A Nonlinear Dynamics Perspective

A NONlINEAR DYNAMICS PERSPECTIVE OF HOlFRAM'S

NEW HlNU OF SCIENCE Volume I1

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE

Editor: Leon O. ChuaUniversity of California, Berkeley

Series A. MONOGRAPHS AND TREATISES

Volume 38: Nonlinear Noninteger Order Circuits & Systems An IntroductionP. Arena, R. Caponetto, L. Fortuna & D. Porto

Volume 39: The Chaos Avant-Garde: Memories of the Early Days of Chaos TheoryEdited by Ralph Abraham & Yoshisuke Ueda

Volume 40: Advanced Topics in Nonlinear Control SystemsEdited by T. P. Leung & H. S. Qin

Volume 41: Synchronization in Coupled Chaotic Circuits and SystemsC. W. Wu

Volume 42: Chaotic Synchronization: Applications to Living SystemsE. Mosekilde, Y. Maistrenko & D. Postnov

Volume 43: Universality and Emergent Computation in Cellular Neural NetworksR. Dogaru

Volume 44: Bifurcations and Chaos in Piecewise-Smooth Dynamical SystemsZ. T. Zhusubaliyev & E. Mosekilde

Volume 45: Bifurcation and Chaos in Nonsmooth Mechanical SystemsJ. Awrejcewicz & C.-H. Lamarque

Volume 46: Synchronization of Mechanical SystemsH. Nijmeijer & A. Rodriguez-Angeles

Volume 47: Chaos, Bifurcations and Fractals Around UsW. Szemplinska-Stupnicka

Volume 48: Bio-Inspired Emergent Control of Locomotion SystemsM. Frasca, P. Arena & L. Fortuna

Volume 49: Nonlinear and Parametric PhenomenaV. Damgov

Volume 50: Cellular Neural Networks, Multi-Scroll Chaos and SynchronizationM. E. Yalcin, J. A. K. Suykens & J. P. L. Vandewalle

Volume 51: Symmetry and ComplexityK. Mainzer

Volume 52: Applied Nonlinear Time Series AnalysisM. Small

Volume 53: Bifurcation Theory and ApplicationsT. Ma & S. Wang

Volume 54: Dynamics of Crowd-MindsA. Adamatzky

Volume 55: Control of Homoclinic Chaos by Weak Periodic PerturbationsR. Chacn

Volume 56: Strange Nonchaotic AttractorsU. Feudel, S. Kuznetsov & A. Pikovsky

Volume 57: A Nonlinear Dynamics Perspective of Wolfram's New Kind of ScienceL. O. Chua

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE\ Series Editor: Leon 0. Chua

Series A VoI. 57

A N O N l l N E A R DYNAMICS PERSPECTIVE OF WOlFRAM'S

NEW KIND OF SCIENCE Volume I1

Leon 0 Chua University of California a t Berkeley, USA

vp World Scientific N E W J E R S E Y LONDON SINGAPORE BElJlNG S H A N G H A I HONG KONG TAIPEI C H E N N A I

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ISBN-13 978-981-256-977-6(Vol. I)ISBN-10 981-256-977-4 (Vol. I)

ISBN-13 978-981-256-976-9(Vol. II)ISBN-10 981-256-976-6 (Vol. II)

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A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAMS NEW KIND OF SCIENCEVolume II

To

Amalia, Ariella, Diana, Dimitri, Jake, Louisa, and Sophia

v

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May 7, 2007 11:38 bkrev

A PRE-PUBLICATION BOOK REVIEW

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAMS

NEW KIND OF SCIENCE

(in 2 Volumes)

Leon O Chua(University of California at Berkeley, USA)World Scientific Series on Nonlinear Science,Series A - Vol. 57

vol. 1, 396 pp. Pub. date: June 2006ISBN 981-256-977-4Vol. 2, 580 pp pp. Pub. date: June 2007Set ISBN: 981-256-642-2

Our over-riding goal is to introduce cellular automata from the perspective of nonlinear dynamics for thelay readers unfamiliar with cellular automata, writes Prof. Leon Chua. The book is a colorful presentationindeed, which will please everyone with fresh ideas and attractive illustrations. The text is not about cellularautomata, it is about a tiny but fundamentally complex class of one-dimensional binary-state three-cellneighborhood cellular automata.

The journey starts with Boolean-cube representation of the cell state transition rules. Every rule isshown by a cube such that every state of three-cell neighborhood is uniquely represented by a vertex ofthe cube. Vertices take values, 0 or 1, of cell-state transition function over corresponding states of theneighborhood. A concept of linear separability of functions and indices of complexity are introduced then.

Complexity index of a transition function is a minimal number of parallel planes necessary to separatevertices of the Boolean-cube representing the function into the clusters of the same values. All functions areclassified by three possible values of complexity index. It is illustrated that functions with complexity index

Appeared in Journal of Cellular Automata, Volume 2, Number 3. [Reproduced with permission from the journal publisher.]

vii

May 7, 2007 11:38 bkrev

viii A Nonlinear Dynamics Perspective of Wolframs New Kind of Science

one exhibit a repetitive behavior while those with index two support mobile self-localizations, gliders, andnon-trivially interacting propagating patterns. Few functions having complexity index three, it is claimed,have complex at the degree of unpredictability behavior.

Introducing a universal neuron is a culmination of the first volume. A universal neuron is a scalar non-linear differential equation which generates each of 256 rules. The equation has eight parameters, whichcan be interpreted as synaptic weights hence the name. The parameters of the neuron-equation relate tocomplexity as follows. One needs just four parameters to represent a cell-state transition function withcomplexity index one, six parameters to represent a function with index two, and all eight parameters arenecessary for functions with index three.1 An abstract interpretation of universal neuron parameters andcell-state transition functions in terms of genotype is provided, illustrated in the book, could be useful infuture studies on evolution of cellular automata.

First volume ends with chapter Predicting the Unpredictable, where 256 rules are partitioned into88 global equivalence classes. Two rules are globally equivalent if they have identical nonlinear dynamicsfor all initial input patterns. Complexity index one is typical for 38 classes, index two for 41 classes, andjust nine classes2 have highest index of complexity.

Second volume proceeds along, in Chuas words, . . . a paradigm shift in research in Cellular Automata,which has hitherto been either empirical or highly abstract. Our approach is both analytical and con-structive, made possible by our discovery of an explicit unified formula for characteristic functions,which was derived from an associated nonlinear differential equation, or a non-linear difference equation

form.There we enjoy complete characterization of behavior of studied cellular automata in terms of attractors

and invariant orbits. Main findings include exact classification of invertible and non-invertible rules (withperiod one to three), selecting of Bernoulli rules. Also complete table of rules is provided which can beused to predict automaton global behavior from any initial configuration.

Out of 256 rules, 112 non-periodic rules remarkably obey an explicit generalized Bernoulli shift formula,thereby allowing precise prediction of the global (time-asymptotic) dynamics. This fundamental result mayhave a substantial impact on future research in cellular automata. Indeed, the remaining 18 equivalenceclasses (including rules 30, 90, 110, etc.) also exhibit a more complex form of Bernoulli shift reminiscentbut topologically different from the 112 Bernoulli rules reported so far.

It is also incited universal rules exhibit 1/f power spectrum,3 which is widely accepted as a signatureof dynamical complexity in many disciplines, including humanities and arts.

Then author invites us to share his findings on fractal geometry of the characteristics function, explicitformulas for generation of characteristics functions from binary bit-strings, geometrical and analytic prop-erties of characteristics functions. We also become acquainted with identification and classification ofnon-constructible configurations and fixed points. At this point the author introduces isle of Eden, aconfiguration, whose the only predecessor is the configuration itself, and which is a fixed point in globalevolution of cellular automaton.4

Rest of the volume deals with time-reversibility and invertibility of cell-state transition rules. Theseare studied by analyzing, sometimes with the help of generalized Bernoulli maps, attractors of the rulesglobal dynamics. Dynamics of each attractor of a time-reversible rule is mirrored, in space and time, by itsbilateral twin rule. Over half of the rules, 170 out of 256, are time-reversible in Chuas framework; other86 rules are irreversible in the sense that attractors mirror each other only in space not time.

1At this point one can be sceptical about representation potential of generative complexity at the global dynamics level, LeonChua however provides certain demonstrative examples to strengthen the idea.2Exact structure of the highest-complexity classes, in Wolfram coding, is {27, 83, 39, 53), {29, 71}, {46, 116, 139, 209}, {58,114, 163, 177}, {78, 92, 121, 197}, {105}, {150}, {172, 228, 202, 216}, {184, 226}.3Such particular spectrum in cellular automata may be a result of glider interaction and reproduction.4As Leon Chua poetically said, time really stood still on an isle-of-Eden, as in a black hole.

