a nonlinear dynamics perspective of … › ... › part5_vol15_no12_2005.pdfpart v: fractals...

January 12, 2006 14:18 01477

Tutorials and Reviews

International Journal of Bifurcation and Chaos, Vol. 15, No. 12 (2005) 3701–3849c© World Scientific Publishing Company

A NONLINEAR DYNAMICS PERSPECTIVEOF WOLFRAM’S NEW KIND OF SCIENCE.

PART V: FRACTALS EVERYWHERE

LEON O. CHUA, VALERY I. SBITNEV and SOOK YOONDepartment of Electrical Engineering and Computer Sciences,

University of California at Berkeley, CA 94720, USA

Received April 5, 2005; Revised July 10, 2005

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unifiedglobal representation for all 256 one-dimensional Cellular Automata local rules. Except for eightrather special local rules whose global dynamics are described by an affine (mod 1 ) function ofonly one binary cell state variable, all characteristic functions exhibit a fractal geometry whereself-similar two-dimensional substructures manifest themselves, ad infinitum, as the number ofcells (I + 1) → ∞.

In addition to a complete gallery of time-1 characteristic functions for all 256 local rules,an accompanying table of explicit formulas is given for generating these characteristic functionsdirectly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potentialapplications of these fundamental formulas, we prove rigorously that the “right-copycat” localrule 170 is equivalent globally to the classic “left-shift” Bernoulli map. Similarly, we prove the“left-copycat” local rule 240 is equivalent globally to the “right-shift” inverse Bernoulli map.

Various geometrical and analytical properties have been identified from each characteristicfunction and explained rigorously. In particular, two-level stratified subpatterns found in mostcharacteristic functions are shown to emerge if, and only if, b1 �= 0, where b1 is the “synapticcoefficient” associated with the cell differential equation developed in Part I.

Gardens of Eden are derived from the decimal range of the characteristic function of eachlocal rule and tabulated. Each of these binary strings has no predecessors (pre-image) and hastherefore no past, but only the present and the future. Even more fascinating, many local rulesare endowed with binary configurations which not only have no predecessors, but are also fixedpoints of the characteristic functions. To dramatize that such points have no past, and no future,they are henceforth christened “Isles of Eden”. They too have been identified and tabulated.

Keywords : Cellular neural networks, CNN; cellular automata; Turing machine; universal com-putation; Bernoulli shift; 1/f power spectrum; global equivalence classes attractors; invariantorbits; Garden of Eden; Isle of Eden; characteristic function; fractal geometry; fractals.

1. Characteristic Functions: GlobalRepresentation of Local Rules

The fundamental concept of the time-1 character-istic function

χ1N : [0, 1] → [0, 1] (1)

of a local rule N is defined in Part IV [Chua et al.,2005] as a function from the unit interval [0, 1] into

itself which uniquely maps each input binary string{x0, x1, x2, . . . , xI} represented in decimal form

φ =I∑

i=0

2−(i+1)xi (2)

into an output binary string

χ1N (φ) =

I∑i=0

2−(i+1)yi (3)

3701

January 12, 2006 14:18 01477

3702 L. O. Chua et al.

Cell(I-1)

Cell(I-2) Cell

ICell

0Cell

1Cell

2

Cell(i-1)

Celli

Cell(i+1)

1tiu −tiu

1tiu +

Local1

outputtiu +

input Symbolic Truth Table

1117

0116

1015

0014

1103

0102

1001

0000 0β

1β2β

3β

4β

5β6β7β

11 1( , , )

t t t ti i i iu u uu +

− += N

1117

-1116

1-115

-1-114

11-13

-11-12

1-1-11

-1-1-10 0γ1γ2γ3γ4γ

5γ6γ

7γ

RuleN

Numeric Truth Table

1tiu −

tiu 1

tiu +

1tiu +

1tiu −

tiu 1

tiu +

1tiu +

4

6 7

5

1

32

0 21 = 2

27 = 12826 = 64

22 = 4 23 = 8

24 = 16

20 = 1

25 = 32(1,-1,1)

(-1,-1,1)

(1,1,1)

(-1,1,1)(-1,1,-1)

(1,1,-1)

(-1,-1,-1)

(1,-1,-1)

1tiu

1tiu

tiu

4

6 7

5

1

32

0 21 = 2

27 = 12826 = 64

22 = 4 23 = 8

24 = 16

20 = 1

25 = 32(1,-1,1)

(-1,-1,1)

(1,1,1)

(-1,1,1)(-1,1,-1)

(1,1,-1)

(-1,-1,-1)

(1,-1,-1)

−

+

tiu

Fig. 1. (a) A one-dimensional Cellular Automata (CA) made of (I + 1) identical cells with a periodic boundary condition.Each cell “i” is coupled only to its left neighbor cell (i − 1) and right neighbor cell (i + 1). (b) Each cell “i” is described by alocal rule N , where N is a decimal number specified by a binary string {β0, β1, . . . , β7}, βi ∈ {0, 1}. (c) The symbolic truthtable specifying each local rule N , N = 0, 1, 2, . . . , 255. (d) By recoding “0” to “−1”, each row of the symbolic truth tablein (c) can be recast into a numeric truth table, where γk ∈ {−1, 1}. (e) Each row of the numeric truth table in (d) can berepresented as a vertex of a Boolean Cube whose color is red if γk = 1, and blue if γk = −1.

January 12, 2006 14:18 01477

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part V 3703

NT

ΣΦ Φ

[ ]0, 1 [ ]0, 11

Nχ

Σ

Fig. 2. A commutative diagram establishing a one-to-onecorrespondence between TN and χ1

N.

in decimal representation, as I → ∞, where{y0, y1, y2, . . . , yI} is the output binary string deter-mined from the local rule N of the one-dimensionalcellular automata under a periodic boundary con-dition, as shown in Fig. 1. Every input binarystring is assumed to be finite but whose lengthcan be chosen to be arbitrarily large. For prac-tical calculation, the domain of the characteristicfunction χ1

Nconsists of only a large but finite

number of equally-spaced real numbers inside [0, 1].In the limit I → ∞, the domain of χ1

Ncoin-

cides with the entire unit interval [0, 1]. In thiscase, every point φ ∈ [0, 1] corresponds uniquelyto an infinite binary string. Conversely, each infi-nite binary string corresponds to a unique pointon [0, 1]. This one-to-one correspondence betweeneach binary string in the set Σ of all infinite binarystrings and each real number in [0, 1] is depictedin Fig. 2, where φ is defined via Eq. (2). Observethat since the time-1 characteristic function χ1

N

maps each infinite binary string into anotherinfinite binary string after one iteration of the localrule N , it is a global representation.1

1.1. Deriving explicit formula forcalculating χ1

N

Recall from Eq. (8) of [Chua et al., 2004] that theoutput of each local rule N can be calculated fromthe formula

(4)

where the eight parameters {z2, c2, z1, c1, z0, b1,b2, b3} determining each local rule N is given in

Table 4 of [Chua et al., 2003]. Since χ1N

is definedin terms of binary variables xi ∈ {0, 1}, let us applythe conversion relationship Eq. (4) from [Chuaet al., 2005] namely,

xi =12

(ui + 1) (5)

and define the step function

(6)

to rewrite Eq. (4) into

(7)

where

z′0 � 12[z0 − (b1 + b2 + b3)]

z′1 � 12z1

z′2 � 12z2

(8)

Substituting Eq. (7) for yi = xt+1i in Eq. (3),

and deleting the superscripts, we obtain the fol-lowing explicit formula for calculating the charac-teristic function χ1

Nfor any local rule N , N =

0, 1, 2, . . . , 255:

(9)

where {z′2, c2, z′1, c1, z

′0, b1, b2, b3} are defined in

Eq. (8) and Table 4 of [Chua et al., 2003].Applying Eq. (9) to all 256 local rules, we

obtain the explicit formulas listed in Table 1 forcalculating the corresponding time-1 characteris-tic functions χ1

N, N = 0, 1, 2, . . . , 255, where the

binary string begins from φ = 0 corresponding to

{0, 0, 0, · · · 0},↑ ↑ ↑ ↑x0 x1 x2 xI

1We can define a time-k characteristic function χkN

: [0, 1] → [0, 1] in exactly the same way where the output string is

calculated after every “k” iterations under rule N .

January12,

200614:18

01477

Table 1. Explicit formulas for calculating characteristic functions χ1N

in terms of binary strings {x0, x1, x2, . . . , xI}. Each row is partitioned into four equal parts,

where each part is color coded either in blue, if the characteristic function has no stratification (see Tables 2 and 5), or in pink, otherwise.

3704

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3705

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3706

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3707

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3708

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3709

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3710

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3711

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3712

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3713

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3714

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3715

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3716

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3717

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3718

January 12, 2006 14:18 01477

Table

1.

(Continued

)

3719

January 12, 2006 14:18 01477


to φ = 1 corresponding to

{1, 1, 1, · · · 1}.↑ ↑ ↑ ↑x0 x1 x2 xI

1.2. Graphs of characteristicfunctions χ1

N

For future reference, we have plotted the time-1 characteristic functions χ1

Nfor all 256 local

rules with I = 65, and displayed them in Table 2showing only 201 points for each rule to avoid clut-ter. In other words, each graph of χ1

Nin Table 2

shows only 201 values of χ1N

, each one calcu-lated from a 66-bit binary string. To enhance clar-ity, every pair of adjacent points in each graphare plotted as a small “red” square “ ” anda small “blue” square “ ” on top of alternat-ing red and blue color bars emanating from eachvalue of φ ∈ [0, 1] corresponding to the 201uniformly distributed points with spacing ∆φ =0.005.

A careful examination of Table 2 reveals thatadjacent pairs of points of χ1

Nare either loca-

ted in “close proximity” of each other, or theyexhibit an “abrupt jump” from each other. We willhenceforth refer to those subintervals where adja-cent red and blue squares are close to each other as“smooth”, and those exhibiting “abrupt jumps” as“discontinuous”. A careful analysis of these subin-tervals reveal that they extend over a minimumrange of ∆φ = 0.25 for all 256 rules. For exam-ple, only the second subinterval φ ∈ [0.25, 0.50) ofrule 2 is discontinuous. On the other hand, the firstand second subintervals [0, 0.25) and [0.25, 0.50) ofrule 3 are discontinuous. For rule 110 , we findonly the fourth subinterval [0.75, 1.00] is discontin-uous. For rule 30 , we find all four subintervals[0, 0.25), [0.25, 0.50), [0.50, 0.75) and [0.75, 1.00) arediscontinuous. Since these properties are quite use-ful for understanding the global dynamics of localrules, we have divided the area to the right ofthe equality sign of each characteristic functionχ1

Nin Table 1 into the above four corresponding

equal parts, and painted each part with a light bluebackground color if the corresponding subintervalhas smooth adjacent red and blue squares, or ina light pink background color if adjacent red andblue squares exhibit discontinuous jumps from eachother.

