September 16, 2008 16:22 00408

Fractals, Vol. 16, No. 4 (2008) 1–12c© World Scientific Publishing Company

BELLS GALORE: OSCILLATIONS ANDCIRCLE-MAP DYNAMICS FROM SPACE-FILLING

FRACTAL FUNCTIONS

CARLOS E. PUENTE,∗,‡ ANDREA CORTIS†and BELLIE SIVAKUMAR∗

∗Department of Land, Air and Water ResourcesUniversity of California, Davis

127 Veihmeyer Hall, Davis, CA 95616, USA†Earth Sciences Division, Lawrence Berkeley Laboratory

Berkeley, CA 94720, USA‡cepuente@ucdavis.edu

Received November 22, 2007Accepted March 11, 2008

AbstractThe construction of a host of interesting patterns over one and two dimensions, as trans-formations of multifractal measures via fractal interpolating functions related to simple affinemappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonlyyield limiting Gaussian distribution measures (bells), there are also situations (depending onthe affine mappings’ parameters) in which there is no limit. Specifically, the one-dimensionalcase may result in oscillations between two bells, whereas the two-dimensional case may giverise to unexpected circle map dynamics of an arbitrary number of two-dimensional circularbells. It is also shown that, despite the multitude of bells over two dimensions, whose meansdance making regular polygons or stars inscribed on a circle, the iteration of affine maps yieldsexotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to themany cases that converge to a single bell.

Keywords :

1

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2 C. E. Puente et al.

1. INTRODUCTION

Recently, a multitude of interesting patterns (overone, two and three dimensions) defined as trans-formations of multifractal measures via fractalinterpolating functions and the iterations of affinemappings have been uncovered.1–4 Within thisframework, it has also been established that Gaus-sian (normal distribution) patterns over one andtwo dimensions appear when the fractal functionsfill up space,5–7 and that a wide variety of exotickaleidoscopes decompose circular two-dimensionalbells.8,9

The purpose of the present article is to showthat, while the Gaussian limit is the most commoncase, there are also interesting cases (depending onthe signs of some affine mapping parameters) inwhich there is no limiting convergence to a singleGaussian bell, but rather oscillations among a mul-titude of patterns that closely approximate bells.It is shown that whereas in the one-dimensionalcase oscillations between two bells happen in onespecific way, in the two-dimensional case oscilla-tory behavior among several two-dimensional cir-cular bells may occur in two different ways thatlead either to a roundabout motion or a star-shapedcriss-crossed movement of an arbitrary number n ofbells, for n > 2.

The organization of this paper is as follows. Givenfirst is the mathematical construction of the inter-esting patterns over one dimension that may beconstructed via the notion of projections and theiteration of affine mappings. Next, the conditionsneeded to define a limiting Gaussian pattern areexplained followed by the specific scenario thatwould give rise to the limiting oscillation betweentwo bells. Having established the results in onedimension, the article then studies generalizationsto two-dimensional Gaussians to identify the rel-evant scenarios that define circle-map dynamicsamong a host of circular bells that jointly sketchbeautiful kaleidoscopes of such ever-rotating attrac-tors. The article ends with a summary and withsome final remarks.

2. AFFINE MAPPINGS,FRACTAL FUNCTIONS ANDRELATED MEASURES

The graph G of a fractal interpolating function,from x to y and passing by N + 1 points on theplane {(xn, yn);x0 < · · · < xN , n = 0, 1, . . . , N}, isdefined as the unique attractor of N affine maps as

follows:10

wn

(xy

)=(

an 0cn dn

)(xy

)+(

en

fn

),

n = 1, . . . , N, (1)

with the scaling parameters dn such that |dn| < 1,and with the other parameters, an, cn, en, and fn,satisfying the initial conditions:

wn

(x0

y0

)=(

xn−1

yn−1

), wn

(xN

yN

)=(

xn

yn

),

n = 1, . . . , N. (2)