May 7, 2007 11:38 bkrev

A Pre-Publication Book Review ix

A test for time-reversibility of attractor is designed, and considered in relation with an idea that havingattractor and its mirror we can move between time periods and thus mimic cosmological phenomena incellular automata.

The book appeals to wide auditorium. Apart of hard-core cellular automatists, those studying innon-linear sciences, electronic engineering, mathematics and logics, complexity and emergent phenomena,and possibly even chemistry and biology will certainly discover exciting concepts, analogies and researchtools in this refreshing text. Anyone from freshmen to elderly academics will find parts interesting tothem. The volumes are somewhat special and exciting because they posses a unique Chua brand andshow gradual development of ideas and concepts in an educational and entertaining hence mathematicallyrigorous manner.

Andrew Adamatzky (University of the West of England, Bristol, UK )Editor Journal of Cellular Automata

1st ReadingMay 7, 2007 6:31 contents

CONTENTS

Preface vii

Prologue xi

Volume I

Chapter 1. Threshold of Complexity 1

1. Introduction 1

2. Cellular Automata is a Special Case of CNN 3

3. Every Local Rule is a Cube with Eight Colored Vertices 4

4. Every Local Rule is a Code for Attractors of a Dynamical System 6

4.1. Dynamical system for rule 110 78

4.2. There are eight attractors for each local rule 79

5. Every Local Rule has a Unique Complexity Index 83

5.1. Geometrical interpretation of projection and

discriminant w() 83

5.2. Geometrical interpretation of transition points of

discriminant function w() 84

5.3. Geometrical structure of local rules 86

5.4. A local rule with three separation planes 87

5.5. Linearly separable rules 95

5.6. Complexity index 96

xi


xii A Nonlinear Dynamics Perspective of Wolframs New Kind of Science

5.7. Every local rule is a member of an equivalence class 102

5.8. Making non-separable from separable rules 110

5.9. Index 2 is the threshold of complexity 111

Chapter 2. Universal Neuron 113

1. Firing and Quenching Patterns 113

2. A Universal Neuron 119

3. Gallery of One-Dimensional Cellular Automata 121

4. Genealogic Classification of Local Rules 190

4.1. Primary and secondary firing patterns 190

4.2. Partitioning 256 local rules into 16 gene families 192

4.3. Each gene family has 16 gene siblings 192

5. The Double-Helix Torus 198

5.1. Algorithm for generating all 16 local rules

belonging to each gene family 198

5.2. 8/24 Distribution pattern in gene siblings 198

5.3. Coding local rules on a double helix 206

6. Explaining and Predicting Pattern Features 210

6.1. Gallery of gene family patterns 210

6.2. Predicting the background 210

Chapter 3. Predicting the Unpredictable 229

1. Introduction 229

2. Local versus Global Equivalence 233

3. Predicting the Unpredictable 241

3.1. Paritioning 256 local rules into 89 global

equivalence classes 241

3.2. The Vierergruppe V: Key for defining globaltransformations T, T, and T 248

3.3. Proof of global equivalence 259

4. Predicting the Predictable 264

4.1. The rotation group R 2654.2. Local equivalence classes 270

4.3. Finding all rotations which map any N Snm toany N Snm 295

4.4. Truth-table mapping matrices for the rotation group R 2964.5. Combining transformations from the Vierergruppe V and

the rotation group R 338


Contents xiii

4.6. Laterally symmetric interaction group for local

equivalence classes 338

4.7. Mapping parameter vectors between rule 110 and

its locally-equivalent rules 338

5. Concluding Remarks 359

References 361

Index (for Volume I) 363

Volume II

Chapter 4. From Bernoulli Shift to 1/f Spectrum 369

1. Introduction 369

1.1. Computing all 256 rules from one CA difference equation 371

2. Mapping Local Rules onto Global Characteristic Functions 371

2.1. CA characteristic functions 372

2.2. Algorithm for plotting the graph of CA characteristic

functions 372

2.3. A glimpse of some time- characteristic functions N

373

3. Transient Regimes and Attractors 379

3.1. Mapping CA attractors onto time- maps 382

3.2. A gallery of time-1 maps and power spectrum 387

3.3. Three general properties of time-1 maps 452

3.4. Invertible time- maps 453

4. Period-k Time-1 Maps: k=1, 2, 3 454

4.1. Period-1 rules 454



4.4. Invariant orbits 463

5. Bernoulli -Shift Rules 463

5.1. Gallery of Bernoulli -shift rules 463

5.2. Predicting the dynamic evolution from {, } 4755.3. Two limiting cases: Period-1 and palindrome rules 487

5.4. Resolving the multivalued paradox 491


xiv A Nonlinear Dynamics Perspective of Wolframs New Kind of Science

6. Predictions from Power Spectrum 494

6.1. Characteristic features of Bernoulli rules 494

6.2. Turing-universal rules: { 110 , 124 , 137 , 193 } exhibit1/f power-frequency characteristics 499


Chapter 5. Fractals Everywhere 509

1. Characteristic Functions: Global Representation of

Local Rules 509

1.1. Deriving explicit formula for calculating 1N

511

1.2. Graphs of characteristic functions 1N

528

1.3. Deriving the Bernoulli map from 1170

528

1.4. Deriving inverse Bernoulli map from 1240

528

1.5. Deriving affine (mod 1) characteristic functions 593

2. Lameray Diagram on on 1N

Gives Attractor Time-1 Maps 596

2.1. Lameray diagram of 170 597







3. Characteristic Functions are Fractals 610

4. Predicting the Fractal Structures 618

4.1. Two-level fractal stratifications 618

4.1.1. Stratification prediction procedure 623

4.1.2. Examples illustrating stratification prediction

procedure 623

4.1.3. {1,2,3,4} stratified families 6254.2. Rules having no fractal stratifications 630