1.3. Deriving the Bernoulli mapfrom χ1

170

As an application of the explicit formulas listedin Table 1, let us apply the characteristic func-tion χ1

170of 170 to an (I + 1)-bit binary string

{x0, x1, x2, . . . , xI} with decimal representation

φ =I∑

i=0

2−(i+1)xi (10)

to obtain

χ1170 (φ) =

I∑i=0

2−(i+1)xi+1 =I+1∑j=1

2−jxj + x0 − x0

= 2

[I+1∑j=0

2−(j+1)xj

]− x0

={

2φ, φ < 0.52φ − 1, φ ≥ 0.5

(11)

as I → ∞.It follows from Eq. (11) that the characteris-

tic function χ1170

of the rule 170 converges to thewell-known Bernoulli map [Billingsley, 1978]

χ1170

(φ) = 2φ mod1 (12)

as I → ∞.Since the output of each pixel “i” of rule 170

in Table 1 is given simply by yi = xi+1, the localrule 170 consists of simply copying the “state” ofthe pixel “i + 1” of the right-neighboring pixel. Wewill henceforth call rule 170 the right-copycat rule.The graph of the right-copycat rule 170 is shownin Fig. 3.

1.4. Deriving inverse Bernoullimap from χ1

240

Let us apply the characteristic function χ1240

of240 from Table 1 to the (I + 1)-bit binary string φdefined in Eq. (10) to obtain

χ1240 (φ) =

I∑i=0

2−(i+1)xi−1 =12

I−1∑j=−1

2−(j+1)xj

=12

[I−1∑j=0

2−(j+1)xj

]+

12xI

=12φ +

12xI (13)

as I → ∞.

January12,

200614:18

01477

Table 2. Gallery of characteristic functions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

3

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

0

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

2

2

3721

January12,

200614:18

01477

Table 2. (Continued )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

5

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

7

7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

4

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

6

6

3722

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

9

9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

11

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

8

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

10

10

3723

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

13

13

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

15

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

12

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

14

14

3724

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

17

17

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

19

19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

16

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

18

18

3725

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

21

21

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

23

23

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

20

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

22

22

3726

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

25

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

27

27

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

24

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

26

26

3727

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

29

29

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

31

31

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

28

28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

30

30

3728

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

33

33

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

35

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

32

32

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

34

34

3729

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

37

37

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

39

39

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

36

36

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

38

38

3730

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

41

41

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

43

43

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

40

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

42

42

3731

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

45

45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

47

47

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

44

44

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

46

46

3732

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

49

49

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

51

51

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

48

48

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

50

50

3733

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

53

53

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

55

55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

52

52

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

54

54

3734

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

57

57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

59

59

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

56

56

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

58

58

3735

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

61

61

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

63

63

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

60

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

62

62

3736

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

65

65

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

67

67

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

64

64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

66

66

3737

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

69

69

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

71

71

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

68

68

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

70

70

3738

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

73

73

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

75

75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

72

72

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

74

74

3739

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

77

77

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

79

79

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

76

76

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

78

78

3740

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

81

81

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

83

83

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

80

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

82

82

3741

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

85

85

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

87

87

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

84

84

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

86

86

3742

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

89

89

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

91

91

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

88

88

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

90

90

3743

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

93

93

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

95

95

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

92

92

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

94

94

3744

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

97

97

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

99

99

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

96

96

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

98

98

3745

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

101

101

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

103

103

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

100

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

102

102

3746

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

105

105

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

107

107

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

104

104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

106

106

3747

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

109

109

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

111

111

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

108

108

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

110

110

3748

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

113

113

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

115

115

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

112

112

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

114

114

3749

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

117

117

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

119

119

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

116

116

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

118

118

3750

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

121

121

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

123

123

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

120

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

122

122

3751

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

125

125

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

127

127

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

124

124

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

126

126

3752

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

129

129

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

131

131

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

128

128

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

130

130

3753

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

133

133

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

135

135

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

132

132

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

134

134

3754

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

137

137

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

139

139

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

136

136

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

138

138

3755

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

141

141

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

143

143

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

140

140

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

142

142

3756

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

145

145

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

147

147

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

144

144

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

146

146

3757

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

149

149

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

151

151

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

148

148

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

150

150

3758

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

153

153

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

155

155

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

152

152

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

154

154

3759

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

157

157

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

159

159

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

156

156

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

158

158

3760

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

161

161

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

163

163

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

160

160

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

162

162

3761

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

165

165

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

167

167

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

164

164

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

166

166

3762

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

169

169

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

171

171

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

168

168

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

170

170

3763

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

173

173

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

175

175

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

172

172

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

174

174

3764

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

177

177

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

179

179

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

176

176

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

178

178

3765

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

181

181

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

183

183

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

180

180

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

182

182

3766

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

185

185

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

187

187

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

184

184

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

186

186

3767

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

189

189

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

191

191

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

188

188

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

190

190

3768

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

193

193

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

195

195

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

192

192

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

194

194

3769

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

197

197

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

199

199

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

196

196

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

198

198

3770

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

201

201

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

203

203

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

200

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

202

202

3771

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

205

205

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

207

207

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

204

204

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

206

206

3772

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

209

209

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

211

211

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

208

208

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

210

210

3773

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

213

213

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

215

215

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

212

212

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

214

214

3774

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

217

217

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

219

219

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

216

216

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

218

218

3775

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

221

221

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

223

223

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

220

220

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

222

222

3776

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

225

225

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

227

227

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

224

224

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

226

226

3777

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

229

229

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

231

231

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

228

228

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

230

230

3778

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

233

233

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

235

235

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

232

232

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

234

234

3779

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

237

237

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

239

239

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

236

236

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

238

238

3780

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

241

241

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

243

243

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

240

240

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

242

242

3781

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

245

245

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

247

247

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

244

244

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

246

246

3782

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

249

249

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

251

251

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

248

248

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

250

250

3783

January12,

200614:18

01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

253

253

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

255

255

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

252

252

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

254

254

3784

January 12, 2006 14:18 01477


χ

0 0.5 1

1

0.51170

1 7 0φ

Fig. 3. The characteristic function of the “right-copycat”rule 170 converges to the Bernoulli map.

Since xI ∈ {0, 1} is the rightmost bit of thebinary string {x0, x1, x2, . . . , xI}, it follows that thegraph of the characteristic function χ1

240(φ) con-

sists of two parallel branches, parameterized by thelast binary bit xI , as shown in Fig. 4.

Since the output of each pixel “i” of rule 240in Table 1 is given simply by yi = xi−1, the localrule 240 consists of simply copying the “state”of the pixel “i − 1” of the left-neighboring pixel.We will henceforth call rule 240 the left-copycatrule.

An examination of the graph of the characteris-tic functions χ1

170in Fig. 3 and χ1

240in Fig. 4 shows

that they are symmetrical with respect to the maindiagonal. In other words, the two graphs are inverseof each other. Observe that although the graphof χ1

240in Fig. 4 appears to be a double-valued

function, it is actually a well-defined single-valued

0 0.5 1

1

0.51240χ

2 4 0φ

Fig. 4. The characteristic function of the “left-copycat” rule240 converges to the inverse Bernoulli map.

function for all finite I, for xI uniquely specifieswhether the upper branch (if xI = 1), or the lowerbranch (if xI = 0) should be selected.

1.5. Deriving affine (mod 1)characteristic functions

We have shown in Secs. 1.3 and 1.4 that thecharacteristic functions χ1

170and χ1

240are affine

(mod 1) functions for finite I. An examination ofTable 2 reveals that there are only eight affine(mod 1) characteristic functions, namely, χ1

0, χ1

15,

χ151

, χ185

, χ1170

, χ1204

, χ1240

and χ1255

. The explicitformula for each of these characteristic functionscan be easily derived from Table 1 as in Secs. 1.3and 1.4. Table 3 lists the explicit formulas defin-ing these eight affine (mod 1 ) local rules and theircorresponding global characteristic function. Thegraph of each characteristic function χ1

Nis shown

in Table 4.

Remark 1.1. Since there exist infinitely many dis-tinct sets of parameters {z2, c2, z1, c1, z0, b1, b2, b3}[Chua et al., 2003] from which the explicit for-mula Eq. (9) represents the same characteristicfunction χ1

N, N = 0, 1, 2, . . . , 255, the equa-

tions listed in Table 1 represent only one of manyequivalent explicit formulas. In fact, we need to listexplicit formulas for only 128 local rules since thesimple transformation in the following propositionallows us to generate corresponding explicit formu-las for the remaining 128 rules.

Proposition 1.1. Given an explicit formula

χ1N =

I∑i=0

2−(i+1) {f(xi−1, xi, xi+1)} (9a)

for local rule N , the following corresponding equa-tion gives an explicit formula for local rule N ′ ∆=255−N :

χ1255–N =

I∑i=0

2−(i+1) {−f(xi−1, xi, xi+1)} (9b)

Proof. Equation (9b) follows directly from Eq. (9a)and the identity

{−f} = 1 − {f}, (9c)

as can be verified by direct substitution. �

January 12, 2006 14:18 01477


Table 3. Explicit formulas defining the eight affine (mod 1) local rules and their associated characteristic functions. The barabove a binary bit means taking its complement, i.e. 0 = 1 and 1 = 0.

Explicit formula forLocal Rule

Formula

Affine(mod 1) Rule Number

( )10

0φ =χ1 0nix + =

1N

χN

255

240

204

170

85

51

15

0

( )11 5

11 , i f 0

21 1

, i f 12 2

I

I

x

x

φ

φφ

− + ==

− + =

χ

( )15 1

1φφ = − +χ

( ) 1

18 51

2 1 , if 0

2 2 , if 1

x

x

φ

φφ

− + == − + =

χ

( )11 7 0

2 m o d 1φφ =χ

( )12 4 0

1, i f 0

21 1

, if 12 2

I

I

x

x

φ

φφ

==

+ =χ

( )12 0 4

φφ =χ

( )12 5 5

1φ =χ

11

n ni ix x+

−=

1n ni ix x+ =

11

n ni ix x+

+=

11

n ni ix x+

+=

1n ni ix x+ =

11

n ni ix x+

−=

1 1nix + =

January 12, 2006 14:18 01477


Table 4. Characteristic functions χ1N

of Affine (mod 1) Rules.

0 0.5 1

1

0.510

χ

0φ

0 0.5 1

1

0.5115

χ

15φ

0 0.5 1

1

0.5151

χ

5 1φ

0 0.5 1

1

0.51170

χ

1 7 0φ

0 0.5 1

1

0.51255

0χ

2 5 5φ

0 0.5 1

1

0.5185

χ

8 5 φ

0 0.5 1

1

0.51204

χ

2 0 4φ

0 0.5 1

1

0.51240

χ

2 40φ

January 12, 2006 14:18 01477


Example 1.1. Given the following explicit formula

χ1110 =

I∑i=0

2−(i+1)

{−1 +

∣∣∣∣(+1xi−1 + 2xi

− 3xi+1 − 12

)∣∣∣∣}

(9d)

for N = 110 , we apply Eq. (9b) to obtain thefollowing explicit formula

χ1145 =

I∑i=0

2−(i+1)

{+1 −

∣∣∣∣(+1xi−1 + 2xi

− 3xi+1 − 12

)∣∣∣∣}

(9e)

for N ′ = 255− 110 = 145 .

Note that the above formula is different fromthe one listed in Table 1, since the formulas in Table1 are constructed independently of Proposition 1.1.