Equations (2) ensure that the attractor G existsand that it contains the initial interpolating points.Such conditions also allow computing the parame-ters an, cn, en, and fn in terms of the scalings dn

and the coordinates of the interpolating points, viasimple linear equations.10 At the end, a unique, andhence deterministic, set G is found that turns outto have a fractal dimension D ∈ [1, 2) that is givenby the solution of:11∑

|dn|aD−1n = 1, if

∑|dn| > 1 (3)

or that equals 1 otherwise.In a practical setting, the graph of a fractal func-

tion is obtained sampling the unique attractor pointby point, starting the process at a point already inG and progressively iterating the affine maps wn

according to, for example, the outcomes of inde-pendent “coin” tosses.11 As this process is carriedout, it happens that a unique invariant measure isalso induced over G that reflects how the attractoris being filled up. The existence of such a measureallows computing unique (and fully deterministic)projections over the coordinates x and y (say dxand dy) that turn out to have irregular shapes asfound in a variety of applications in geophysics andbeyond.2−4

Figure 1 shows an example of these ideas for afractal function that passes by the three points {(0,0), (1/2, −0.35), (1, −0.2)}, when the scalings of thetwo affine maps are d1 = −0.8 and d2 = −0.6. Inaddition to the graph of the attracting fractal func-tion f , the figure includes the implied projectionsdx and dy of the unique measure over G when thecorresponding mappings w1 and w2 are iterated (15million times) according to a 30–70% proportion,using independent pseudo-random numbers, start-ing the process from the mid-point (1/2,−0.35).

As there is a lack of dependence of the coordi-nate y on x, i.e. as implied by the zero value in thefirst component of the affine mappings [Eq. (1)],

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Oscillations and Circle-Map Dynamics 3

Fig. 1 From a multifractal measure dx to a derived measure dy via a fractal interpolating function f . (The scale in x is from0 to 1, and the one in y is from −0.38 to 0.06. The vertical scales in dx and dy are not given but both measures are normalizedso that they add up to one).

the measure dx is simply a deterministic bino-mial multifractal.12 The measure dy, in turn, beingrelated to dx via the deterministic self-affine frac-tal function, is just the derived measure of dx viathe function f and is, hence, computed looking atall possible heights y and adding the corresponding“probabilities” from “events” that emanate from x.3

As is seen in Fig. 1, the ideas lead to very inter-esting and “random-looking” measures dy, whichas in the above example resemble, for instance, arainfall data set as a function of time.13 As multi-fractal measures have been found relevant in stud-ies of turbulence,14 the projection sets given bythese ideas, which turn out to perform a non-trivial fractional integration of a simple parentmultifractal measure over x, may be assigned aninterpretation as reflections or transformations ofturbulence.15

3. THE PLANE-FILLING CASESAND THE GAUSSIAN LIMIT

When the interpolating points are equally spaced inx, all coefficients an happen to be equal to 1/N andhence the fractal dimension D [Eq. (3)] tends to 2when the magnitudes of all the scalings dn’s tend toone. This implies, considering all possible sign com-binations on such scalings, that there are 2N routestoward obtaining plane-filling fractal interpolatingfunctions.

Figure 2 shows an example of such limitingnotions for a fractal function that passes by thethree points {(0, 0), (1/2, 1), (1, 0)}, when the scal-ings of the two required affine maps are d1 = 0.999and d2 = −0.999 (the “plus-minus” case) andwhen the two maps are iterated (50 million times)according to a 70–30% proportion. As may be seen,a Gaussian distribution appears as a projection

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4 C. E. Puente et al.

Fig. 2 From a multifractal measure dx to a derived Gaussian measure dy via a space-filling fractal interpolating function f .

from a simple multifractal measure, indicating thatthe plane-filling fractal function has the ability tofilter all the spikiness in dx to produce, quite sur-prisingly, an ever-smooth dy.

This Gaussian limit turns out to be universal,as the same plane-filling fractal interpolating func-tion gives a bell from an arbitrary iteration scheme.7

For the aforementioned three interpolating points,if the two mappings are iterated according to a p−qproportion, with q = 1 − p, then the resulting bellsimply has a mean µ equal to p. The result is yetmore general as one may replace a simple binomialmultifractal dx by any measure defined over a con-tinuum on x, to find yet another derived Gaussianlimit.7

For the uniform measure in x, i.e. the 50–50% proportion with three equally-spaced pointsin x, some analytical results have been alreadyestablished.7 In this scenario, the mean and

variance are µ = 1/2 and σ2 = 1/(12 − 12z2),where d1 = −d2 = z and z tending to one. Asthe fractal function fills-up the plane when z tendsto one, i.e. as its range grows from −∞ to ∞, theimplied variance also goes to infinity. But, as shownin Table 1, computed using the Maple V symboliclanguage package with a precision of 250 digits, thefirst 12 standardized moments of the derived mea-sure over y, indeed converge, in an orderly manner,to the moments of the standard bell (as included inthe last column). A formal proof of the Gaussianresult for this case has already been presented andrelies on showing (by induction) that all standard-ized moments of dy converge to the moments of thestandard Gaussian measure.7