4.3. Origin of the fractal structures 630

5. Gardens of Eden 634

6. Isle of Eden 655



Contents xv

Chapter 6. From Time-Reversible Attractors to the Arrow of Time 657

1. Recap on Time- Characteristic Functions and Return Maps 658

2. Rule 62 Has Four Distinct Topological Dynamics 660

2.1. Period-1 attractor 1( 62 ) and its basin tree

I+1[1( 62 )] 667

2.2. Period-3 isle of Eden orbits 2( 62 ) 675

2.3. Period-3 attractors 3( 62 ) and their basin trees

I+1[3( 62 )] 675

2.4. Bernoulli -shift attractors 4( 62 ) and their basin trees

I+1[4( 62 )] 676

3. Concept of a Time-Reversible

Attractor 700

4. Time-Reversible Rules 705

4.1. Relationship between invertible and time-reversible attractors 706

4.2. Time reversible does not imply invertible 706

4.3. Time-reversible implies invertible if it is not period-1 707

4.4. Table of time-reversible rules 707

5. There are 84 Time-Reversible Bernoulli -Shift Rules 715

5.1. There are 42 time-reversible Bernoulli -shift rules

(with || = 1, = 2 > 0, and = 1) having onlyone Bernoulli attractor 715

5.2. Four canonical Bernoulli shift maps 732

5.3. There are eight time-reversible time-2 Bernoulli -shift

rules (with || = 1, = 2 > 0, and = 2) having only oneBernoulli attractor 732

5.4. There are 32 time-reversible Bernoulli -shift rules with

two invertible attractors 732

5.5. Composition of 84 time-reversible Bernoulli rules 732

6. What Bit Strings Are Allowed in an Attractor or Invariant

Orbit? 765

6.1. Laws governing period-1 bit strings 765



6.4. Laws governing Bernoulli -shift bit strings 792

6.4.1. Shift left or shift right by one bit 792

6.4.2. Unfolding Bernoulli orbits in complex plane 816


xvi A Nonlinear Dynamics Perspective of Wolframs New Kind of Science

6.4.3. Shift left or shift right by one bit and followed by

complementation 816

6.4.4. Shift left or shift right by one bit every two

iterations 827

6.4.5. Time-reversible rules with two Bernoulli

attractors 827

6.4.6. Time-irreversible rules with two Bernoulli

attractors 827

6.4.7. Time-irreversible rules with three Bernoulli

attractors 827

6.4.8. Deriving bit string laws for globally-equivalent rules

is trivial 827

7. Mathematical Foundation of Bernoulli -Shift Maps 867

7.1. Exact formula for time-1 Bernoulli maps for rules 170 ,

240 , 85 , or 15 867

7.1.1. Exact formula for Bernoulli right-copycat

shift map 170 867

7.1.2. Exact formula for Bernoulli left-copycat

shift map 240 871

7.1.3. Exact formula for Bernoulli shift map for 85 873

7.1.4. Exact formula for Bernoulli shift map for 15 874

7.1.5. Exact formula for Bernoulli shift maps with

= 2 (left shift), or = 2 (right shift), = 1

and = 1 874

7.1.6. Exact formula for Bernoulli shift maps for 184 875

7.2. Exact formula for time- Bernoulli maps for rules 170 ,

240 , 85 , or 15 876

7.2.1. Geometrical interpretation of time- maps 876

7.2.2. Exact formula for time-2 Bernoulli left-shift map for

rule 170 881

7.2.3. Exact formula for time-2 Bernoulli right-shift map for

rule 240 884

7.2.4. Exact formulas for generalized Bernoulli maps 888

7.2.5. Analytical proof of period-3 Isle of Eden 2( 62 )

with I = 4 as subshift of time-1 Bernoulli -shift map

(with = 1) n = 2n1 mod (I) for rule 170 889


Contents xvii

7.2.6. Analytical proof of time-2 map of 2( 62 ) with

I = 4 is a subshift of time-2 Bernoulli -shift map

(with = 1) of Eq. (85) of rule 240 8927.2.7. Analytical proof of time-1 map of 4( 62 ) with

I = 4 is a subshift of time-1 Bernoulli -shift map

(with = 2) of Eq. (63) of rule 170 893

7.3. 4( 62 ) is a subshift of ( 240 ) 895

8. The Arrow of Time 899

8.1. Rule 6 899

8.2. Rule 9 899

8.3. Rule 25 899

8.4. Rule 74 910


9.1. Attractors of 206 local rules 910

9.2. Time reversality 911

9.3. Paradigm shift via nonlinear dynamics 913

Errata for Volume I 934

Epilogue 935

References 937

Index (for Volumes I and II) 939

Chapter 4

FROM BERNOULLI SHIFT TO 1/f SPECTRUM

By exploiting the new concepts of CA characteristic functions and their associated attractortime- maps, a complete characterization of the long-term time-asymptotic behaviors of all 256one-dimensional CA rules are achieved via a single probing random input signal. In particular,the graphs of the time-1 maps of the 256 CA rules represent, in some sense, the generalized Greensfunctions for Cellular Automata. The asymptotic dynamical evolution on any CA attractor,or invariant orbit, of 206 (out of 256) CA rules can be predicted precisely, by inspection. Inparticular, a total of 112 CA rules are shown to obey a generalized Bernoulli -shift rule,which involves the shifting of any binary string on an attractor, or invariant orbit, either tothe left, or to the right, by up to 3 pixels, and followed possibly by a complementation of theresulting bit string.

The most intriguing result reported in this paper is the discovery that the four Turing-universal rules 110 , 124 , 137 , and 193 , and only these rules, exhibit a 1/f power spectrum.

1. Introduction

The basic notations and concepts underlying thistutorial stem from [Chua, 1998; Chua & Roska,2002] and from Part I [Chua et al., 2002], Part II[Chua et al., 2003], and Part III [Chua et al., 2004].Throughout this paper we are concerned exclusivelywith 2-state one-dimensional cellular automata con-sisting of I + 1 cells, i = 0, 1, 2, . . . , I with periodicboundary conditions, as depicted in Fig. 1(a). Eachcell i interacts only with its nearest neighbors i 1and i+ 1, as depicted in Fig. 1(b). Here ui1, ui andui+1 denote the three inputs needed to compute thesingle output yi by a three-input nonlinear function

yi = N(ui1, ui, ui+1) (1)

Boolean computations by this function are executedaccording to the truth table depicted in Fig. 1(c).

Each of the eight binary bits 0, 1, . . . , 7 in therightmost column of this figure is equal to either0 or 1. There are 256 distinct combinations ofzeros and ones among the eight binary bits0, 1, . . . , 7, each one defining a unique Booleanfunction of three binary variables. One-to-one cor-respondence of each of these 256 Boolean functionswith its associated decimal number

N =7

k=0

k2k (2)

determines a local rule N of the cellular automa-ton. Each coefficient 0, 1, . . . , 7 is uniquelyidentified, via its coordinates (ui1, ui, ui+1) fromFig. 1(d) as a vertex of the Boolean cube shown inFig. 1(e). A vertex k is colored in blue if k = 0 andin red if k = 1. For example, the colored vertices in

369

370 A Nonlinear Dynamics Perspective of Wolframs New Kind of Science

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 {0, 1, . . . , 7}, i {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 k {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 k = 1, and blue if k = 1.

Chapter 4: From Bernoulli Shift to 1/F Spectrum 371

Fig. 1(e) correspond to the local rule

N = 0 20 + 1 21 + 1 22 + 1 23 + 0 24

+1 25 + 1 26 + 0 27 = 110. (3)

Depending on the context, each variable ui1, ui,ui+1, or yi may assume a symbolic Boolean value0 or 1, as depicted in Fig. 1(c), or a numericvalue, 1 or +1, as depicted in Fig. 1(d). Thesymbolic and numeric representations are related toeach other as follows1:

numeric variable = 2 Boolean variable 1(4)

Boolean variable =12

(numeric variable + 1

)(5)

In particular, the real variables 0, 1, . . . , 7in Fig. 1(d) are related to the Boolean variables0, 1, . . . , 7 in Fig. 1(c) via the formula

k = 2k 1 (6)Substituting Eq. (6) into Eq. (2), we obtain the fol-lowing equivalent local rule number

N =12

(255 +

7k=0

k2k)

(7)

1.1. Computing all 256 rules fromone CA difference equation

The cellular automaton evolves in discrete timesteps t = 0, 1, 2, . . . . The output of the ith cell (innumeric representation) can be calculated analyti-cally from the following nonlinear difference equa-tion [Chua et al., 2004] involving eight parameters:

CA Difference Equation 1 : uti {1, 1}ut+1i = sgn{z2 + c2|(z1 + c1|(z0 + b1uti1

+ b2uti + b3uti+1)|)|}

(8)

It is indeed remarkable that one equation sufficesto define all 28 = 256 Boolean functions of threevariables ui1, ui, and ui+1 by simply specify-ing eight real numbers. Even more remarkable isthat the CA Difference equation (8) is robust inthe sense that the eight parameter values definingeach local rule N form a dense set. One set ofparameters {z2, c2, z1, c1, z0, b1, b2, b3} for realizing

each one of the 256 local rules is listed in Table 4of Part II [Chua et al., 2003]. The eight parame-ters {z2, c2, z1, c1, z0, b1, b2, b3} in this equation canbe used to derive the coefficients k in Fig. 1(c),k = 0, 1, . . . , 7 via the formula:

k =12(1 + sgn{z2 + c2|(z1 + c1|(z0 + b1uk,i1

+ b2uk,i + b3uk,i+1)|)|}) (9)where the numeric coefficients uk,i1, uk,i and uk,i+1are given by row k of the truth table in Fig. 1(d).The state variables uti1, u

ti, and u

ti+1 in Eq. (8)

must be expressed in numeric values 1 and +1.Since this paper (Part IV) will be devoted

exclusively to Boolean variables xi {0, 1}, it ismore convenient to express Eq. (8) in terms of xivia Eq. (4); namely,

CA Difference Equation 2 : xti {0, 1}xt+1i =

12(1 + sgn{z2 + c2|(z1 + c1|(z0 + b1xti1

+ b2xti + b3xti+1)|)|})

where z0 12[z0 (b1 + b2 + b3)], z1

12z1,

z2 12z2

(10)

2. Mapping Local Rules onto GlobalCharacteristic Functions

Given any local rule N , N = 0, 1, 2, . . . , 255,and any binary initial configuration (or initial statewhen used in the context of nonlinear dynamics)

x(0) = [x0(0), x1(0), , xI1(0), xI(0)] (11)

for a one-dimensional Cellular Automaton with I+1cells [see Fig. 1(a)], where xi(0) {0, 1}, we canassociate uniquely the Boolean string x(0) withthe binary expansion (in base 2) of a real num-ber 0 x0x1 xI1xI on the unit interval [0, 1];namely,

x [x0x1 xI1xI ] 0 x0x1 xI1xI (12)

1The Boolean variable is considered as a real number in Eqs. (4) and (5), or in any equation involving real-variable (i.e. nonlogic)operations.


where the decimal equivalent of Eq. (12) is given by

=I

i=0

2(i+1)xi (13)

We will often need to consider also the bilateralimagex(0)= [xI(0), xI1(0), . . . , x1(0), x0(0)] = T(x(0))

(14)

henceforth called the backward Boolean string asso-ciated with the forward Boolean string x fromEq. (12), where T is the (I + 1)-dimensional leftright transformation operator defined in Table 13 of[Chua et al., 2004], namely,

T[x0x1 xI1xI ] = [xIxI1 x1x0] (15)Each backward Boolean string x in Eq. (14) mapsinto the real number defined by

x 0 xIxI1 x1x0 (16)where the decimal equivalent of Eq. (16) is given by

=I

i=0

2(I+1)+ixi (17)

wherex [xIxI1 x1x0]

2.1. CA characteristic functions

For a one-dimensional CA with I+1 cells, there aren

= 2Idistinct Boolean strings, where I = I +1.