Proposition 1.2. The graphs of the characteris-tic functions for χ1

Nand χ1

255–Nare symmetric

with respect to the horizontal axis χ1N

= 0.5. Inparticular,

χ1N (φ) = 1 − χ1

255–N (φ) (9f)

Proof. Equation (9f) follows directly from Propo-sition 1.1 and the identity (9c). �

Remark 1.2. It would be instructive for the readerto verify the graphs of χ1

Nin Table 2 for N = 128,

129, . . . , 255 can be obtained by flipping the graphsof χ1

Nfor N = 127, 126, 125, . . . , 0, respectively,

about the φN -axis, and then translating themupward by ∆χ1

N= 1.

Remark 1.3. It follows from [Chua et al., 2004]that only 89 characteristic functions, out of 256, inTable 1, give qualitatively distinct global dynamics.All other characteristic functions can be generatedtrivially by applying the three global transforma-tions T†, T and T∗ from Klein’s Vierergruppe.

Example 1.2. Applying T†, T and T∗ to Eq. (9d),we obtain:

χ1124 = T†[χ1

110 ] =I∑

i=0

2−(i+1)

{−1 +

∣∣∣∣(−3xi−1

+ 2xi + 1xi+1 − 12

)∣∣∣∣}

(9g)

χ1137 = T[χ1

110 ] =I∑

i=0

2−(i+1)

{+1 −

∣∣∣∣(−1xi−1

− 2xi + 3xi+1 − 12

)∣∣∣∣}

(9h)

χ1193 = T∗[χ1

110 ] =I∑

i=0

2−(i+1)

{+1 −

∣∣∣∣(+3xi−1

− 2xi − 1xi+1 − 12

)∣∣∣∣}

(9i)

Remark 1.4. By choosing the characteristic functionof only one local rule from each of the 15 local equiv-alence classes listed in the left column of Table 20of [Chua et al., 2004], we can apply the appropriaterotation transformation listed in Table 1 of [Chuaet al., 2004], and use Proposition 1.1 to derive anexplicit characteristic function formula for each ofthe remaining 241 local rules.

2. Lameray Diagram on χ1N

GivesAttractor Time-1 Maps

We have demonstrated in [Chua et al., 2005] thatthe dynamics on each attractor of a local rule N isuniquely characterized by the forward time-1 returnmap

ρ1[N ] : φn−1 �→ φn (14)

starting from any point on the attractor. For the69 period-1 rules listed in Tables 3 and 4 of [Chuaet al., 2005], such time-1 maps consist of onlyone point on the main diagonal of the charac-teristic function χ1

N, and is therefore trivial. For

the 25 period-2 rules listed in Tables 7 and 8, suchtime-1 maps consist of two points on the main diag-onal of χ1

N, and is also trivial. Similarly, the time-1

map of the four period-3 rules listed in Table 9 con-sist of three points, as illustrated in Fig. 11 of [Chuaet al., 2005].

The most interesting and nontrivial time-1maps considered so far are the 112 generalizedBernoulli στ -shift rules listed in Tables 10–12 of[Chua et al., 2005]. The time-1 maps of these rules,as well as those corresponding to the remaining 50rules listed in Tables 17 and 18 of [Chua et al., 2005]consist in general of an uncountable [Devaney, 1992]number of points φ ∈ [0, 1], assuming I → ∞.

An instructive way to analyze the dynamics ofsuch time-1 maps is to plot the Lameray (cobweb)diagram starting from any generic initial point, andobserve how its “cobweb” loci evolves from this

January 12, 2006 14:18 01477


initial point on the characteristic function χ1N

. Letus study some of these rules.

2.1. Lameray diagram of 170

The dynamic pattern of the first 65 iterations ofrule 170 (with complexity index κ = 1) startingfrom φ0 = 0.0253584 is shown in Fig. 5(a). The first12 iterations of the Lameray diagram constructedfrom the characteristic function χ1

170are shown in

Fig. 5(b) for ease of visualization. The continuediterations of the Lameray diagram for 170 up ton = 63 are shown in Fig. 5(c). The color codeon top of Fig. 5(a) denotes the iteration numbern = 0, 1, 2, . . . , 63.

A comparison of the loci (corner points not onthe diagonal) in Fig. 5(c) with the forward time-1 map of 170 in Table 2 (p. 1106) of [Chua et al.,2005] clearly shows that they all fall on the graph oftime-1 map. Indeed, as n → ∞, this loci is seen (notshown) to converge on the complete graph of theforward time-1 map of 170 . Indeed, in this exam-ple, all points on the Lameray diagram, includingany initial point, are seen to fall on the associatedattractor. Note also that all points on the loci ofthe Lameray diagram of any rule N must be a sub-set of the associated characteristic function χ1

N, by

construction.


The dynamic pattern of the first 65 iterations of rule240 (with complexity index κ = 1) starting fromφ0 = 0.0253584, is shown in Fig. 6(a). The first12 iterations of the Lameray diagram constructedfrom the characteristic function χ1

240are shown in

Fig. 6(b) for ease of visualization. The continuediterations of the Lameray diagram for 240 up ton = 63 are shown in Fig. 6(c).

A comparison of the loci in Fig. 6(c) with theforward time-1 map of 240 in Table 2 (p. 1124) of[Chua et al., 2005] clearly shows that they all fall onthe graph of the time-1 map. Indeed, as n → ∞, thisloci is seen (not shown) to converge on the completegraph of the forward time-1 map of 240 . Indeed, inthis example, all points on the Lameray diagram,including any initial point, are seen to fall on theassociated attractor.


The dynamic pattern of the first 65 iterations ofrule 2 (with complexity index κ = 1) starting from

φ0 = 0.64368 is shown in Fig. 7(a). The first six iter-ations of the Lameray diagram constructed from thecharacteristic function χ1

2are shown in Fig. 7(b)

and the continued iterations up to n = 63 are shownin Fig. 7(c).

A comparison of the loci in Fig. 7(c) with theforward time-1 map of 2 in Table 2 (p. 1064) of[Chua et al., 2005] shows that except for the ini-tial point (which did not fall on the attractor butbelongs to the basin of attraction of the associatedattractor), all other points of the loci fall on thetime-1 map of 2 .

To demonstrate this result holds for any otherinitial point belonging to the same basin of attrac-tion, Fig. 8 shows the Lameray diagram of 2 (start-ing from a different initial point φ0 = 0.343184)over n = 25, 30, 35, 50, 75, 100, 150, 200 and 250iterations, respectively. Again except for the initialpoint, the loci of the Lameray diagram is seen toconverge on the forward time-1 map of 2 in Table 2of [Chua et al., 2005].


The dynamic pattern of the first 65 iterations ofrule 3 (with complexity index κ = 1) startingfrom φ0 = 0.64368 is shown in Fig. 9(a). The firstten iterations of the Lameray diagram constructedfrom the characteristic function χ1

3are shown in

Fig. 9(b) and the continued iterations up to n = 63are shown in Fig. 9(c).

A comparison of the loci in Fig. 9(c) with theforward time-1 map of 3 in Table 2 (p. 1064) of[Chua et al., 2005] shows that except for the ini-tial point (which did not fall on the attractor butbelongs to its basin of attraction), all other pointsof the loci fall on the time-1 map of 3 .

To demonstrate this result holds for any otherinitial point belonging to the same basin of attrac-tion, Fig. 10 shows the Lameray diagram of 3(starting from a different initial point φ0 =0.556412) over n = 5, 10, 15, 20, 35, 50, 100, 150and 250 iterations, respectively. Again, except forthe initial point, the loci of the Lameray diagram isseen to converge on the forward time-1 map of 3in Table 2 of [Chua et al., 2005].


The dynamic pattern of the first 65 iterations ofrule 46 (with complexity index κ = 3) startingfrom φ0 = 0.895314 is shown in Fig. 11(a). The first

January12,

200614:18

01477

0 10 20 30 40 50 60

60

50

40

30

20

10

0

n

i

9876543210

N = 170 , φ = 0.02535840 0 63

n = 120 0.5 1

0

0.5

1

φ

φ

n-1

n

2

3

4

5

6

7

8

9

n = 630 0.5 1

0

0.5

1

φ

φ

n-1

n

Fig. 5. (a) Dynamic pattern of 170 from φ0 = 0.0253584. (b) Lameray diagram for first 12 iterations. (c) Lameray diagram over 63 iterations.

3790

January12,

200614:18

01477

0 10 20 30 40 50 60

60

50

40

30

20

10

0

n

i

9876543210

N = 240 , φ = 0.02535840 0 63

n = 120 0.5 1

0

0.5

1

φ

φ

n-1

n

1

2

3

4

5

6

7

8

9

n = 630 0.5 1

0

0.5

1

φ

φ

n-1

n

Fig. 6. (a) Dynamic pattern of 240 from φ0 = 0.0253584. (b) Lameray diagram for first 12 iterations. (c) Lameray diagram over 63 iterations.

3791

January12,

200614:18

01477

0 10 20 30 40 50 60

60

50

40

30

20

10

0

n

i

N = 2 , φ = 0.643680 0 63

n = 60 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 630 0.5 1

0

0.5

1

φ

φ

n-1

n

Fig. 7. (a) Dynamic pattern of 2 from φ0 = 0.64368. (b) Lameray diagram for first six iterations. (c) Lameray diagram over 63 iterations.

3792

January 12, 2006 14:18 01477


N = 2 , φ = 0.3431840 0 250

n = 250 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 300 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 350 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 750 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 1000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 1500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 2000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 2500 0.5 1

0

0.5

1

φ

φ

n-1

n

Fig. 8. Nine snapshots of the Lameray diagram of 2 starting from φ0 = 0.343184.

ten iterations of the Lameray diagram constructedfrom the characteristic function χ1

46are shown in


A comparison of the loci in Fig. 11(c) with theforward time-1 map of 46 in Table 2 (p. 1075)of [Chua et al., 2005] shows that except for theinitial point (which did not fall on the attractor but

belongs to its basin of attraction), all other pointsof the loci fall on the time-1 map of 46 .

To demonstrate this result holds for any otherinitial point belonging to the same basin of attrac-tion, Fig. 12 shows the Lameray diagram of 46(starting from a different initial point φ0 =0.888034) over n = 7, 15, 25, 50, 75, 100, 150, 200and 250 iterations, respectively. Again, except for

January12,

200614:18

01477

0 10 20 30 40 50 60

60

50

40

30

20

10

0

n

i

N = 3 , φ = 0.643680 0 63

n = 100 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 630 0.5 1

0

0.5

1

φ

φ

n-1

n

Fig. 9. (a) Dynamic pattern of 3 from φ0 = 0.64368. (b) Lameray diagram for first ten iterations. (c) Lameray diagram over 63 iterations.

3794

January 12, 2006 14:18 01477


N = 3 , φ = 0.5564120 0 250

n = 50 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 100 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 150 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 200 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 350 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 1000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 1500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 2500 0.5 1

0

0.5

1

φ

φ

n-1

n


the initial point, the loci of the Lameray diagram isseen to converge on the forward time-1 map of 46in Table 2 of [Chua et al., 2005].


