magnifications we can also see picture similar to Fig. 6. It can be thus concluded that the fractal (11) is approximately self-affine.
4. Closing remarks In all examples presented in the last section the transition from rectifiable to nondifferential behaviour was observed (strictly speaking the transition takes place in N ! 1 limit but it can be anticipated after examination of the first N terms in expansion (6)). The resolution at which the transition occurs can be controlled by the parameter L in Example 1, by parameter r in Example 2, and by parameters r, b in Example 3. The transitions occur at larger magnifications for increasing value of L and for decreasing value of r. It follows that the wave functions W 1(x, 0) defined in (4) will have 2 the same properties for f(m) = bk, k!, b k as fractals discussed in Examples 1–3. Time evolution of such wavefunctions W1(x, t) was investigated in Ref. [11,12] where it was shown that the probability density of Weierstrass-type wave functions exhibited fractal nature (this result applies also to our Example 1). Computing the probability density 2 P(x, t) = jW1(x, t) j2 of the wave function (4) for f(m) = k!, bk and isolating the time independent part Px(x) as in [12] it is possible to show using the BelovÕs theorem, Theorem 1, that Px(x) is nowhere differentiable. This transition is similar to classical to quantum transition for quantum mechanical path. Important difference consists in transition occurring for the wave function rather than the average distance hDli.
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