Let denote the state space made of the collectionof all n

Boolean strings. Each local rule N induces

a global map

TN : (18)

where each state x is mapped into exactly onestate TN (x) . Since each state x corre-sponds to one, and only one, point [0, 1] viaEq. (13), it follows that the global map (18) inducesan equivalent map N from the set of all ratio-nal numbers R[0, 1] over the unit interval [0, 1] intoitself; namely,

N : R[0, 1] R[0, 1] (19)

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

henceforth called the CA characteristic function ofN . The one-to-one correspondence between theglobal map TN and the CA characteristic functionN is depicted in the diagram shown in Fig. 2,where denotes the transformation of the state xinto the decimal function defined in Eq. (13). Thisdiagram is said to be commutative because

TN = N (20)where denotes the composition operation.

Observe that in the limit where I , thestate space coincides with the collection of allbi-infinite strings extending from to , and

limI

R[0, 1] = [0, 1] (21)In this general case, the CA characteristic func-tion N is defined on every point (i.e. real number) [0, 1], thereby including all irrational numbersas well [Niven, 1967].

2.2. Algorithm for plotting the graph ofCA characteristic functions

Since the domain of the CA characteristic functionN of any local rule N (for finite I) consists of asubset of rational numbers in the unit interval [0, 1],a computer program for constructing the graph ofthe characteristic function N can be easily writtenas follow:

Step 1. Divide the unit interval [0, 1] into a finitenumber of uniformly-spaced points, called a lineargrid, of width . For the examples in Sec. 2.3, wechoose = 0.005.

Step 2. For each grid point j [0, 1], identifythe corresponding binary string sj .Step 3. Determine the image sj of sj underN , i.e. find sj = TN (sj) via the truth table of N .


Step 4. Calculate the decimal equivalent of sj viaEq. (13).

Step 5. Plot a vertical line through the abscissaN = j with height equal to sj.

Step 6. Repeat steps 15 over all (1/) + 1grid points. For the examples in Sec. 2.3, there are(1/0.005) + 1 = 201 grid points.

For reasons that will be clear later, it is some-times more revealing to plot the th iteratedvalue

sj = TN (sj) TN TN TN

times

(sj) (22)

of sj, instead of TN (sj), at each grid point j [0, 1]. For obvious reasons, such a function iscalled a time- CA characteristic function andwill henceforth be denoted by

N. Using this

terminology, the algorithm presented above can beused to plot the graph of the time-1 CA charac-teristic function 1

Nof any local rule N . The same

algorithm applies mutatis mutandis, for plottingthe graph of the time- characteristic function

N

as well.To enhance readability, it is desirable to plot

theM = (1/)+1 vertical lines of N

in alternat-ing red and blue colors, henceforth referred to as redand blue -coordinates red and blue, respectively.The tip of each vertical line gives the value of

N

corresponding to each coordinate. The systemof red and blue lines is defined via the followingsimple algorithm.

For any I, partition all (I+1)-bit binary stringsinto a red group and a blue group. All members ofthe red group have a 0 as their rightmost bit.The blue group then consists of all (I+1)-bit binarystrings with a 1 as their rightmost bit. Each grouphas therefore exactly one half of the total number(M = 2I+1) of distinct strings, namely, 2I .

Since the end (rightmost) bit of each blue [0, 1] is equal to a 1, by construction, it followsthat the largest value of blue is greater than thelargest value of red by exactly 1/2I+1. This meansthat the rightmost vertical line must have color blue,and tends to = 1 as I . The rightmost blueline is, for plotting purpose, drawn through = 1.We can then divide the interval [0, 1] into (1/)+1grid points, where is the prescribed resolution.All characteristic functions in Figs. 37 are drawnwith = 0.005, where a tiny red or blue square isdrawn around the tip of each vertical line for ease of

identification. In other words, the distance betweeneach red line and its adjacent blue line is equal to0.005.

Although higher precision can be easily imple-mented by a computer, the limited printer resolu-tion will cause adjacent red and blue lines to mergethrough ink diffusion for < 0.005.

To construct Figs. 37, we chose = 0.005,start = 0 00 00

65 0s1, and end = 0 11 11

65 1s1.

Our choice leads to exactly 100 red verticallines (located at 0.005, 0.015, 0.025, . . . , 0.995,red = 0.01) with binary base-2 expansion red =0 123 650, i {0, 1}, which interleavewith 101 blue vertical lines (located at 0.000, 0.01,0.02, . . . , 1.00, blue = 0.01) with binary base-2expansion blue = 0 1

2

3 651, i {0, 1}.

The 201 red and blue lines shown in the char-acteristic functions in Figs. 37 represent only theirapproximate positions on [0, 1] because the resolu-tion of their exact positions is determined by thevalue of I, which is chosen to be 65 in Figs. 37. Thismeans that our state space is coarse grain andcontains only 266 distinct 66-bit binary strings, eachone representing a unique rational number on [0, 1],of which only 201 are actually drawn in these figuresto avoid clutter. Since an arbitrary rational num-ber on [0, 1] requires an arbitrarily large (thoughfinite) value of I for an exact base-2 expansion (i.e.I in Eq. (13)), a fine grain characteristic func-tion

Nwhich includes all possible rational num-

bers [0, 1] in its domain would be impracticalto plot on paper, or even store on any computermemory. However, the characteristic functions (cal-culated with I = 65) shown in Figs. 3 to 7 are ade-quate for most purposes. Increasing the value of I isequivalent to sandwiching more vertical lines inbetween the existing lines drawn in these figures.

2.3. A glimpse of some time-characteristic functions

N

Let us take a glance at some representative exam-ples of CA characteristic functions. For brevity, wewill henceforth refer to time-1 CA characteristicfunctions simply as characteristic functions.

Example 1. 1128

The graph of the characteristic function 1128

of128 is shown in Fig. 3(a). This is among the


(a)

(b)

Fig. 3. Time-1 CA characteristic functions 1128

and 1200

for local rules 128 and 200 , respectively. Although only 201

points (enclosed by tiny squares) are shown, the abscissa ( coordinate) of each point is calculated with a 66-bit stringresolution.


(a)

(b)


and 1240




(a)

(b)


and 1110




(a)

(b)


and time-2 CA characteristic function 251

for local rule 51 . Although

only 201 points (enclosed by tiny squares) are shown, the abscissa ( coordinate) of each point is calculated with a 66-bitstring resolution.


(a)

(b)


and time-3 CA characteristic function 362

for local rule 62 . Although

only 201 points (enclosed by tiny squares) are shown, the abscissa ( coordinate) of each point is calculated with a 66-bitstring resolution.


simplest characteristic functions. Observe that novertical lines intersect the unit-slope main diagonalexcept at 128 = 000 and 128 = 100 (where thebar over a sequence of binary bits denotes repeti-tion of these bits ad infinitum). These two period-1fixed points give rise to a homogeneous 0 dynamicpattern D 128 [000] (homogeneous blue color) in theformer, and a homogeneous 1 dynamic patternD 128 [100] (homogeneous red color) in the latter.The qualitative dynamics of these two orbits, how-ever, are dramatically different. The orbit fromD 128 [000] is an attractor in the sense of nonlin-ear dynamics [Alligood et al., 1996] because it hasa nonempty basin of attraction B, which in thiscase consists of all points in the closed-open unitinterval [0, 1).

The orbit from 128 = 100 is an example ofboth an invariant orbit, and a Garden of Eden, tobe defined in Sec. 3.

Example 2. 1200


of200 is shown in Fig. 3(b). In this case, observethat there are many vertical lines which terminateexactly on the main diagonal. There are thereforemany period-1 fixed points which imply the pres-ence of many period-1 attractors. This is character-istic of local rules belonging to Wolframs class 1rules [Wolfram, 2002]. We will return to this classof attractors in Sec. 3.