The dynamic pattern of the first 65 iterations ofrule 110 (with complexity index κ = 2) starting

from φ0 = 0.40653 is shown in Fig. 13(a). The firstfive iterations of the Lameray diagram constructedfrom the characteristic function χ1

110are shown in


A comparison of the loci in Fig. 13(c) with theforward time-1 map of 110 in Table 2 (p. 1091) of[Chua et al., 2005] shows that except for some initial

January12,

200614:18

01477

0 10 20 30 40 50 60

60

50

40

30

20

10

0

n

i

N = 46 , φ = 0.8953140 0 63

n = 100 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 630 0.5 1

0

0.5

1

φ

φ

n-1

n


3796

January 12, 2006 14:18 01477


N = 46 , φ = 0.8880340 0 250

n = 70 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 150 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 250 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 750 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 1000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 1500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 2000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 2500 0.5 1

0

0.5

1

φ

φ

n-1

n


set of points (which may not fall on the attractorassociated with the loci but belongs to its basin ofattraction), all other points of the loci fall on thetime-1 map of 110 associated with the correspond-ing attractor.

To demonstrate this result holds for any otherinitial point belonging to the same basin of attrac-tion, Fig. 14 shows the Lameray diagram of 110

(starting from a different initial point φ0 =0.149766) over n = 5, 10, 15, 25, 50, 75, 200, 350and 500 iterations, respectively. Again, except forsome initial set of points not belonging to the cor-responding attractor of 110 , the loci of the Lam-eray diagram is seen to converge to the forwardtime-1 map of 110 in Table 2 of [Chua et al.,2005].

January12,

200614:18

01477

0 10 20 30 40 50 60

60

50

40

30

20

10

0

n

i

N = 110 , φ = 0.406530 0 63

n = 50 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 630 0.5 1

0

0.5

1

φ

φ

n-1

n

Fig. 13. (a) Dynamic pattern of 110 from φ0 = 0.40653. (b) Lameray diagram for first five iterations. (c) Lameray diagram over 63 iterations.

3798

January 12, 2006 14:18 01477


N = 110 , φ = 0.1497660 0 500

n = 50 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 100 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 150 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 250 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 750 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 2000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 3500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 5000 0.5 1

0

0.5

1

φ

φ

n-1

n



The dynamic pattern of the first 65 iterations ofrule 30 (with complexity index κ = 2) startingfrom φ0 = 0.895314 is shown in Fig. 15(a). Thefirst ten iterations of the Lameray diagram con-structed from the characteristic function χ1

30are

shown in Fig. 15(b), and the continued iterationsup to n = 63 are shown in Fig. 15(c).

A comparison of the loci in Fig. 15(c) with theforward time-1 map of 30 in Table 2 (p. 1071) of[Chua et al., 2005] shows that except for some ini-tial set of points (which do not fall on the attractorassociated with the loci but belong to its basin of

January12,

200614:18

01477

0 10 20 30 40 50 60

60

50

40

30

20

10

0

n

i

N = 30 , φ = 0.8953140 0 63

n = 100 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 630 0.5 1

0

0.5

1

φ

φ

n-1

n


3800

January 12, 2006 14:18 01477


attraction), all other points of the loci fall on thetime-1 map of 30 associated with the correspond-ing attractor.

To demonstrate this result holds for any otherinitial point belonging to the same basin of attrac-tion, Fig. 16 shows the Lameray diagram of 30(starting from a different initial point φ0 =

0.149766) over n = 7, 14, 21, 50, 75, 100, 200,300 and 500 iterations, respectively. Again exceptfor some initial set of points not belonging to thecorresponding attractor of 30 , the loci of theLameray diagram is seen to converge to the for-ward time-1 map of 30 in Table 2 of [Chuaet al., 2005].

N = 30 , φ = 0.1497660 0 500

n = 70 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 140 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 210 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 500 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 750 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 1000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 2000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 3000 0.5 1

0

0.5

1

φ

φ

n-1

n

n = 5000 0.5 1

0

0.5

1

φ

φ

n-1

n


January 12, 2006 14:18 01477


3. Characteristic Functions areFractals

A careful examination of the characteristic func-tions χ1

Nin Table 2 would reveal that each graph

of χ1N

is composed of many subpatterns which areself-similar in the sense that each can be rescaledby appropriate horizontal and vertical scaling fac-tors so that it coincides with a part of the compos-ite pattern. Let us illustrate this fractal geometry[Barnsley, 1988] with some examples.

Example 3.1 [Characteristic function χ12]. Let us

repeat the characteristic function χ12

from Table 2in Fig. 17(a) by passing a curve through the smallred and blue squares while deleting the red andblue vertical bars to avoid clutter. We will hence-forth refer to this curve as the graph of χ1

N.

Let us examine two subpatterns of this graph.Subpattern 1 covers the “cyan” rectangular areabounded by φ 2 ∈ [0.5, 1.0] and χ1

2∈ [0, 0.2]. To

show that this subpattern can be used as a tem-plate, which upon rescaling by appropriate hori-zontal and vertical scaling factors, can reproduceexactly corresponding portions of the graph atarbitrarily small scales, we have enlarged it assubpattern 1© in Fig. 17(a), by a factor of 2 in bothhorizontal and vertical directions.

Let us observe next that the smaller “cyan”rectangle “2” bounded by φ 2 ∈ [0.25, 0.5] and χ1

2∈

[0.5, 0.6] can be enlarged by a horizontal scale = 22

and a vertical scale = 22 to obtain the subpattern2© which is identical to the “template” 1© directlyabove it.

Let us examine next the smaller cyan rect-angle “3” in subpattern 2© bounded by φ 2 ∈[0.28125, 0.3125] and χ1

2∈ [0.5625, 0.575], and

observe that it too coincides with the template 1©upon enlarging it horizontally by 25 and verticallyby 25.

The above process can be repeated any numberof times with appropriate choice of scaling factors,commensurate with the computer word length.

Repeating the above scaling process to thethree small cyan rectangles labeled “4”, “5”, and“6” in Fig. 17(b), we obtain the three subpatternsshown in 4©, 5©, and 6© of Fig. 17(b), each one isagain found to be identical to the original template1© in Fig. 17(a).

We conclude therefore that the graph of χ12

near the origin consists of infinitely many scaledinfinitesimal subpatterns of template 1©. In other

words, the graph of χ12

exhibits a fractal geometryin the sense that it is composed of infinitely manyself-similar subpatterns.

Example 3.2 [Characteristic function χ13]. The

graph of characteristic function χ13

replotted fromTable 2 is shown in Fig. 18. By rescaling the threecyan rectangles labeled “1”, “2”, and “3” by appro-priate scaling factors, we obtain the correspondingsubpatterns 1©, 2© and 3© in Fig. 18. Observe thatthese three subpatterns are all identical. Continuingthis process, we found the graph of χ1

3is composed

of infinitely many scaled copies of template 1©.

Example 3.3 [Characteristic function χ110

]. Thegraph of characteristic function χ1

10replotted from

Table 2 is shown in Fig. 19. Observe the three cyanareas labeled “1”, “2”, and “3” are identical afterappropriate rescaling, as shown in subpatterns 1©,2© and 3© in Fig. 19.



11is replotted

from Table 2 in Fig. 20. Observe the cyan arealabeled “1” and rescaled as template 1© containsinfinitely many scaled copies of itself, as illustratedin subpattern 2© in Fig. 20.



17replot-

ted from Table 2 is shown in Fig. 21. Thefractal geometry of χ1

17is obvious from the rescaled

patterns 1© and 2©.



110replotted from

Table 2 is shown in Fig. 22. The two subpatterns 1©and 2© reveal the fractal geometry of χ1

110.



124replotted from

Table 2 is shown in Fig. 23. The two subpatterns 1©and 2© illustrate the fractal geometry of χ1

124.



137replotted from


137.



193replotted from


193.

January 12, 2006 14:18 01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

2

2

0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

φ

χ1

2

2

0.25 0.3 0.35 0.4 0.45 0.50.5

0.55

0.6

φ

χ1

2

2

0.28125 0.2875 0.29375 0.3 0.30625 0.31250.5625

0.56875

0.575

φ

χ1

2

2

1

1

2

23

3

(a) subpattern 1©: horizontal scaling = 21,vertical scaling = 21

subpattern 2©: horizontal scaling = 22,vertical scaling = 22


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

2

2

0.125 0.15 0.175 0.2 0.225 0.250.25

0.275

0.3

φ

χ1

2

2

0.0625 0.075 0.0875 0.1 0.1125 0.1250.125

0.1375

0.15

φ

χ1

2

2

0.03125 0.0375 0.04375 0.05 0.05625 0.06250.0625

0.06875

0.075

φ

χ1

2

2

4

4

5

5

6

6

(b) subpattern 4©: horizontal scaling = 23,vertical scaling = 23



Fig. 17. Fractal compositions of χ12.

January 12, 2006 14:18 01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

3

3

0.375 0.4 0.425 0.45 0.475 0.50.5

0.525

0.55

φ

χ1

3

3

0.1875 0.2 0.2125 0.225 0.2375 0.250.75

0.7625

0.775

φ

χ1

3

3

0.09735 0.1 0.10625 0.1125 0.11875 0.1250.875

0.88125

0.8875

φ

χ1

3

3

1

1

2

2

3

3

Fig. 18. Fractal compositions of χ13.




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

10

10

0.1885 0.2 0.2125 0.225 0.2375 0.250.375

0.3875

0.4

φ

χ1

10

10

0.09375 0.1 0.10625 0.1125 0.11875 0.1250.1875

0.19375

0.2

φ

χ1

10

10

0.04685 0.05 0.05311 0.05524 0.05937 0.06250.09375

0.09688

0.01

φ

χ1

10

10

1

1

2

2

3

3

Fig. 19. Fractal compositions of χ110

.




January 12, 2006 14:18 01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

11

11

0.625 0.6375 0.65 0.6625 0.675 0.68750

0.01

0.02

0.03

0.04

0.05

0.06

φ

χ1

11

11

0.66409 0.66488 0.66566 0.66644 0.66722 0.6680

0.001

0.002

0.003

φ

χ1

11

11

1

1

2

2


.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

17

17

0.6875 0.7 0.7125 0.725 0.7375 0.750

0.01

0.02

0.03

0.04

0.05

0.06

φ

χ1

17

17

0.731 0.732 0.733 0.7340

0.001

0.002

0.003

φ

χ1

17

17

1

1

2

2


.



January 12, 2006 14:18 01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

110

110

0.5 0.525 0.55 0.575 0.6 0.6250.5

0.55

0.6

0.65

0.7

0.75

φ

χ1

110

110

0.5 0.503125 0.50625 0.509375 0.5125 0.5156250.5

0.50625

0.5125

0.51875

0.525

0.53125

φ

χ1

110

110

1

1

2

2


.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

124

124

0.5 0.525 0.55 0.575 0.6 0.6250.75

0.775

0.8

0.825

0.85

0.875

φ

χ1

124

124

0.5 0.503125 0.50625 0.509375 0.5125 0.5156250.75

0.753125

0.75625

0.759375

0.7625

0.765625

φ

χ1

124

124

1

1

2

2


.



January 12, 2006 14:18 01477


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

137

137

0.9375 0.95 0.9625 0.975 0.9875 10.875

0.9

0.925

0.95

0.975

1

φ

χ1

137

137

0.996094 0.998047 10.992188

0.99375

0.995313

0.996875

0.998438

1

φ

χ1

137

137

1

1 2

2


.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

193

193

0.4375 0.45 0.4625 0.475 0.4875 0.50.1875

0.2

0.2125

0.225

0.2375

0.25

φ

χ1

193

193

0.464844 0.466797 0.468750.199218

0.199999

0.200781

0.201562

0.202343

0.203124

φ

χ1

193

193

1

1

2

2


.