The “minus-plus” case (−d1 = d2 = z) turnsout to yield a fractal interpolating function thatis just the mirror image of the one found via the“plus-minus” case. As a consequence, when z tends

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Oscillations and Circle-Map Dynamics 5

Table 1 Standardized Central Moments of dy for the Plus-Minus Case.Parameters: d1 = −d2 = z and {(0, 0), (1/2, 1), (1, 0)}.

Order z =0.9999 z =0.999999 z =0.99999999 N (0, 1)

3 0.000000000 0.000000000 0.000000000 0.04 2.999365046 2.999993650 2.999999937 3.05 0.000000000 0.000000000 0.000000000 0.06 14.990478199 14.999904750 14.999999048 15.07 0.000000000 0.000000000 0.000000000 0.08 104.866744091 104.998666509 104.999986665 105.09 0.000000000 0.000000000 0.000000000 0.0

10 943.002111830 944.979997736 944.999799975 945.011 0.000000000 0.000000000 0.000000000 0.012 10362.054001516 10394.669964570 10394.996699588 10395.0

to one, a simple multifractal measure also yields aunique bell (but now with mean q = 1− p) and theanalytical results, just mentioned, also hold.

When d1 = d2 = z (the “plus-plus” case) thefractal interpolating function becomes a symmetric“cloud” built by successive mid-point additions ofpowers of z starting with the aforementioned threeinterpolating points.9 This case yields a derivedmeasure eventually defined over the range [0,∞)whose mass progressively centers at infinity. Forsuch a case, and for the 50–50% proportion, onemay compute analytically that, in the limit, themean and the variance are µ = 1/(2 − 2z) andσ2 = 1/(12 − 12z2) (the same variance as before),which both tend to infinity as z tends to one. Thisturns out to be a rather curious case, because thecoefficient of variation (i.e. σ/µ) tends to zero, indi-cating that in the limit the mass concentrates, withprobability one, at infinity. As presented in Table 2,the lower order moments happen to behave in amanner (after standardization) that is progressivelyconsistent with a standard Gaussian distribution,

Table 2 Standardized Central Moments of dy for the Plus-Plus Case. Param-eters: d1 = d2 = z and {(0, 0), (1/2, 1), (1, 0)}.

Order z =0.9999 z =0.999999 z =0.99999999 N (0, 1)

3 −0.032656325 −0.003265983 −0.000326599 0.04 3.001639399 3.000016400 3.000000164 3.05 −0.326658273 −0.032659923 −0.003265986 0.06 15.035259628 15.000352666 15.000003527 15.07 −3.432783098 −0.342932064 −0.034292860 0.08 105.643263393 105.006430686 105.000064307 105.09 −41.260116637 −4.115251502 −0.411514386 0.0

10 956.901068056 945.118861509 945.001188600 945.011 −568.823119975 −56.586203271 −5.658324296 0.012 10628.732737598 10397.330855430 10395.023307907 10395.0

but a complete proof of the Gaussian limit is notavailable due to a lack of simplification on therequired moment formulas.

4. OSCILLATIONS BETWEEN TWOBELLS IN ONE DIMENSION

Given the results presented in the previous section,consideration of the “minus-minus” case (i.e. −d1 =−d2 = z) would also be expected to give a Gaussiandistribution in the limit when z tends to one. Forsuch a case, and for the interpolating points usedbefore, the mean and the variance yield µ = 1/4and σ2 = 1/(12 − 12z2) (yet the same variance asbefore), and it appears that a single bell is foundby considering the first 12 standardized momentsas reported in Table 3. However, close examinationof this case reveals that the limit does not containjust a single bell but rather oscillations between twobells, as follows.

When a graph similar to Fig. 2 (based on 50 mil-lion iterations of suitable maps) is drawn for this

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6 C. E. Puente et al.

Table 3 Standardized Central Moments of dy for the Minus-Minus Case.Parameters: −d1 = −d2 = z and {(0, 0), (1/2, 1), (1, 0)}.