Example 3. 1170


of170 is shown in Fig. 4(a). Note that there are noperiod-1 fixed points except at 170 = 000 and 170 = 100. Observe also the vertices of all verti-cal lines fall on one of two parallel lines with slope =2. This is an example, par excellence, of the classicBernoulli shift [Nagashima & Baba, 1999], a subjectto be discussed at length in Sec. 5.

Example 4. 1240


of240 is shown in Fig. 4(b). There are no period-1fixed points except at 240 = 000 and 240 =100.2 The double-valued appearance is only illu-sory because all red vertical lines terminate onthe lower straight lines of slope = 1/2, and all

blue vertical lines terminate on the upper parallelstraight lines. Since the blue and red vertical linesinterleave but do not intersect each other, 1

240is a

well-defined single-valued function. In fact, a carefulexamination of 1

170and 1

240in Fig. 4 will reveal

that these two piecewise-linear functions are inverseof each other. Subsets of both characteristic func-tions in Fig. 4 are typical of Wolframs class 2 rules.

Example 5. 130


of 30is shown in Fig. 5(a). This complicated characteris-tic is typical of all local rules belonging to Wolframsclass 3 CA rules.

Example 6. 1110


of110 is shown in Fig. 5(b). This rather exoticcharacteristic exhibits many features typical ofWolframs class 4 rules.

Example 7. 151

and 251

The graphs of the time-1 characteristic function1

51and time-2 characteristic function 2

51of

51 are shown in Figs. 6(a) and 6(b), respectively.Observe that while there is only one period-1 fixedpoint in 1

51, every vertical line terminates on the

main diagonal of 251

. This implies that 51 hasa dense set of period-2 invariant orbits. Such localrules will be studied in Sec. 4.4.

Example 8. 162

and 362

The graphs of the time-1 characteristic function1

62and time-3 characteristic function 3

62of

62 are shown in Figs. 7(a) and 7(b), respectively.Observe that while there are no period-1 fixedpoints in 1

62, there are many vertical lines which

landed on the main diagonal of 362

. This impliesthat 62 has many period-3 attractors. Such localrules will be studied in Sec. 4.3.

3. Transient Regimes and Attractors

For a CA with finite I, the state space containsexactly n 2I

distinct states, where I = I + 1.

It follows that given any initial state

x(0) =[x0(0) x1(0) xI1(0) xI(0)

](22)

2The leftmost vertical line should actually be drawn at = 0. Printer resolution precludes our showing the correct value1

240(0.00) = 0.00.


the dynamic pattern DN [x(0)] evolving from theinitial state x(0) under any local rule N musteventually repeat itself with a minimum period T,where the

Attractor period T 2I+1 (23)

depends only on the local rule N , and is indepen-dent of the initial state x(0), assuming x(0) belongsto the basin of attraction of a period-T attractor to be defined below.

Definition 1. Transient Regime and TransientDuration: Given any local rule N , and any ini-tial configuration x(0), let T be the smallest non-negative integer such that

x (T + T) = x (T) (24)

Since x(t), t = 0, 1, 2, . . . , T 1, will never recuragain for all t T, the first T consecutive rows of

the dynamic pattern DN [x(0)] is called the tran-sient regime originating from the initial state x(0)and the time (T1) is called the transient duration.

Definition 2. Period-T Attractor: If T > 1,then the T consecutive rows of DN [x(0)]denoted by

N (x(0)) x(T) x(T + 1) x(T + (T 1)) (25)

is called a period-T attractor of the localrule N originating from the initial configurationx(0). The set B of all initial states x(0) whichtend to the attractor is called the basin of attrac-tion of .

To illustrate the above definitions, consider firstthe dynamic pattern D 62 [xa(0)] shown in Fig. 8(a).For the initial configuration xa (row 0 in Fig. 8(a)),we find T = 51 and T = 3. Hence, the transient

(a) (b)

Fig. 8. Illustrations of the transient regime and transient duration of rule 62 originating from two different initial configu-rations xa and xb.


regime originating from xa of the dynamic patternD 62 [xa] consists of the first 51 rows in Fig. 8(a).The period-3 orbit is clearly seen by the alternatingcolor backgrounds. For the dynamic pattern D 62 [xb]shown in Fig. 8(b), observe that the initial configu-ration xb (row 0 in Fig. 8(b)) gives rise to a longertransient duration T = 83.

However, since xa and xb in Fig. 8 were cho-sen to belong to the basin of attraction of , theperiod T of the periodic orbit in Figs. 8(a) and8(b) must be the same, namely, T = 3, as can beeasily verified by inspection of the dynamic patternin Fig. 8.

For some local rules, the period T can be muchlarger than the transient duration, as depicted inthe two dynamic patterns D 99 [xa] and D 99 [xb] inFigs. 9(a) and 9(b) for local rule N = 99 . Observethat T = 14 and T = 71 for D 99 [xa]. Similarly,T = 43 and T = 71 for D 99 [xb].

In fact, there are local rules such as 110 and30 , and their global equivalence classes, where Tcan tend to infinity as T (for I = ). In suchcases, it is no longer useful to talk about a tran-sient regime and we will simply refer to the entiredynamic pattern DN [x(0)] as an orbit originatingfrom x(0).

The basin of attraction B of an attrac-tor must contain, by definition, at least onepoint not belonging to . It is possible, how-ever, for some periodic orbits to have no basin ofattraction.

Definition 3. Invariant Orbits: An orbit whosebasin of attraction is the empty set is called aninvariant orbit.

It follows from Definition 3 that an invari-ant orbit must have a zero transient duration, i.e.T = 1.

(a) (b)

Fig. 9. Illustrations of the transient regime and transient duration of rule 99 originating from two different initial configu-rations xa and xb.


Proposition 1. Local Equivalent Class S14 isInvariant: The orbits of all six rules { 15 , 51 ,85 , 170 , 204 , 240 } belonging to the local equiv-alence class S14 [Chua et al., 2003] are invariant.

Proof. We will see in Table 2 and Sec. 5.4 that, forfinite I, every point in Fig. 4(a) is a point on a peri-odic orbit of 170 whose dynamics consist of shiftingeach initial string by one pixel to the left. Conse-quently, there is no transient regime in this case andhence all orbits of 170 are invariant orbits. Sincethe shifting operation is preserved under the rota-tional transformations of the local equivalence classS14 listed in Table 25(o) of [Chua et al., 2003], itfollows that all orbits of 15 , 51 , 85 , 204 , and240 are invariant as well.

It has been verified by exhaustive computersimulation that only the six rules belonging to S14are endowed with only invariant orbits. In gen-eral, invariant orbits have noninvariant neighbor-ing orbits. We have seen earlier a special caseof an invariant orbit consisting of only a singlepoint; namely, 128 = 100 in Fig. 3(a). Observethat in addition to having no basin of attraction, 128 = 100 has no preimage (predecessor). Suchspecial initial configuration is called a garden ofEden [Moore, 1962].

Observe that no garden of Eden can be a peri-odic orbit with a period T > 1, otherwise anypoint on the orbit is a predecessor of its next iter-ate. A period-1 garden of Eden is therefore a trulyunique specie worthy of its own name, henceforthdubbed an isle of Eden. Indeed, we can generalizethis unique phenomenon, which does not exist incontinuous dynamical systems (such as ODE), todefine a period-k isle of Eden from the kth iter-ated characteristic function k

Nof N . A gallery

of period-k isles of Eden of all one-dimensional cel-lular automata will be presented in Part V of thistutorial series.

3.1. Mapping CA attractors ontotime- maps

Since invariant orbits are not attractors, they arenot robust in the sense that precisely specified ini-tial states must be used to observe them. Sinceone of the most fundamental problems in nonlin-ear dynamics is to analyze and predict their long-term behaviors as t , we will develop somenovel and effective techniques for analyzing and

predicting global qualitative behaviors of robust CAattractors.

In general, each CA local rule N can exhibitmany distinct attractors i, i = 1, 2, . . . ,, asdemonstrated in Figs. 37. Each attractor repre-sents a distinct operating mode and must be ana-lyzed as a separate dynamical system. In order toexploit the lateral symmetry exhibited by manybilateral pairs N and N T [N ] of local rules,where T denotes the leftright transformationoperation defined in [Chua et al., 2004], it is morerevealing to represent and examine each attractorfrom two spatial directions, namely, a forward (left right) direction and a backward (right left)direction.