January 12, 2006 14:18 01477




30replotted from


30.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

30

30

0 0.01 0.02 0.03 0.04 0.05 0.060

0.02

0.04

0.06

0.08

0.1

0.12

φ

χ1

30

30

0 0.001 0.002 0.0030

0.001

0.002

0.003

0.004

0.005

0.006

φ

χ1

30

30

1

1

2

2


.





135replotted from


135.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

135

135

0 0.01 0.02 0.03 0.04 0.05 0.060.9375

0.95

0.9625

0.975

0.9875

1

φ

χ1

135

135

0 0.002 0.004 0.0060.992188

0.99375

0.995313

0.996875

0.998438

1

φ

χ1

135

135

1

12

2


.



January 12, 2006 14:18 01477




90replotted from


90.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

90

90

0 0.01 0.02 0.03 0.04 0.05 0.060

0.02

0.04

0.06

0.08

0.1

0.12

φ

χ1

90

90

0 0.001 0.002 0.0030

0.001

0.002

0.003

0.004

0.005

0.006

φ

χ1

90

90

1

1

2

2


.





150replotted from


150.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ

χ1

150

150

0 0.001 0.002 0.0030

0.001

0.002

0.003

0.004

0.005

0.006

φ

χ1

150

150

0 0.0001 0.00020

0.0001

0.0002

0.0003

0.0004

φ

χ1

150

150

1

1

2

2


.



January 12, 2006 14:18 01477


4. Predicting the Fractal Structures

The fractal structure of each characteristic func-tion χ1

Nin Table 2 can be analyzed and predicted

using the properties of characteristic functions tobe presented below, and from the coefficient b1 inthe explicit formulas from Table 1.

4.1. Two-level fractal stratifications

Let us partition the unit interval [0,1] into foursubintervals Φ1

∆= [0, 0.25), Φ2∆= [0.25, 0.5), Φ3

∆=[0.5, 0.75), and Φ4

∆= [0.75, 1.0]. It follows from thebinary-to-decimal conversion formula

{x0, x1, x2, . . . , xI} �→ φ =I∑

i=0

2−(i+1)xi (15)

that each binary string belonging to these foursubintervals must have the following form:

{0, 0, x2, x3, x4, . . . , xI} ∈ Φ1

{0, 1, x2, x3, x4, . . . , xI} ∈ Φ2

{1, 0, x2, x3, x4, . . . , xI} ∈ Φ3

{1, 1, x2, x3, x4, . . . , xI} ∈ Φ4

(16)

Let {y0, y1, y2, . . . , yI} denote the image of{x0, x1, x2, . . . , xI} under local rule N :

{x0, x1, x2, . . . , xI}T N�−→{y0, y1, y2, . . . , yI} (17)

where

{y0, y1, y2, . . . , yI} �→ χ1N (φ) (18)

and φ is the decimal representation in Eq. (15),namely,

χ1N

(φ) =I∑

i=0

2−(i+1)yi (19)

Observe that if we let the binary string{x0, x1, x2, . . . , xI} assume all binary combinationsfrom {0, 0, 0, . . . , 0} to {1, 1, 1, . . . , 1} in Eq. (17),and calculate the corresponding decimal valuefrom Eq. (19), we would obtain the coordinates(φN , χ1

N) for plotting the characteristic functions

in Table 2.Observe also that the first binary bit “y0” of

Eq. (17) is given by the first step function (i =0) in Table 1, as defined by Eq. (7). The valueof y0 is therefore determined by the three binary

bits

(x−1, x0, x1) = (xI , x0, x1) (20)

in view of the periodic boundary condition x−1 = xI

in Fig. 1(a). In other words, we have:

Property 4.1. The first binary bit y0 of χ1N

ofEq. (18) depends in general on the last binary bitxI of the input binary string of Eq. (15).

Since the first binary bit y0 of Eq. (18) con-tributes the largest component 2−1 = 0.5 (ify0 = 1), and since the last binary bit xI alter-nates between “0” and “1” as we increase φN fromφN = 0 to φN = 1 in Table 2, it follows that,depending on the formula in Table 1 for local ruleN , the characteristic function χ1

Nmay exhibit a

discontinuous jump in χ1N

equal to ∆χ1N

= 0.5.This implies that certain subintervals of χ1

Nmay

exhibit vertical jumps equal to 0.5 between adjacentred and blue bars in Table 2, resulting in a two-levelstratification of χ1

N.

In view of Property 4.1, the characteristic func-tion χ1

Nin Table 2 may exhibit a discontinuous

jump by an amount equal to ∆χ1N

= 0.5 overdifferent subintervals Φ1, Φ2, Φ3 and Φ4, respec-tively. An examination of the graph of each charac-teristic function χ1

Nin Table 2 shows that within

each subinterval Φi, one of the following groupingsof small red squares (resp. blue squares) on topof each thin red bar (resp. blue bar) apply to allsmall squares located within the same subintervalΦi, i = 1, 2, 3, 4 (this information is coded in colorred, blue or violet, in Table 5):

Group 1. All small red squares have χ1N

≥ 0.5and all small blue squares (in the case where thereare two-level stratifications) have χ1

N< 0.5. In this

case, we will paint the upper rectangle of N incolumn Φi of Table 5 in red, and the rectangle belowit will be painted blue.

Group 2. The small red and blue squares arelocated opposite to those of Group 1. In this case,the upper rectangle of Table 5 will be painted blue,and the lower rectangle will be painted red.

Group 3. There is no stratification and all smalladjacent red and blue squares have χ1

N≥ 0.5. In

this case, only the upper rectangle is painted in vio-let color,2 while the lower rectangle is left blank.

2We have chosen the violet color as the nearest approximation of the color combination between the red and blue colors.

January 12, 2006 14:18 01477


Table 5. Stratification of characteristic functions over subintervals Φ1 ∈ [0, 0.25), Φ2 ∈ [0.25, 0.5), Φ3 ∈ [0.5, 0.75), andΦ4 ∈ [0.75, 1].

30

Φ4Φ3Φ2Φ1N

0

1

2

3

4

5

31

29

28

27

26

25

24

23

22

21

20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

62

Φ4Φ3Φ2Φ1N

32

33

34

35

36

37

63

61

60

59

58

57

56

55

54

53

52

51

50

49

48

47

46

45

44

43

42

41

40

39

38

January 12, 2006 14:18 01477



94

Φ4Φ3Φ2Φ1N

64

65

66

67

68

69

95

93

92

91

90

89

88

87

86

85

84

83

82

81

80

79

78

77

76

75

74

73

72

71

70

126

Φ4Φ3Φ2Φ1N

96

97

98

99

100

101

127

125

124

123

122

121

120

119

118

117

116

115

114

113

112

111

110

109

108

107

106

105

104

103

102

January 12, 2006 14:18 01477



158

Φ4Φ3Φ2Φ1N

128

129

130

131

132

133

159

157

156

155

154

153

152

151

150

149

148

147

146

145

144

143

142

141

140

139

138

137

136

135

134

190

Φ4Φ3Φ2Φ1N

160

161

162

163

164

165

191

189

188

187

186

185

184

183

182

181

180

179

178

177

176

175

174

173

172

171

170

169

168

167

166

January 12, 2006 14:18 01477



222

Φ4Φ3Φ2Φ1N

192

193

194

195

196

197

223

221

220

219

218

217

216

215

214

213

212

211

210

209

208

207

206

205

204

203

202

201

200

199

198

254

Φ4Φ3Φ2Φ1N

224

225

226

227

228

229

255

253

252

251

250

249

248

247

246

245

244

243

242

241

240

239

238

237

236

235

234

233

232

231

230

January 12, 2006 14:18 01477


Group 4. The small red and blue squares arelocated opposite to those of Group 3. In this case,only the lower rectangle is painted in violet color,while the upper rectangle is left blank.

4.1.1. Stratification prediction procedure

The color painted in each of the two rectangleslocated in each subinterval Φi of the characteristicfunction χ1

Nof Table 5 is obtained by inspection

of the graph of χ1N

in Table 2. We will now showhow these colors can be predicted directly from the“firing patterns” [Chua et al., 2003] of each ruleN ; namely, from the “three-bit pattern” associ-ated with each red vertex of the Boolean cube rep-resenting N in Fig. 1(e).

Our goal is to determine the color of thefirst output bit y0 from these firing patterns. Inparticular, y0 depends on the three binary bits{x−1, x0 and, x1}, as shown in Fig. 30, where x−1

= xI in view of the periodic boundary conditionin Fig. 1(a). The “color” of x0 and x1 in Fig. 30is chosen from the subinterval Φi as defined by thefirst two columns of Eq. (16); namely,

Φi x0 x1

Φ1 0 0Φ2 0 1Φ3 1 0Φ4 1 1

The color of x−1 in Fig. 30 is chosen as follows:

1. x−1 = 0, if xI = 0

Using the bar-coloring scheme described inp. 1049 of [Chua et al., 2005], this corresponds to

1x− 0x 1x

0y

NT

Fig. 30. The left-most output bit y0 of χ1N

is “1” if

{x−1, x0, x1} = {xI , x0, x1} is a firing pattern of N .Otherwise, y0 = 0.

the case where the right-most bar of χ1N

in Table 2is a red bar.

2. x−1 = 1, if xI = 1

This corresponds to the case where the right-most bar of χ1

Nin Table 2 is a blue bar.

Since both x−1 and y0 can assume a “0” or a“1”, there are four possible outcomes:

Outcome 1 : x−1 = 0 , y0 = 0

In this case, all bars belonging to subinterval Φi arecolored red and lie below χ1

N= 0.5.

Outcome 2 : x−1 = 0 , y0 = 1

In this case, all bars belonging to subinterval Φi arecolored red and lie above χ1

N= 0.5.

Outcome 3 : x−1 = 1 , y0 = 0

In this case, all bars belonging to subinterval Φi arecolored blue and lie below χ1

N= 0.5.

Outcome 4 : x−1 = 1 , y0 = 1

In this case, all bars belonging to subinterval Φi arecolored blue and lie above χ1

N= 0.5.

4.1.2. Examples illustrating stratificationprediction procedure

Let us illustrate the above “stratification predic-tion” procedure with some examples.

Example 4.1 [Predicting Stratification for χ116

].The Boolean cube for rule 16 is reproduced fromTable 1 of [Chua et al., 2003] in Fig. 31(a). Sinceonly vertex 4© is painted red in this Boolean cube,there is only one firing pattern → 1 00 , as shown in Fig. 31(b).

For each subinterval Φi, i = 1, 2, 3, 4, we insertthe corresponding x0 and x1 as specified in thepreceding table. We then assign x−1 = 0 (for redbars) on the left column and x−1 = 1 (for blue bars)on the right column for each Φi. The color of y0 ineach case is then determined from the firing pat-terns in Fig. 31(b). In this case, y0 = 1 only in (row1, column 2). Since y0 = 0 in the left column of Φ1

and y0 = 1 in the right column of Φ1, it follows thatχ1

16has two stratifications over the subinterval Φ1,

where the red bars are below χ116

= 0.5 and the bluebars are above χ1

16= 0.5.