Order z =0.9999 z =0.999999 z =0.99999999 N (0, 1)

3 0.000002939 0.000000003 0.000000000 0.04 2.999080244 2.999990800 2.999999908 3.05 0.000029380 0.000000029 0.000000000 0.06 14.986209090 14.999862001 14.999998620 15.07 0.000308244 0.000000309 0.000000000 0.08 104.807032780 104.998068023 104.999980680 105.09 0.003694899 0.000003704 0.000000004 0.0

10 942.107516384 944.971020552 944.999710200 945.011 0.050733888 0.000050923 0.000000051 0.012 10347.314696658 10394.521843197 10394.995218301 10395.0

case (not shown), the corresponding limiting bell-like measure exhibits noticeably larger oscillationsaround the mode than what is found for the “plus-minus” case. At first, this discrepancy appears tobe a matter of a lack of precision in the calculationsdue to a small number of “9”s in say z = 0.999, butfurther studies revealed that such was not the case.

If the mean from all points within the fractalinterpolating function are computed, that is by fol-lowing the binary tree of all successive (and equallyspaced) additions up to a given level n, one encoun-ters oscillatory behavior according to the formula:

µn =2n + z(−2z)n

(2n+1 + 1)(z + 1). (4)

Hence, in the limit, when z tends to one, there hap-pen to be two bells whose means oscillate between0 and 1/2, and therefore give an apparent averageof 1/4.

A similar study of the other sign combinationcases (i.e. by layers) reveals convergence to a sin-gle mean, but the “minus-minus” case indeed yieldsunexpected oscillations, irrespective of the interpo-lating points used in the construction. As the non-standardized variance increases to infinity and asthe difference in means remains finite for such acase, the resulting measure appears to be a singlebell but this turns out not to be so, despite theinformation to the contrary implied by the first fewmoments in Table 3.

5. EXTENSIONS TO THREEDIMENSIONS

The expressions presented in Eqs. (1) and (2) maybe extended to higher dimensions so that they pro-duce attracting fractal functions living in three

dimensions. Specifically, the graph G of a fractalinterpolating function, from x into the plane y − zand passing by N + 1 points in three-dimensionalspace {(xn, yn, zn);x0 < · · · < xN , n = 0, 1, . . . , N},is defined as the unique attractor of N affine mapsas follows:11

wn

x

yz

=

an 0 0

cn dn hn

kn ln mn

x

yz

+

en

fn

gn

,

n = 1, . . . , N,

(5)

such that

An =(

dn hn

ln mn

)=

(r(1)n cos θ

(1)n −r

(2)n sin θ

(2)n

r(1)n sin θ

(1)n r

(2)n cos θ

(2)n

)

(6)

has L2-norm (i.e. the square root of the maximumeigenvalue of AT

n An) less than 1 and subject to theinitial conditions

wn

x0

y0

z0

=

xn−1

yn−1

zn−1

, wn

xN

yN

zN

=

xn

yn

zn

,

n = 1, . . . , N.

(7)

As these generalized fractal functions, with frac-tal dimensions now ranging from 1 to 3, are com-puted via iterations, such a process defines, onceagain, a unique measure over the new graph G thatmay then be used to calculate a joint derived mea-sure dyz over the plane y − z based on a simplemultifractal measure dx over x.

Figure 3 shows an example, consonant withFig. 1, of a three-dimensional fractal function that

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Oscillations and Circle-Map Dynamics 7

Fig. 3 From a multifractal measure dx to a joint derived measure dyz (with marginals dy and dz) via a fractal interpolatingfunction f from x to y − z, shown in the x − y and x − z planes.

passes by the points {(0, 0, 0), (1/2, 1, 1), (1, 0, 0)}and with parameters An [Eq. (6)] given by,

A1 =

(0 1

2

12 0

), A2 =

(0 −1

2

12 0

), (8)

that generates an interesting joint derived mea-sure dyz and non-trivial marginal measures dy anddz, that is over the y and z directions. As before,these sets are obtained iterating the correspondingmappings w1 and w2 [Eq. (5)] based on 15 mil-lion iterations according to pseudo-random num-bers, following a 70–30% proportion that also yieldsa simple binomial multifractal measure dx over x.