Since each CA attractor is periodic (forfinite I) with some period T, it is usually rep-resented by displaying T consecutive binary bitstrings s1, s2, . . . , sT , as illustrated in Figs. 8 and9. In order to exploit the analytical tools from non-linear dynamics [Alligood et al., 1996; Shilnikovet al., 1998], it is essential that we transcribe theserather unwieldy pictorial data into an equivalentnonlinear time series. Such a one-to-one transcrip-tion is precisely defined by Eqs. (13) and (17) viathe commutative diagram shown in Fig. 2. Hence,each forward Boolean string x is mapped bijectivelyonto a real number [0, 1] via Eqs. (12) and(13). Similarly, each backward Boolean string

x(0)

is mapped bijectively onto a real number [0, 1]via Eqs. (16) and (17). Each period-T attrac-tor defined by a pattern made of T consecutiveBoolean strings is therefore mapped onto a forwardtime series

= [0, 1, 2, . . . , T ], i [0, 1], (26)henceforth called a forward orbit, and a backwardtime series

= [0, 1,

2, . . . ,

T

], i [0, 1], (27)

henceforth called a backward orbit, where the lengthof each time series (resp. period of each orbit) isequal to T.

It follows from the definition of the period ofan attractor that T = 1 for all period-1 attrac-tors in Fig. 3(b), T = 2 for all period-2 attrac-tors in Fig. 6(b), and T = 3 for all period-3 attractors in Fig. 7(b). Attractors associated withlocal rules belonging to Wolframs Classes 3 and 4can have an extremely large period T, a numbergreater than the number of elementary particles inthe universe even for a relatively small I = 100.


From a computational perspective, such time seriesis effectively infinite in length.

The qualitative dynamics associated with anattractor can often be uncovered and understood byplotting the following two attractor-induced time-maps [Alligood et al., 1996] associated with the for-ward time series and the backward time series ,respectively:

Forward time- map

[N ] : n n(28)

Backward time- map

[N ] : n n(29)

For each local rule N , the forward time-map and the backward time- map

are

defined explicitly via the time- characteristicfunction

Nas follow:

(n ) = N (n ) (30)

(n ) = N (n ) (31)

Explicit coordinates (n , n) and (n , n)

for plotting each point of the forward time- map [N ] and the backward time- map

[N ], are

listed in Table 1 for = 1, 2, and 3.When = 1, the resulting time-1 maps

[Alligood et al., 1996; Hirsch & Smale, 1974] 1[N ]and

1[N ] are sometimes called first-return maps

in the literature because they behave like Poincarereturn maps [Poincare, 1897]. Figure 10 shows thePoincare first-return map interpretation of four for-ward time-1 maps 1[200], 1[51], 1[62], and 1[170]of CA rules 200 , 51 , 62 , and 170 , repectively.In each case, the Poincare cross-section is the sameunit square [0, 1] [0, 1] we have encountered earlierin our definition of the CA characteristic function1

Nin Eq. (19). Only one out of many attractors is

shown for each time-1 map in Fig. 10.In Fig. 10(a), only one period-1 attractor of rule

200 is shown (labeled as point 1). The domainof the time-1 map 1[200] in this trivial case con-sists of only the single point { 1}, and all iterates

map trivially onto the fixed point 1. One can inter-pret point 1 as the point where a planet intersectsan imaginary Poincare cross-section once everyrevolution.

Figure 10(b) shows a period-2 attractor (outof many others) of local rule 51 . The orbit ofthe circulating planet intersects the Poincare cross-section at two points. The domain of the time-1map 1[51] is { 1, 2} where 1( 1) 2 and1( 2) 1.

Figure 10(c) shows a period-3 attractor of localrule 62 . The circulating orbit is seen to inter-sect the Poincare cross-section at three points. Thedomain of the time-1 map 1[62] consists of { 1,2, 3} where 1( 1) 2, 1( 2) 3, and1( 3) 1.

Figure 10(d) shows a Bernoulli 1-shift attrac-tor (to be discussed in depth in Sec. 5) of local rule170 where the domain of the time-1 map 1[170]consists of all points on the two parallel lines withslope equal to 2 for the case I = . For finiteI > 60, the attractor consists of almost all points onthese two lines separated by tiny gaps < 1018and is therefore not discernible. The domain in thecase I = consists of the entire unit interval[0, 1]. Only a few iterates ( 1, 2, 3, . . . , 6) areshown to avoid clutter. One can associate the com-plicated orbit in Fig. 10(d) with the trajectory ofa comet, which in this case would visit almost allpoints on these two parallel lines, as originally envi-sioned by Poincare.

In so far as the qualitative dynamics is con-cerned, it suffices to examine the evolution of thetime-1 map induced by the orbit in Fig. 10. To illus-trate this important insight discovered by Poincare,let us examine the forward time-1 map of a period-3attractor of 62 consisting of points 1, 2, 3 inFig. 11(a), as well as the associated backward time-1map in Fig. 11(b) consisting of points 1, 2 and 3.

In order to illustrate what we mean by thefundamental principle which asserts that CA rulesbelonging to the same global equivalence class m[Chua et al., 2004] must have identical qualitativedynamics, we also show the forward time-1 map for118 in Fig. 11(c), and the backward time-1 mapfor 118 in Fig. 11(d), where 118 and 62 belongto same equivalence class 222 derived in [Chua et al.,2004]. Since 118 and 62 are related by a leftrighttransformation operator T, i.e. T [62] = 118 , itfollows from the theory of global equivalence classdeveloped in [Chua et al., 2004], that the two rules62 and 118 have identical qualitative behaviors.


Table 1. Explicit coordinates (n , n) for defining time- maps for = 1, 2, 3.


Fig. 10. Poincare return map interpretation of four forward time-1 maps. (a) Period-1 map 1[200]. (b) Period-2 map 1[51].(c) Period-3 map 1[62]. (d) Bernoulli 1-shift map 1[170].

In particular, they have, qualitatively, the sametransient regimes, the same attractors, and thesame invariant orbits, modulo a bijection. More-over, their dynamics must also be mapped onto eachother, as depicted by the diagram shown in Fig. 11.This well-known geometrical construction is calleda Lameray diagram [Shilnikov et al., 1998], namedafter the French mathematician Lameray who firstdiscovered its pedagogical value in the eighteenthcentury. It is also called a cobweb diagram [Alligoodet al., 1996] because it resembles the web spun bya spider.

Important Observation

Every Point on the forward time-1 map 1 :n1 n, or the backward time-1 map 1 :n1 n, of any CA rule N is a point onthe characteristic function 1

N.

In other words, the CA characteristic function1

Nis a complete and global representation of N .

It is complete because it contains all informationneeded to derive the dynamic evolutions from anyinitial state by simply drawing a Lameray (cobweb)diagram directly on 1

N! It is global because each

point on 1N

codes for an entire configuration oflength I + 1, and not just for one pixel if the localrule were used instead. Clearly, the points definingthe time-1 maps 1[N ] and

1[N ] are subsets of the

points defining the characteristic function 1N.

It follows from the above observation that everypoint on a time- map of N is a point on the time-characteristic function

N.

Remarks

1. If we imagine the three points on the time-1 map 1[62] in Fig. 11(a) as points on aunimodal function (e.g. logistic map) [Alligood


Fig. 11. Cobweb diagram showing the evolution of 62 and 118 from any state of a period-3 attractor in forward time (a andc), and backward time (b and d). Points k and k denotes corresponding instants of time.

et al., 1996], then we can associate this particu-lar period-3 attractor of 62 as a period-3 pointof a continuous map f : [0, 1] [0, 1] whichwe know is chaotic because period-3 implieschaos [Alligood et al., 1996].

2. It follows from Remark 1 above that every for-ward and backward time-1 map exhibited inTable 2 of Sec. 3.2 can be interpreted as aperiod-T attractor of a continuous map f :[0, 1] [0, 1] over the unit interval [0, 1].

3. It follows from Remark 2 above that for everyCA rule N , N = 0, 1, 2, . . . , 255, and finite I,we can construct a continuous one-dimensionalmap fN : [0, 1] [0, 1] which has a period-Tpoint coinciding with a period-T attractor, orinvariant orbit, of rule N .

4. It follows from Remark 3 above that since allattractors, or invariant orbits, of a CA rule Nare disjoint sets of points over [0, 1], we canalways construct a polynomial PN (x), x [0, 1],


which passes through all of these points. Infact, we can invoke the canonical representa-tion from [Chua & Kang, 1977] to derive anexplicit equation PN (x) involving the absolutevalue function as the only nonlinearity, such thatPN (x) passes through the union of all pointsassociated with all attractors i, i = 1, 2, . . . ,of N . The continuous real-valued functionPN (x ) : [0, 1] [0, 1], henceforth called a RuleN induced function, contains each attractor,i, i = 1, 2, . . . ,, of N as a period-Ti point.Since PN (x) can be constructed to include alsoattractors not observed from N , it clearly hasmuch richer nonlinear dynamics. Hence, for finiteI, all attractors and invariant orbits of each ofthe 256 CA rules can be imbedded into a sin-gle continuous real-valued function over the unitinterval [0, 1].