January 12, 2006 14:18 01477


16T

Firing Pattern

Φ4

Φ3

Φ2

Φ1

1x− 0x 1x

0 0 0

00y

1x− 0x 1x

0 0 1

00y

1x− 0x 1x

0 1 0

0y

1x− 0x 1x

0 1 1

00y

1x− 0x 1x

1 0 0

10y

1x− 0x 1x

1 0 1

00y

1x− 0x 1x

1 1 0

0y

1x− 0x 1x

1 1 1

00y

162 3

1

5

0

4

6 7

0 0

16T

16T

16T

16T

16T

16T

16T

16T

Fig. 31. Predicting stratification for rule 16 . (a) Boolean cube for 16 . (b) Firing patterns for 16 . (c) Stratificationdetermination data.

January 12, 2006 14:18 01477


Since y0 = 0 for all other cases in Fig. 31(c),it follows that there are no stratifications insubinterval Φ2, Φ3 and Φ4, and all red and bluebars in these three subintervals are located belowχ1

16= 0.5, as is indeed the case in Table 2.

Example 4.2 [Predicting Stratification for χ12].

The Boolean cube for rule 2 is shown in Fig. 32(a).It has only one firing pattern → 00 1 , as shown in Fig. 32(b). The correspond-ing stratification determination data is shown inFig. 32(c). From these data, we conclude thatonly subinterval Φ2 has a stratification in χ1

2

where all red bars lie above, and all blue barslie below χ1

2= 0.5. Since y0 = 0 in the other

three subintervals Φ1, Φ3 and Φ4, all red andblue bars must lie below χ1

2= 0.5. Observe

that this prediction is consistent with Table 2, asexpected.

Example 4.3 [Predicting Stratification for χ14].

The Boolean cube for rule 4 is shown in Fig. 33(a).It has only one firing pattern → 0 1 0 ,as shown in Fig. 33(b). In this case, only subinter-val Φ3 in Fig. 33(c) has a stratification in χ1

4where

all red bars must lie above χ14

= 0.5, and all bluebars must lie below χ1

4= 0.5. All red and blue bars

in the other three subintervals Φ1, Φ2 and Φ4 must

lie below χ14

= 0.5, because y0 = 0. Again, thisprediction is consistent with Table 2.

Example 4.4 [Predicting Stratification for χ1110

].The Boolean cube for rule 110 is shown inFig. 34(a). It has five firing patterns, →0 0 1 , → 0 1 0 , →0 1 1 , → 1 0 1 , → 11 0 , as shown in Fig. 34(b). The stratificationdetermination data in Fig. 34(c) shows only subin-terval Φ4 has a stratification where all red bars mustlie above χ1

110= 0.5, and all blue bars must lie

below it.Since y0 = 0 in both columns of Φ1, it follows

that all red and blue bars must lie below χ1110

= 0.5.On the other hand, since y0 = 1 in both

columns of Φ2 and Φ3, it follows that all red andblue bars in subintervals Φ2 and Φ3 must lie aboveχ1

110= 0.5. All of these predictions are consistent

with Table 2, as expected.

4.1.3. {Φ1,Φ2,Φ3,Φ4} — stratified families

If we examine the color of the rectangles of the foursubintervals Φ1, Φ2, Φ3 and Φ4 in Table 5 and codeeach violet rectangle by a “0” binary bit, and allother rectangles by a “1” binary bit, we will dis-cover the first 16 rules follow the four-bit binarynumber system as shown below, henceforth calledthe stratified family “0”.

N ψ 1Φ 2Φ 3Φ 4Φ EquivalentDecimal Number

0 0ψ 0 0 0 0 0

1

2

3

4

5

6

7

8

9

10

11ψ 1 0 0

22ψ 0 1 0

33ψ 1 1 0

44ψ 0 0 1

55ψ 1 0 1

66ψ 0 1 1

77ψ 1 1 1

88ψ 0 0 0 1

99ψ 1 0 0 1

1010ψ 0 1 0 1

1111ψ 1 1 0 1 11

1212ψ 0 0 1 1 12

1313ψ 1 0 1 1 13

1414ψ 0 1 1 1 14

Stratified

Family

“0”

1515ψ 1 1 1 1 15

0

0

0

0

0

0

0

Observe that the color background of Table 1 is determined from this coding scheme, where “0” iscoded “light blue”, while “1” is coded “light pink” in Table 1.

January 12, 2006 14:18 01477


2T

Firing Pattern

Φ4

Φ3

Φ2

Φ1

1x− 0x 1x

0 0 0

00y

1x− 0x 1x

0 0 1

10y

1x− 0x 1x

0 1 0

0y

1x− 0x 1x

0 1 1

00y

1x− 0x 1x

1 0 0

00y

1x− 0x 1x

1 0 1

00y

1x− 0x 1x

1 1 0

0y

1x− 0x 1x

1 1 1

00y

0 0

2T

2T

2T

2T

2T

2T

2T

2T

22 3

1

5

0

4

6 7

Fig. 32. Predicting stratification for rule 2 . (a) Boolean cube for 2 . (b) Firing patterns for 2 . (c) Stratificationdetermination data .

January 12, 2006 14:18 01477


4T

Firing Pattern

Φ4

Φ3

Φ2

Φ1

1x− 0x 1x

0 0 0

00y

1x− 0x 1x

0 0 1

00y

1x− 0x 1x

0 1 0

0y

1x− 0x 1x

0 1 1

00y

1x− 0x 1x

1 0 0

00y

1x− 0x 1x

1 0 1

00y

1x− 0x 1x

1 1 0

0y

1x− 0x 1x

1 1 1

00y

1 0

4T

4T

4T

4T

4T

4T

4T

4T

42 3

1

5

0

4

6 7


January 12, 2006 14:18 01477


2 3

1

5

0

4

6 7

110

Φ4

Φ3

Φ2

Φ1

1x− 0x 1x

0 0 0110

T

00y

1x− 0x 1x

0 0 1110

T

10y

1x− 0x 1x

0 1 0110

T

10y

1x− 0x 1x

0 1 1110

T

10y

1x− 0x 1x

1 0 0110

T

00y

1x− 0x 1x

1 0 1110

T

10y

1x− 0x 1x

1 1 0110

T

10y

1x− 0x 1x

1 1 1110

T

00y

110T

110T

110T

110T

Firing Patterns

110T


January 12, 2006 14:18 01477


Observe that no two members of the stratifiedfamily “0” have identical four-bit binary patterns.Since there are only 16 distinct combinations of fourbinary bits, the stratified family “0” had alreadyconsumed all of them and it is clear that the remain-ing rules in Table 5 must consist of repetitions

of these patterns. A careful analysis of the next16 rules reveals, however, that the four-bit binarypatterns of the next 16 rules {16, 17, . . . , 31} aremere permutations of the stratified family “0”,as illustrated below (henceforth called stratifiedfamily “1”):

N ψ 1Φ 2Φ 3Φ 4Φ EquivalentDecimal Number

16 0ψ 1 0 0

171ψ 0 0 0

182ψ 1 1 0

193ψ 0 1 0

204ψ 1 0 1

215ψ 0 0 1

226ψ 1 1 1

237ψ 0 1 1

248ψ 1 0 0

259ψ 0 0 0

2610ψ 1 1 0

2711ψ 0 1 0

2812ψ 1 0 1

2913ψ 0 0 1

3014ψ 1 1 1

Stratified

Family

“1”

3115ψ 0 1 1

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

0

3

2

5

4

7

6

9

8

11

10

13

12

15

14

Observe that the 4-bit code ψ = {Φ1,Φ2,Φ3,Φ4} of stratified family “1” and stratified family “0” arerelated by a 16 × 16 permutation matrix; namely,

ψ0(1)ψ1(1)ψ2(1)ψ3(1)ψ4(1)ψ5(1)ψ6(1)ψ7(1)ψ8(1)ψ9(1)ψ10(1)ψ11(1)ψ12(1)ψ13(1)ψ14(1)ψ15(1)

=

0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0

︸︷︷︸M1,0

ψ0(0)ψ1(0)ψ2(0)ψ3(0)ψ4(0)ψ5(0)ψ6(0)ψ7(0)ψ8(0)ψ9(0)ψ10(0)ψ11(0)ψ12(0)ψ13(0)ψ14(0)ψ15(0)

January 12, 2006 14:18 01477


It turns out that the remaining rules can bepartitioned in a similar way into a stratified family“2”, “3”, . . . ,“15”. Together, these 16 families corre-spond exactly to the 16 rules listed in the 16 pages ofTable 1, respectively. Let us summarize this remark-able organization as follows:

Property 4.2. The 16 rules listed in each of the16 pages of Table 1 form a stratified family whosefour-bit binary patterns ψ(k), k = 0, 1, 2, . . . , 15, arerelated to each other by a 16×16 permutation matrix

ψ(k) = Mk,l ψ(l)

Table 6 lists 16 matrices which transform ψ(0) fromstratified family “0” into ψ(k) of stratified family“k”, k = 0, 1, 2, . . . , 15.

The four-bit binary patterns of the last eightrules of each stratified family “k” can be obtainedby copying the four-bit binary patterns of the firsteight rules and then complementing the fourth bitcorresponding to Φ4. Observe that, in Table 5, thefour color codes of all Φi’s of each local rule N isequivalent to the vertically-switched color codes ofall Φi’s of the corresponding local rule 255 − N asillustrated below:

,

For example, the color code of each Φi of 0 ,24 , 120 and 220 in Table 5 can be obtainedby switching vertically the color code of the

corresponding Φi of 255 , 231 , 135 and 35 ,respectively.

4.2. Rules having no fractalstratifications

Since the occurrence of two-level stratificationscomes from the term b1x−1 [i = 0 in Eq. (7)] inTable 1, it follows that no such stratification canoccur in χ1

Nif b1 = 0 for rule N . An examina-

tion of Table 2 shows that there are only 16 ruleswith b1 = 0. These 16 xI -insensitive rules are listedin Table 7. Observe that these 16 rules can be gen-erated from the following formula:

N = 17β , β = 0, 1, 2, . . . , 15

This relationship can be derived from Table 6. Inparticular, since the right-most bit xI -insensitiverules corresponds to ψi = {Φ1,Φ2,Φ3,Φ4} ={0, 0, 0, 0}, it follows from Tables 1 and 6 thatψi = {0, 0, 0, 0} is repeated after integer multipliesof 17, and hence all xI -insensitive rules N mustsatisfy N = 17k, k = 0, 1, 2, . . . , 15.

These 16 rules are composed of the correspond-ing (k + 1)th members in the stratified family “k”,k = 0, 1, 2, . . . , 15. For example, 0 is the first mem-ber of the stratified family “0”, 17 is the secondmember of the stratified family “1”, . . . , and 225 isthe sixteenth member of the stratified family “15”.