As may be appreciated, these ideas lead to ahost of interesting, complex and “random-looking”patterns dyz, which resemble sets found in geo-physical applications such as rainfall-radar spa-tial patterns and contaminant plumes in groundwaters.2,4,16 Surprisingly, the transformation of aturbulent-based multifractal measure via a “simple”

three-dimensional fractal interpolating functionencompasses the complex geometries of a class ofnatural sets and such a result hence suggests thatthe notions may one day be useful in devising adeterministic language for complexity.4,15

6. SPACE-FILLING CASES ANDTHE TWO-DIMENSIONALGAUSSIAN LIMIT

Increasingly space-filling fractal interpolating func-tions may be obtained as ‖An‖2 → 1 for all n. Thisleads, in polar coordinates [Eq. (6)], to the condi-tions |r(j)

n | → 1, j = 1, 2; θ(1)n → θ

(2)n + kπ, for any

integer k.6 This yields 4N possible paths towardsspace-filling functions when considering all possiblesign combinations on the new scaling parametersr(j)n , as defined in Table 4 for N = 2 affine mappings.Figure 4 shows an example of such limiting

notions for a fractal function that passes by the

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8 C. E. Puente et al.

Table 4 Sign Combinations on Scaling Parametersfor Three-Dimensional Fractal Interpolating Func-tions Based on Two Affine Mappings.

Case r(1)1 r

(2)1 r

(1)2 r

(2)2 Case r

(1)1 r

(2)1 r

(1)2 r

(2)2

1 + + + + 9 − + + −2 + + + − 10 − − + +3 + − + + 11 − + − +4 + + − + 12 + − − −5 − + + + 13 − − + −6 + − + − 14 − + − −7 + + − − 15 − − − +8 + − − + 16 − − − −

aforementioned three points {(0, 0, 0), (1/2, 1, 1),(1, 0, 0)}, when the parameter matrices An take onthe values,

A1 =(

0 0.9990.999 0

), A2 =

(0 −0.999

0.999 0

),

(9)

Fig. 4 From a multifractal measure dx to a derived joint circular Gaussian measure dyz via a space-filling fractal interpolatingfunction f .

and when the two maps are iterated (15 mil-lion times) according to a 70–30% proportion. Asmay be seen, a circular joint Gaussian distributionappears as a projection from a simple multifrac-tal measure, hence generalizing what was pre-viously encountered with two-dimensional fractalfunctions.

Even though a complete proof of the Gaussian-ity of the bell shown in Fig. 4 is not available yetas formulas defy simplification, it shall be shownthat not all the sign combination cases in Table 4result in unique bells over two dimensions. Leavingaside cases that show oscillations, as the “minus-minus” case in two dimensions, for a later section,the following are the trends that we have encoun-tered for groups of sign combinations defined basedon Table 4, and for the specific situation whenθ1 = θ

(1)1 = θ

(2)1 and θ2 = θ

(1)2 = θ

(2)2 .

Grouping the 16 cases on the scalings into fourgroups: A = Cases[1, 7, 10, 16], B = Cases[2, 4,13, 15], C = Cases[3, 5, 12, 14], and D = Cases[6, 8,

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Oscillations and Circle-Map Dynamics 9

9, 11], it may be shown numerically (via suitableiterations) that the coefficient of correlation, ρ, oftheir corresponding bells are as follows,

Group A: ρ = 0, except when θ1 = kπ, θ2 = lπ thatgive ρ = 1 or ρ = −1, for k, l integers.

Group B: ρ = 0, except when θ1 = kπ that yieldsarbitrary correlation ρ ∈ [−1, 1], which dependson θ2 and the interpolating coordinates, for kinteger.

Group C: ρ = 0, except when θ2 = lπ that givesarbitrary correlation ρ ∈ [−1, 1], which dependson θ1 and the interpolating coordinates, for linteger.

Group D: ρ = 0, except when θ = θ1 = θ2 + kπthat gives arbitrary correlation ρ ∈ [−1, 1], whichdepends on θ and the interpolating coordinates, fork integer.

Further numerical calculation of means and vari-ances, over both the y and z components, via itera-tions as well as by levels (as previously done on thetwo-dimensional case), reveals that the most com-mon derived measures generated by the space-fillingthree-dimensional fractal functions are indeed sin-gle circular bells (ρ = 0) that have finite means, andvariances that grow to infinity as the magnitude ofthe scalings tends to one.