3.2. A gallery of time-1 maps andpower spectrum

The qualitative dynamics and long-term asymp-totic behaviors of each attractor of a local ruleN can often be predicted from one or more ofits time- maps [N ], = 1, 2, . . . . In fact, atotal of 224 out of 256 local rules have attrac-tors that resemble those shown in Fig. 10, or theircompositions. For an in-depth study of some ofthese rules in Sec. 5, and in Part V, as well asfor future reference, a gallery of the forward time-1map 1[N ] and the backward time-1 map

1[N ]

of up to three distinct attractors are exhibitedin Table 2. For local rules with several qual-itatively different attractors, their time-1 mapsare printed in different colors. Unlike in Figs. 10and 11, the points are not labeled to avoidclutter.

For each rule N in Table 2, the forward time-1map 1[N ] is printed in the left column and thebackward time-1 map 1[N ] is printed in the rightcolumn. All points with the same color (red, blue,or green) pertain to an attractor of the same color.

The power spectrum of the forward time series of Eq. (26) associated with the red forward time-1 map is calculated using the Mathcad soft-ware and printed in the middle column.3 We will seein Sec. 6 that the power spectrum reveals additionalvaluable and insightful information which cannot beextracted from time- maps.

Table 2 contains 256 three-component frames,henceforth referred to in this paper as CA attrac-tor vignettes, corresponding to the 256 local rules.Each vignette N provides a signature of the type ofattractors inhabiting a CA local rule N . Except forthe six local rules 15 , 51 , 85 , 170 , 204 , and240 (to be discussed in Sec. 4.4), whose dynamicpatterns are invariant orbits, all other vignettescontain information on robust CA attractors.

The simplest vignette shows the time-1 map (inred) of only one attractor (e.g. vignette 2 ). In thiscase, the power spectrum pertains to the forwardtime-1 map 1[2] depicted in the left column. Wewill show in Sec. 6 that some spectrum harborsadditional albeit nonrobust dynamic modes.

Vignette 11 of Table 2 shows two time-1maps (colored in red and blue, respectively) corre-sponding to two distinct types of attractors, calledBernoulli attractors, to be analyzed in Sec. 5. In thiscase, the power spectrum pertains to the red for-ward time-1 map 1[11] depicted in the left column.

Vignette 25 shows three time-1 maps (coloredin red, blue, and green, respectively) correspond-ing to three distinct types of attractors to be ana-lyzed in Sec. 5. In this case, the power spectrum,as always, pertains to the red forward time-1 map1[25] depicted in the left column.

An exception to our 3-color code applies totime-1 maps of rules with a continuum of period-1and period-2 attractors. Since such attractors arequalitatively similar, only a dull blue color is usedto indicate various clusters of period-1 and period-2points. In addition, the location of two typicalperiod-1 points are identified as solid dots (paintedin light red and light blue color) in both forward andbackward time-1 maps of such period-1 attractors(e.g. 1[4] and

1[4] for N = 4). Note that the back-

ground color of the power spectrum of all period-1time-1 maps are painted yellow with only a bold redline emerging at f = 1 signifying the absence of anyother frequency components.

Similarly, two typical pairs of solid points arepainted red and blue at the precise locations wherethey are located in both forward and backwardtime-1 maps of period-2 attractors for those rulesharboring a continuum of period-2 attractors (e.g.N = 5, 51, etc). All other period-2 points form clus-ters and are painted in dull blue color. The powerspectrum of all period-2 time-1 maps consists of abold red line located at f = 1/2.

3The power spectrum of the corresponding backward time-1 map is qualitatively identical and is therefore redundant.


Table 2. Gallery of forward time-1 maps 1[N ] and backward time-1 maps 1[N ] for attractor 1(red), 2(blue), and

3(green).


Table 2. (Continued )


All power spectra in Table 2 are calculated withI = 450. In some more intricate cases, such as110 , 54 , etc., a larger value of I 900 is used.An (I + 1)-bit random bit string generated by aBorland Delphi random function software is usedas our initial configuration (i.e. initial state). Thisbit string is not repeatable in view of its randomnature. Since a sufficiently long random bit-stringshould in principle contain all possible combinationsdistributed over different portions of the string, wecan expect that most of the robust modes of eachlocal rule N will emerge in the subsequent itera-tions. Indeed, all vignettes in Table 2 are repeatablewith different random bit strings.

In the case where there are multiple attrac-tors with widely-separated basins of attractions wemust repeat our simulations with different care-fully chosen initial states. We usually choose initialconfigurations containing various periodic subcon-figurations of different periods. It is important thatsuch choices do not provoke the nonlinear dynam-ics from escaping into another basin of attraction.To enhance our chances of uncovering most of therobust modes, we usually found it useful to ran-domize the periodicity and relative positions of thevarious subconfigurations.

To obtain a reliable power spectrum atvery low-frequency ranges, we have significantlyextended our simulation time for some rules, suchas 110 , 137 , etc., in order to obtain a sufficientlylong time series of length up to n = 216 = 65536.Such lengthy simulations also call for a correspond-ing increase in I because f = 1/(I + 1) repre-sents the lowest observable frequency component.For rules in Table 2 which exhibit a 1/f -spectrum,namely, the four universal computing rules 110 ,124 , 137 , and 193 [Chua et al., 2004] discussedin Sec. 6.2, the determination of their low-frequencyspectra in Table 2 requires an immense amount ofsimulation times.

3.3. Three general properties oftime-1 maps

Following are some fundamental relationshipsexhibited by time-1 maps between various localrules. Let N T [N ] denote the local ruleobtained by applying the leftright transformationoperator T to N [Chua et al., 2004]. We will hence-forth call N the lateral twin of N , and vice versa.

The twin rules (N,N ), N = 0, 1, 2, . . . , 255, areglobally equivalent and listed in Table 1 of [Chuaet al., 2004]. It is therefore not surprising that theirforward and backward time-1 maps are related.

Time-1 map Property 1: Dual mappingCorrespondence

(1) The forward time-1 map 1[N ] of N is identicalto the backward time-1 map 1[N

] of N :

1[N ] = 1[N

] (32)

(2) The forward time-1 map 1[N ] of N is iden-tical to the backward time-1 map 1[N ] of N :

1[N ] = 1[N ] (33)

The proof follows from Eqs. (13), (17), (28)and (29).

As an example, compare vignette 110 and itslateral twin vignette 124 in Table 2. Observe theleft frame of vignette 110 and the right frame ofvignette 124 are identical. Similarly, the left frameof vignette 124 and the right frame of vignette 110are also identical.

It is instructive for the reader to verify the Dualmapping Correspondence by comparing the twinvignettes of all rules in Table 2, thereby obtaininga constructive, albeit less rigorous, proof.

If N is bilateral in the sense that N T [N ] = N , i.e. N is a fixed point of the leftright transformation T, then we have the followingCorollary:

Time-1 map Property 2: Bilateral mappinginvariance

The forward time-1 map 1[N ] and the backwardtime-1 map 1[N ] of any bilateral CA rule N areidentical.

There are 64 bilateral CA rules. They are listedin Table 8 of [Chua et al., 2004]. For all of theserules, their vignettes in Table 2 have identical leftand right frames (e.g. 0 , 1 , 4 , 5 , 18 , 19 , etc.).

Time-1 map Property 3: -rotation mappingsymmetry4

1. The forward time-1 map 1[N ] of N and theforward time-1 map 1[N ] of N

= T [N ]

4Time-1 map property 3 is true for all rules except { 57 , 99 , 184 , 226 }.


are related by a 180 rotation about thecenter.5

2. The backward time-1 map 1[N ] of N and thebackward time-1 map 1[N ] of N T [N ] arerelated by a 180 rotation about the center.

Remarks

1. The time-1 map property 3 can be verified byinspection of Table 2.

2. The above three properties are stated for time-1maps for simplicity. The same properties holdalso for time- maps for all .

3. We have verified, by computer simulations, thatthe above three properties are consistent with alltime-1 maps listed in Table 2. It is truly remark-able that such consistency is achieved by usingonly one random configuration probing stringfor each attractor.

4. Our computer simulation results have pro-vided a resounding validation of Wieners bril-liant insight of using random signals as probesfor nonlinear system characterizations [Wiener,1958].