Property 4.3. The following 16 characteristic func-tions have no stratifications:

Single-Level Fractalsχ1

0, χ1

17, χ1

34, χ1

51χ1

68, χ1

85, χ1

102, χ1

119,

χ1136

, χ1153

, χ1170

, χ1187

, χ1204

, χ1221

, χ1238

, and χ1255

4.3. Origin of the fractal structures

We will now uncover a basic mechanism responsi-ble for the presence of fractals in the characteris-tic functions shown in Table 2. Let us examine aninfinitesimally small neighborhood of the origin ofχ1

Nby considering the input string

0, 0, . . . , 0︸︷︷︸first k entries

, xk, xk+1, . . . , xI

�→ φ

=I∑

i=0

2−(i+1)xi (21)

If we multiply φ by 2k in Eq. (21), we would obtain

φ • 2k =I∑

i=k

2−(i+1)+kxi =I−k∑j=0

2−(j+1)xj (22)

where j � (i−k). Observe that, as I → ∞, Eq. (22)converges to the binary string in Eq. (15). It fol-lows that no matter how close the binary stringin Eq. (21) is to the origin, we can always rescaleit to recover the original string in Eq. (15). More-over, except possibly for the (k + 1)th term xk = 1,all other terms xk+1, xk+2, . . . , xI in Eq. (21) will

January 12, 2006 14:18 01477


Table 6. List of 16 permutation matrices Mk,l which transform ψ(0) from stratified family “0” into ψ(k) of stratified family“k”, k = 0, 1, 2, . . . , 15.

January 12, 2006 14:18 01477



January 12, 2006 14:18 01477


Table 7. List of 16 right-most bit xI -insensitive local rules.

255238221204

187170153136

1191028568

5134170

generate the same corresponding output binary bityi, i = k + 1, k + 2, . . . , I, because the same localrule N is used. Hence, by rescaling χ1

Naccord-

ingly, we can recover the corresponding originalgraph. To determine the appropriate scaling factor,it is necessary to examine the output binary bit yi,i = 0, 1, 2, . . . , k − 1 because it may not be zero fortwo reasons:

(i) yk−1 = 1 if

(xk−2, xk−1, xk) = (0, 0, 1)T N�−→ 1 (23)

In other words, if is a firing pattern [Chuaet al., 2003] of Rule N , or equivalently, if vertex1© in the Boolean cube in Fig. 1(e) is painted in redcolor.

(ii) yi = 1 if

(xi−1, xi, xi+1) = (0, 0, 0)T N�−→ 1 (24)

for i = 0, 1, 2, . . . , k − 2. In other words, ifis a firing pattern of Rule N , or

equivalently, if vertex 0© in the Boolean cube inFig. 1(e) is painted in red color. Let us examinenow the consequences of the occurrence of one orboth of these two cases. There are four possiblescenarios:

Scenario 1. (0, 0, 0) is not a firing pattern3 but(0, 0, 1) is a firing pattern of N , and xI = 0.

An example satisfying scenario 1 is Rule 2 ,where vertex 0© is blue but vertex 1© is red. In thiscase, yk−1 = 1 and the output binary pattern has

the form0, 0, . . . , 0, yk−1︸︷︷︸

first k terms

, yk, yk+1, . . . , yI

�→ χ1

N

=I∑

i=k−1

2−(i+1)yi (25)

Multiplying the right side of Eq. (25) by 2(k−1) weobtain

χ1N (2k−1) =

I∑i=k−1

2−(i+1)+(k−1)yi

=I−k+1∑

j=0

2−(j+1)yj

=I∑

i=0

2−(i+1)yi, as I → ∞ (26)

Hence, by multiplying the output string in Eq. (25)by the factor 2(k−1), we obtain Eq. (26), which isidentical to Eq. (18) as I → ∞. In other words,in Scenario 1, the characteristic function χ1

Narbi-

trarily near the origin can be rescaled to obtain thecorresponding original graph by multiplying φN by2k and χ1

Nby 2(k−1).

Scenario 2. (0, 0, 0) is not a firing pattern but(1, 0, 0) is a firing pattern of N , and xI = 1 withb1 �= 0.

In this case, y0 = 1 in view of Eq. (19), and theoutput binary string assumes the form:

{1, 0, 0, . . . , 0, yk−1, yk, yk+1, . . . , yI}

�→ 12

+I∑

i=k−1

2−(i+1)yi (27)

It follows from Eq. (27) that

χ̃1N � χ1

N − 12

=I∑

i=k−1

2−(i+1)yi

=I∑

i=0

2−(i+1)yi, as I → ∞ (28)

as in Eq. (26). Hence, apart from a translation by∆χ1

N= 1/2, the infinitesimal pattern near the

origin can be rescaled to coincide with the corre-sponding original pattern.

3An examination of the Boolean cubes in Table 1 of [Chua et al., 2003] shows (0, 0, 0) is not a firing pattern for rules with aneven number: i.e. N = 2n, n = 0, 1, 2, . . . , 128.

January 12, 2006 14:18 01477


Scenario 3. (0, 0, 0) is a firing pattern of N , andxI = 0.

In this case, yi, i = 0, 1, 2, . . . , k − 2; namely, 1, 1, . . . , 1,︸︷︷︸

first k−1 terms

yk−1, yk, yk+1, . . . , yI

�→ χ1

N (29)

where

χ1N =

k−2∑i=0

2−(i+1) +I∑

i=k−1

2−(i+1)yi

= 1 − 2−(k−1) +I∑

i=k−1

2−(i+1)yi (30)


χ̃1N � 1 − χ1

N = 2−(k−1) −I∑

i=k−1

2−(i+1)yi (31)

Multiplying both sides of Eq. (31) by the scalingfactor 2(k−1), we obtain

χ̃1N • 2(k−1) = 1 −

I∑i=k−1

2−(i+1)+(k−1)yi

= 1 −I∑

i=0

2−(i+1)yi, as I → ∞ (32)

as in Eq. (26).Since Eq. (32) is an affine transformation of

Eq. (26), it follows that the fractal pattern arbitrar-ily near the origin is preserved upon scaling, apartfrom an affine transformation.

Scenario 4. (0, 0, 0) is a firing pattern of N , but(1, 0, 0) is not a firing pattern of N , and xI = 1with b1 �= 0.

In this case, y0 = 0, in view of Eq. (18); namely,0, 1, 1, . . . , 1, yk−1︸︷︷︸

first k terms

, yk, yk+1, . . . , yI

�→ χ1

N (33)

where

χ1N =

I∑i=1

2−(i+1)yi

=k−2∑i=1

2−(i+1) +I∑

i=k−1

2−(i+1)yi

=12− 2−(k−1) +

I∑i=k−1

2−(i+1)yi (34)


χ̃1N �

(1 − χ1

N

)− 1

2

= 2−(k−1) −I∑

i=k−1

2−(i+1)yi

= 2−(k−1) −I∑

i=0

2−(i+1)yi, as I → ∞ (35)

as in Eq. (26).Since Eq. (34) is an affine transformation of

Eq. (26), it follows that the fractal pattern arbitrar-ily near the origin is preserved upon scaling, apartfrom an affine transformation.

5. Gardens of Eden

A cursory inspection of the graphs of the character-istic functions χ1

Ndisplayed in Table 2 reveals that

most of these graphs exhibit a discontinuous jumpover a finite range ∆χ1

N= χ1

N(φ+

jump)−χ1N

(φ−jump)

at various discrete points φjump ∈ [0, 1] where allpoints within the interval (χ1

N(φ+

jump),χ1N

(φ−jump))

have no preimage under χ1N

. Such a characteristicfunction is therefore not a surjective (onto) func-tion. Points within such intervals are special casesof the following interesting class of initial bit-stringconfigurations.

Definition 5.1. Garden of Eden of N . An (I +1)-bit binary string {x0, x1, x2, . . . , xI} is saidto be a garden of Eden of a local rule N iff itdoes not have a predecessor under the local ruletransformation TN .

It follows from Definition 5.1 that a garden ofEden φ0 �

∑Ii=0 2−(i+1)xi of N can never occur as

a point on an orbit of N arising from some initialbit-string configuration whose decimal equivalent isdifferent from φ0. Hence, a garden of Eden has nopast, but only present and future.4

4The name garden of Eden was first introduced by Moore in the context of von Neumann’s self-reproducing automata [Moore,1962].

January 12, 2006 14:18 01477


Table 8. A compendium of all gardens of Eden (colored in green) over the range [0, 1] of χ1N

.

χ =10

χ =11

χ =12

χ =13

χ =14

χ =15

χ =16

χ =17

χ =18

χ =19

χ =110

χ =111

χ =112

χ =113

χ =114

χ =115

χ =116

χ =117

χ =118

χ =119

χ =120

χ =121

χ =122

χ =123

χ =124

χ =125

χ =126

χ =127

χ =128

χ =129

χ =130

χ =131

χ =132

χ =133

χ =134

χ =135

χ =136

χ =137

χ =138

χ =139

χ =140

χ =141

χ =142

χ =143

χ =144

χ =145

χ =146

χ =147

χ =148

χ =149

χ =150

χ =151

χ =152

χ =153

χ =154

χ =155

χ =156

χ =157

χ =158

χ =159

χ =160

χ =161

χ =162

χ =163

January 12, 2006 14:18 01477



χ =164

χ =165

χ =166

χ =167

χ =168

χ =169

χ =170

χ =171

χ =172

χ =173

χ =174

χ =175

χ =176

χ =177

χ =178

χ =179

χ =180

χ =181

χ =182

χ =183

χ =184

χ =185

χ =186

χ =187

χ =188

χ =189

χ =190

χ =191

χ =192

χ =193

χ =194

χ =195

χ =196

χ =197

χ =198

χ =199

χ =1100

χ =1101

χ =1102

χ =1103

χ =1104

χ =1105

χ =1106

χ =1107

χ =1108

χ =1109

χ =1110

χ =1111

χ =1112

χ =1113

χ =1114

χ =1115

χ =1116

χ =1117

χ =1118

χ =1119

χ =1120

χ =1121

χ =1122

χ =1123

χ =1124

χ =1125

χ =1126

χ =1127

January 12, 2006 14:18 01477



χ =1128

χ =1129

χ =1130

χ =1131

χ =1132

χ =1133

χ =1134

χ =1135

χ =1136

χ =1137

χ =1138

χ =1139

χ =1140

χ =1141

χ =1142

χ =1143

χ =1144

χ =1145

χ =1146

χ =1147

χ =1148

χ =1149

χ =1150

χ =1151

χ =1152

χ =1153

χ =1154

χ =1155

χ =1156

χ =1157

χ =1158

χ =1159

χ =1160

χ =1161

χ =1162

χ =1163

χ =1164

χ =1165

χ =1166

χ =1167

χ =1168

χ =1169

χ =1170

χ =1171

χ =1172

χ =1173

χ =1174

χ =1175

χ =1176

χ =1177

χ =1178

χ =1179

χ =1180

χ =1181

χ =1182

χ =1183

χ =1184

χ =1185

χ =1186

χ =1187

χ =1188

χ =1189

χ =1190

χ =1191

January 12, 2006 14:18 01477



χ =1192

χ =1193

χ =1194

χ =1195

χ =1196

χ =1197

χ =1198

χ =1199

χ =1200

χ =1201

χ =1202

χ =1203

χ =1204

χ =1205

χ =1206

χ =1207

χ =1208

χ =1209

χ =1210

χ =1211

χ =1212

χ =1213

χ =1214

χ =1215

χ =1216

χ =1217

χ =1218

χ =1219

χ =1220

χ =1221

χ =1222

χ =1223

χ =1224

χ =1225

χ =1226

χ =1227

χ =1228

χ =1229

χ =1230

χ =1231

χ =1232

χ =1233

χ =1234

χ =1235

χ =1236

χ =1237

χ =1238

χ =1239

χ =1240

χ =1241

χ =1242

χ =1243

χ =1244

χ =1245

χ =1246

χ =1247

χ =1248

χ =1249

χ =1250

χ =1251

χ =1252

χ =1253

χ =1254

χ =1255

January 12, 2006 14:18 01477


Table 9. A compendium of period-n isles of Eden of N .