There are, however, some noteworthy exceptionsalong the lines θ1 = kπ, θ2 = kπ and θ1 = θ2 + kπ,for an integer k, that generalize what was reportedfor the one-dimensional bell(s). The specific trendsalong such lines are as follows.

1. For θ = θ1 = kπ: (a) all cases in groups A andC and cases 8 and 9 (from group D) give a sin-gle circular bell having finite means; (b) cases 2and 6 (from groups B and D, respectively) yield asingle bell whose mean converges to (∞,∞), whilecases 4 and 11 (also from groups B and D) resultin another single bell whose mean converges to(∞,−∞); and (c) cases 13 and 15 (from group B),which generate non-circular bells as a function of θ,give two oscillating bells, as in the one-dimensionalcase.

2. For θ = θ2 = kπ: (a) all cases in groups A andB and cases 8 and 9 give a single circular bell withfinite means; (b) cases 3 and 6 result in a single bellcentered at (∞,∞), whereas cases 5 and 11 yieldbells with means (−∞,∞); and (c) cases 12 and 14result in two oscillating elliptical bells.

3. For θ = θ1 = θ2 + kπ: (a) all cases in groupsB and C and cases 7 and 10 give a single circularbell with finite means; (b) cases 6 and 11 yield sin-gle bells whose mean tends respectively to (∞,∞)and (∞,−∞); (c) cases 8 and 9, yielding nonzerocorrelation, correspond (contrary to the results justreported for the other θ lines) to single bells withfinite means; and (d) cases 1 and 16 appear to con-verge to a single bell with finite means, but closeexamination of the results reveal that they containoscillatory behavior that encompasses 2π/θ bells, asexplained in the next section.

7. MULTIPLE BELLS AND THEIRCIRCLE MAP DYNAMICS

When all scalings r(j)n are positive (case 1) or nega-

tive (case 16), and when the angles θ1 = θ(1)1 = θ

(2)1

and θ2 = θ(1)2 = θ

(2)2 are such that θ = θ1 = θ2 +kπ,

for k integer, one encounters seemingly only a sin-gle circular bell, but detailed calculations by levelsindicate that there are oscillations among a host ofbells depending on the angle θ.

As illustrated in Fig. 5 for the “all plus” casewith angles θ equal to 2π over 3, 4, 5, 6, 7, and8, the bell’s center travels following a circle, cen-tered at the apparent single mean, and in conso-nance with the simple circle map φn+1 = φn + θ.As seen, the radius of such a circle increases as θdecreases, for the line joining the points (0, 0) and(1/2, 1/2) is always inscribed within such a circle.When 2π/θ = n, the circle map generates a regularpolygon having n sides and hence such a case cor-responds to n bells that cycle (counter-clockwise)in such a circle. When θ does not divide 2π, case 1generates infinitely many bells whose means travelin a circle.

Figure 6 shows what happens level by level inthe “all minus” case. As may be inferred, such a signcombination case also yields oscillations among sev-eral bells, now consonant with the simple dynamicson a circle, φn+1 = φn + π − θ, which result some-times in simple polygons (for large angles θ) butmost often stars inscribed on a finite circle thatcontains, once again, the line joining the points(0, 0) and (1/2, 1/2). When 2π/θ = n and n > 6all patterns are stars and the simple map travelswithin the circle (clockwise) yielding sharp oscilla-tory behavior on the means of the implied n bells.As before, this case yields infinitely many bells whenθ does not divide 2π.

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10 C. E. Puente et al.

0 0.2 0.4

0

0.2

0.4

0.6

−0.5 0 0.5

0

0.2

0.4

0.6

0.8

1

−0.5 0 0.5

0

0.5

1

−1 −0.5 0 0.5

0

0.5

1

1.5

−0.5 0 0.5

0

0.5

1

1.5

−1 −0.5 0 0.5

0

0.5

1

1.5

Fig. 5 Oscillations on bell’s means computed by layers for “all positive” scalings case. The angle θ takes on the values 2π/3,2π/4, 2π/5, 2π/6, 2π/7, and 2π/8 (shown left to right and top to bottom).