Table 9-1(a). Period-1 isles of Eden.

* Here I +1 must be divisible by 3.

134 140 142 148 150 156 158

162 166 170 172 174 176 180

182 184 186 188 190 196 198

204 206 212 214 220 222 226

228 230 234 236 238 240 242

244 246 248 250 252 254

128 130 132 134 136 138 140

142 144 146 148 150 152 154

156 158 160 162 168 170 176

184 186 192 194 196 198 200

202 204 206 208 210 212 214

216 220 224 226 240 242

4 6 7 12 14 15 20

21 22 23 30 31 68 84

85 86 87 132 134 135 140

142 143 148 149 150 151 158

159 196 204 206 207 212 213

214 215 220 221 222 223

72 73 76 104 105 108 109

200 201 204 205 233 236 237

72 76 200 204 236

200 204 205 236 237

45 101 173 204 205 229

74 75 88 89 204

3

8*

7*

6

5

4

2

1

Rules Endowed with Isle of EdenPeriod-1 Isle of Edenn

January 12, 2006 14:18 01477


Table 9-1(b). Additional period-1 isles of Eden.

any combination ofwith red string for any I :

any combination ofwith blue string for any I :

4 204

4 204

4 204

204 223

204 223

204 223

200 204

200 204

200 204

204 236

204 236

204 236

204

any combination ofwith blue string for any I :

any combination ofwith red string for any I :

9

10

11

12

13

14

15

16

17

18

19

any pattern and any I21

20


January 12, 2006 14:18 01477


Table 9-2. Period-2 isles of Eden.

3

any pattern and any I4

2

1


13 15 27 29 35 41 43

49 51 57 59 69 71 77

79 85 93 97 99 105 107

113 115 121

32 33 34 35 40 41 42

43 48 49 51 59 96 97

104 105 106 107 112 113 115

120 121 123 168 169 170 171

187 224 225 232 233 234 235

240 241 243 248 249 251

18 19 22 23 50 51 54

55 146 147 150 151 178 179

182 182

51

For n = 1, Period-2 Isles of Eden can exist for any I.

For n = 2, Period-2 Isles of Eden can exist for any I + 1 that is divisible by 2.

For n = 3, Period-2 Isles of Eden can exist for any I + 1 that is divisible by 4.

Remarks :

January 12, 2006 14:18 01477



3

4

2

1


Period-3 Isles of Eden can exist for any I + 1 that is divisible by 3.

2 3 42 43 75 99 106

107 130 131 170 171 202 203

226 227 234 235

16 17 57 89 112 113 120

121 144 145 184 185 216 217

240 241 248 249

40 41 42 43 44 45 56

57 62 63 168 169 170 171

172 184 190 191

96 97 98 99 100 101 112

113 118 119 224 225 226 228

240 241 246 247

January 12, 2006 14:18 01477



3

4

2

1



10 15 170 175

80 85 240 245

15

85

January 12, 2006 14:18 01477



6

159

215

20

82 83 87 114 115 119

26 27 31 58 58 63

3 7 35 39 163 167

5

4

6

7

3

2

1


January 12, 2006 14:18 01477



113 121 185 240 241 249

43 107 170 171 227 235

96 97 98 112 113 240

17 21 49 53 117 181

40 41 42 43 56 170

15 43

85 113

12

11

13

14

10

9

8


January 12, 2006 14:18 01477



any pattern at I = 53

2

1


14 15 142 143

84 85 212 213

170 240


January 12, 2006 14:18 01477



3

4

2

1



40 41 42 43 169 170

96 97 112 113 225 240

43 106 107 170 171 235

113 120 121 240 241 249

January 12, 2006 14:18 01477



3

4

2

1


40 41 42 43 56 169 170

96 97 98 112 113 225 240

43 106 107 170 171 227 235

113 120 121 185 240 241 249

January 12, 2006 14:18 01477




14 15 142 143

84 85 212 2136

5


January 12, 2006 14:18 01477



40 41 42 43 169 170


96 97 112 113 225 240

43 106 107 170 171 235

113 120 121 240 241 249

3

4

2

1


January 12, 2006 14:18 01477



40 41 42 43 169 170

96 97 112 113 225 240

43 106 107 170 171 2353

2

1


January 12, 2006 14:18 01477



6

5

4


113 120 121 240 241 249


14 15 142 143

84 85 212 213

January 12, 2006 14:18 01477



40 41 42 43 169 170

96 97 112 113 225 240

43 106 107 170 171 235

3

2

1


January 12, 2006 14:18 01477



113 120 121 240 241 249

40 41 42 43 169 170

96 97 112 113 225 240

6

5

4


January 12, 2006 14:18 01477


Table 9-11(c). Additional period-11 isles of Eden.

43 106 107 170 171 235

113 120 121 240 241 249


8

7


Observe that any binary string

{x0, x1, x2, . . . , xI} �→ φ0 =I∑

i=0

2−(i+1)xi (36)

which has no preimage under χ1N

(i.e. φ0 does notbelong to the range of χ1

N) is a garden of Eden of

N . In other words, φ0 ∈ [0, 1] is a garden of Edenof N if there does not exist a φ−1 ∈ [0, 1] suchthat φ0 = χ1

N(φ−1), where φ−1 �= φ0. Note that

this property is only a sufficient condition for φ0

to be a garden of Eden. In the next section, we willsee that there exist some rather special points which

violate this property, but is nevertheless a garden ofEden because it satisfies Definition 5.1.

A compendium of all gardens of Eden belongingto each local rule N is listed in Table 8. Each pointφ0 ∈ [0, 1] printed in green along the χ1

Naxis does

not have a preimage under χ1N

and is therefore agarden of Eden.

6. Isle of Eden

For most rules N , there exist some specialperiod-1 points which have no predecessors in thesense that no orbits from other initial bit-string

January 12, 2006 14:18 01477


Table 10. Rules without period-k isles of Eden, k < 12.

255253239231219218211209

199197195193189165164161

157155153141139137133129

127126125124122117116111

1101031029594929190

8178706766656461

6052474638373628

25241198510

configurations can converge to such points, andhence they are also gardens of Eden in view ofDefinition 5.1. Moreover, since each of these pointsis a period-1 point, they have a preimage underχ1

N; namely, itself. Such special points (obtained

by exhaustive computer search) are henceforthcalled isles of Eden of N . Observe that an isle ofEden has no past, and no future (in the poetic sensethat time stood still)!

A compendium of all period-1 isles of Eden ofN for binary strings of various length I +1 is listedin Tables 9-1(a) and 9-1(b).

Let us now generalize the concept ofperiod-1 isle of Eden for “time-n” characteristicfunctions χn

N. Since such special period-n points

also have no predecessors under the “nth iterated”map χn

N, they are henceforth called period-n isles

of Eden. Table 9-k lists all period-k isles of Eden,k = 1, 2, 3, . . . , 11, obtained by an exhaustive com-puter search over all binary bit-strings of lengthI + 1 = 8, 9, 10, 11, respectively. In most cases, thesame bit-string configurations can be extended toarbitrarily large I, provided I + 1 is divisible bysome integer specified in the table.

Finally, we remark that based on exhaustivecomputer search, there are 64 local rules that donot have period-k isles of Eden, at least for k < 12.They are listed in Table 10.

7. Concluding Remarks

We have demonstrated that except for the eightaffine (mod 1 ) rules listed in Table 3, the graphof all characteristic functions χ1

Nin Table 2 are

endowed with a fractal structure. Indeed, eventhe graph of the eight affine (mod 1) rules canbe considered to exhibit a degenerate form of frac-tal structure since arbitrarily short segments of the

graph can be made to coincide with correspondingportions of the original graphs by appropriate affinetransformations. We have traced the origin of thesefractals to the decimal representation of binary bit-strings in Eq. (2), as well as to the local rule, whichmust apply to any bit string, regardless of the num-ber of “zeros” in front of it.

The explicit formulas in Table 1 for convert-ing any bit string {x0, x1, x2, . . . , xI} into a realnumber χ1

N∈ [0, 1] is remarkable for its scope of

potential applications. Indeed, these formulas canbe interpreted as a digital-to-analog converter inclosed form. Such formulas could not have beenderived without the explicit “universal” formuladerived in [Chua et al., 2003] for all local rules.

The widespread presence of period-k “isles ofEden” came as a surprise since they certainly haveno counter part in hyperbolic differential equations.To dramatize this phenomenon, we end Part V withour following poetic interpretation of the above newphenomenon: Hidden within the “garden of Eden”,which has no past, one finds immortality in an “isleof Eden”, which has neither past nor future.

Acknowledgments

This paper is supported in part by the MURIcontract no. N00014-03-1-0698, the DURINTcontract no. N00014-01-0741, the MARCO Micro-electronics Advances Research Corporation FENAAward no. 442521/WK57015, the NSF grant CHE-0103447, the UNIVERSITE du SUD, TOULON-VAR, France, the Foundations Francqui Fonds,Belgium, and the ministry of university researchand education, Italy, under the field project numberRBNE012NFW.

References

Barnsley, M. F. [1988] Fractals Everywhere (AcademicPress).

Billingsley, P. [1978] Ergodic Theory and Information(Robert Keirger Publishing Company, Huntington).

Chua, L. O., Yoon, S. & Dogaru, R. [2002] “A nonlineardynamics perspective of Wolfram’s new kind of sci-ence. Part I: Threshold of complexity,” Int. J. Bifur-cation and Chaos 12, 2655–2766.

Chua, L. O., Sbitnev, V. I. & Yoon, S. [2003] “A non-linear dynamics perspective of Wolfram’s new kind ofscience. Part II: Universal neuron,” Int. J. Bifurcationand Chaos 13, 2377–2491.

Chua, L. O., Sbitnev, V. I. & Yoon, S. [2004] “A non-linear dynamics perspective of Wolfram’s new kind of

January 12, 2006 14:18 01477


science. Part III: Predicting the unpredictable,” Int.J. Bifurcation and Chaos 14, 3689–3820.

Chua, L. O., Sbitnev, V. I. & Yoon, S. [2005] “A non-linear dynamics perspective of Wolfram’s new kindof science. Part IV: From Bernoulli shift to 1/f spec-trum,” Int. J. Bifurcation and Chaos 15, 1045–1183.

Devaney, R. L. [1992] A First Course in ChaoticDynamic Systems: Theory and Experiment (Addison-Wesley, Reading, MA).

Moore, E. F. [1962] “Machine models of self-reproduction,” Proc. Fourteenth Symp. Applied Math-ematics, pp. 17–33.

a nonlinear dynamics perspective of … › ... › part5_vol15_no12_2005.pdfpart v: fractals...

Documents