0 0.5 1−1

−0.5

0

0.5

0 0.2 0.4 0.6 0.8

−0.5

0

0.5

0 0.2 0.4 0.6 0.8−0.4

−0.2

0

0.2

0.4

0 0.2 0.4 0.6

−0.2

0

0.2

0.4

0 0.2 0.4 0.6

−0.2

0

0.2

0.4

0 0.2 0.4 0.6

−0.2

0

0.2

0.4

Fig. 6 Oscillations on bell’s means computed by layers for “all negative” scalings case. The angle θ takes on the values 2π/3,2π/4, 2π/5, 2π/6, 2π/7, and 2π/8 (shown left to right and top to bottom).

It should be stressed that cases 1 and 16 onlygive many bells along the line θ = θ1 = θ2 + kπ, foroutside such the dynamics no longer happen along acircle but rather are attracted towards a true centeryielding, at the end, a single bell with finite mean,as reported earlier.

8. EXOTIC DECOMPOSITIONSOF OSCILLATORY BELLS

As previously reported,8,9 when the aforementionedangles associated with the two affine mappings θ1

and θ2 both divide 2π and when such are multiplesof one another, exotic kaleidoscopes of patterns end

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Oscillations and Circle-Map Dynamics 11

Fig. 7 Selected exotic kaleidoscopic patterns inside oscillating bells. “All positive” scalings case (left) with parameters{(0, 0, 0), (1/2, 1, 1), (1, 0, 0)}, θ1 = 2π/11, θ2 = 6π/11 and “all negative” scalings case (right) with parameters {(0, 0, 0),(1/2, 1,−3), (1, 5, 0)}, θ1 = 2π/5, θ2 = π/5. Shown patterns are made up of 20,000 dots.

up decomposing single bells in non-trivial manners.9

It happens that such a behavior turns out to bevalid not only for the cases that define single bellswith finite means but also for the two oscillatorycases explained in the previous section.

As an illustration, Fig. 7 shows examples ofthe beautiful transient patterns that decomposethe oscillatory attractors for the two cases, when thebinary expansion of π is used to guide the iterationsand when the scalings have magnitudes equal to 1−εwith ε = 10−8. Remarkably, despite the cyclic move-ment of the dancing bells, the patterns obtaineddo not exhibit within them oscillations of any kindand (as in cases defining single bells) their succes-sive superposition gives an attractor that appearsto define a single bell.9

9. SUMMARY

This work has illustrated that the transformationof simple multifractal measures, via fractal interpo-lating functions (defined over two and three dimen-sions and computed through the iteration of simpleaffine mappings), gives rise to: (a) interesting andseemingly-random complex patterns if the dimen-sion of the fractal function is “low,” and (b) lim-iting Gaussian or “closely” Gaussian-like measureswhen the fractal functions fill up the space in whichthey live.

It has been explained how the nature of thelimiting Gaussian measures (the bells) turns outto depend on the sign combinations of the scaling

parameters of the affine mappings, as follows. Forthe two-dimensional fractal functions via the useof two mappings, there are four cases. Two ofthem, the “plus-minus” and “minus-plus” cases,may indeed be proven to define a Gaussian limitthat prior to standardization has a finite mean.There is a case, the “plus-plus” one, that appearsalso to converge to a Gaussian, but such is sub-stantially different as it has a mean that convergesto infinity (prior to standardization) and a coeffi-cient of variation that tends to zero. The remaining“minus-minus” case turns out to be surprising, forcalculation of all points within the fractal function,by levels, shows convergence to two oscillating bells(with finite alternating means before standardiza-tion) rather than just one.

For the three-dimensional fractal functions andvia two affine mappings, there are 16 differentsign combination cases. Whereas the most commonbehavior obtained turns out to be convergence toa single circular Gaussian distribution with finitemeans, there are also cases that mimic the resultsobtained with the two-dimensional fractal func-tions. For instance, and similar to the “plus-plus”case, there are cases that appear to lead to sin-gle bells that drift to the corners of the fourquadrants, and there are also sign combinationsthat, as the “minus-minus” case, result in oscilla-tions among several bells. In regards to the lat-ter, it has been shown, analyzing the process bylevels, that the “all plus” and “all minus” casessurprisingly lead to oscillations among an arbitrary

September 16, 2008 16:22 00408

12 C. E. Puente et al.

number of two-dimensional bells that follow circle-map dynamics and that result in lovely transientkaleidoscopic patterns that decompose the limitingattractors.

ACKNOWLEDGMENTS

This work was supported in part by the Director,Office of Science, of the US Department of Energyunder Contract No. DE-AC02-05CH11231.

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