3.4. Invertible time- maps

Since the period T of any attractor of N is thesmallest integer where the orbit repeats itself, notwo points in the domain of the functions [N ]and [N ] can map to the same point, it followsthat both maps [N ] and

[N ] are bijective, and

hence have a well-defined single-valued inverse map[ [N ]]1 and [

[N ]]1, respectively.

More than half (146 out of 256) of allone-dimensional CA rules exhibit the followingimportant mathematical property which makes thenonlinear dynamics of these rules tractable.

Definition 3. Invertible Time- map (I = ): Theforward time- map [N ] : [0, 1] [0, 1] defined inEq. (29) (for = 1) is said to be invertible over[0, 1] iff

[N ] = [ [N ]]1 (34)

Similarly, the backward time- map [N ] : [0, 1] [0, 1] defined in Eq. (30) (for =1) is said to beinvertible over [0, 1] iff

[N ] = [ [N ]]1 (35)

Remarks

1. It is important to keep in mind that each time-map is associated with one, and only one, attrac-tor. We will see in Example 3 below that time-maps corresponding to different attractors of thesame rule N may exhibit different invertibilityproperty.

2. For finite I, the domain of the functions [N ]and [N ] in Definition 3 must be restricted toa subset of all rational numbers on [0, 1].

Geometrical Interpretation of Invertibletime-1 maps

For = 1, the two conditions (34) and (35) areequivalent to the condition that the set of points,henceforth called the graphs of 1[N ], and

1[N ], in

the left and right frames of vignette N , are mirrorimages (i.e. reflection) of each other relative to themain diagonal.

Example 1. Consider vignette 3 of Table 2. Itsleft and right frames have only one color (red).Hence 3 has only one robust attractor.6 Since thegraph of 1[3] on the left and the graph of

1[3] on

the right of vignette 3 are reflections of each otherabout the main diagonal, the time-1 maps 1[3] and1[3] are invertible.

Example 2. Consider vignette 11 . The two colorsin the left and right frames imply that 11 has atleast two robust attractors. But since both graphsof the same color are mirror images about the diag-onal, both pairs of time-1 maps of 11 are invertible.

Example 3. Consider vignette 110 . The red colorgraphs on the left and the right sides of vignette110 are clearly not mirror images of each other.Hence, the forward time-1 map 1[110] and thebackward time-1 map 1[110] are not invertible.

5The symbol T denotes the global complementation operator defined in [Chua et al., 2004]. Indeed, the two forward time-1 maps{1[N ], 1[N ]} form a two-element Abelian group whose group multiplication operation consists of a 180 rotation about thecenter. Similar property applies to the two backward time-1 maps {1[N ], 1[N ]}. Both are examples of the abstract elementgroup C2.6For each vignette in Table 2, we have shown only time-1 maps of robust attractor prototypes. Many rules have attractors thatcan only be observed with specially chosen initial configurations.


Example 4. Finally, consider vignette 62 . Thereare at least two attractors. The graphs of thered color time-1 maps 1[62] and

1[62] con-

sisting of only three red dots7 are not mirrorsymmetric about the main diagonal. It followsthat the red color forward and backward time-1maps of 62 are not invertible. In contrast, theblue color time-1 maps 1[62] and

1[62] consist-

ing of a large ensemble of points exhibit reflec-tion (mirror) symmetry about the diagonal andhence the two blue color time-1 maps of 62 areinvertible.

Remarks

1. A forward time-1 map 1[N ] is invertible if, andonly if, its associated backward time-1 map 1[N ]is invertible.

2. Since the composition between two invertiblefunctions is also an invertible function, it followsthat if a forward time-1 map 1[N ], or a back-ward time-1 map 1[N ], is invertible, then so aretheir associated time- maps [N ] and

[N ],

for any integer .

4. Period-k Time-1 Maps: k= 1, 2, 3

In this section we organize local rules into threeseparate groups based on the global qualitativebehaviors of their time-1 maps, which were derivedfrom random initial configurations. Each time-1map is the outcome of a single random initialstate. Unlike the 256 dynamic patterns presented in[Wolfram, 2002] and [Chua et al., 2003], which haveno predictive ability because the probing inputsignal consists of only a single red center pixel, thetime-1 maps in Table 2 can be used, with completeconfidence, to predict the long-term behaviors dueto any initial configurations. Time-1 maps are, qual-itatively, reminiscent of the classic Greens functionfrom theoretical physics, the impulse response fromlinear circuit and system theory [Chua et al., 1987]and the Brownian motion response a la Wiener[Wiener, 1958], where in all cases, a single testingsignal is enough to predict the response to anyinitial configurations.

4.1. Period-1 rules

Our research on time-1 maps of period-1 attrac-tors has found that there are a total of 93 (out of256) one-dimensional CA rules from Table 2 withrobust period-1 modes in the sense that almost allrandom initial states will converge to a period-1configuration; namely, a fixed point. These 93 rulescan be logically partitioned into four distinct fam-ilies whose members are listed in Tables 3 and 4,respectively. These rules are organized in accor-dance with the theory of global equivalence class mdeveloped in [Chua et al., 2004].8 Since all mem-bers of a given equivalence class m have identicalglobal dynamical behaviors, it suffices to examineand analyze in depth only one member of each class.Since Table 3 contains 45 rules which exhibit invert-ible time-1 maps, we will henceforth refer to theserules as invertible rules for simplicity. These rulesare invertible because their forward time-1 maps1[N ] and backward time-1 maps

1[N ] are identical

with respect to both color and position, along themain diagonal, and hence they satisfy Definition 3in a trivial way. Observe that since there are only20 global equivalence classes in Table 3, only 20out of the 45 invertible rules have qualitatively dis-tinct global dynamical response to arbitrary initialstates, including transient, attractor, and invariantorbit regimes, and their respective basins of attrac-tion (for attractors).

Table 4 contains 24 noninvertible period-1 rulesfrom Table 2. Since they can be partitioned intosix global equivalence classes, only six represen-tative noninvertible period-1 rules warrant an in-depth analysis. Observe that the rules in Table 4 arenoninvertible because each fixed point of 1[N ] inthe left frame of vignette N does not map into thesame point in 1[N ] in the right frame of vignetteN . For example, the red fixed point in the leftframe of vignette 12 and its corresponding fixed-point in the right frame of vignette 12 are two dif-ferent points, and hence are not mirror images ofeach other, relative to the main diagonal.

A careful examination of the vignettes inTable 2 corresponding to the 45 time-1 maps fromTable 3 reveals that there are 12 rules from Table 3which must tend to a homogeneous 0 (colored

7Note that the domain of the two red color time-1 maps 1[62] and 1[62] consists of only three rational numbers, obtained

by projecting the three red points onto [0, 1].8Throughout this paper, each rule N is coded in red, blue or green color, in accordance with the complexity index [Chua et al.,2002] = 1, 2, or 3, respectively.


Table 3. 45 Invertible period-1 rules, among them only 29 are bilateral.


Table 4. 24 Noninvertible period-1 rules.

blue in Fig. 12) attractor and another 12 ruleswhich must tend to a homogeneous 1 (coloredred in Fig. 13), for almost all initial states. These24 homogeneous rules are collected in Tables 5

and 6, respectively. Since all time-1 maps in Table 5consists of a fixed point at n = 000, we canpredict that all dynamic patterns from the 12rules in Table 5 must tend to a homogeneous 0

Fig. 12. All rules belonging to Table 5 tend to a homogeneous blue (0) state, regardless of the initial state, chosen randomly.Each pattern has 67 rows and 11 columns.


Fig. 13. All rules belonging to Table 6 tend to a homogeneous red (1) state, regardless of the initial state, chosen randomly.Each pattern has 67 rows and 11 columns.

Table 5. 12 Homogeneous 0 (blue) Rules. All areinvertible but only four are bilateral.

Table 6. 12 Homogeneous 1 (red) Rules. All are invert-ible but only four are bilateral.


(blue) pattern after T iterations. Simulating these12 rules from a random initial state leads to the 12dynamic patterns shown in Fig. 12, which confirmour prediction of a homogeneous blue steady state.

A similar analysis of the 12 rules in Table 6shows a common fixed point at = 100, whichimplies a homogeneous 1 (red) steady stateresponse, as confirmed by the simulation resultsshown in Fig. 13.

A comparison of Tables 3 and 5 shows thatall nonbilateral (i.e. N = N ) period-1 rules fromTable 3 are members of Tables 5 and 6, whichcan exhibit only trivial homogeneous 0 and 1,respectively, patterns. Hence, all invertible non-homogeneous period-1 rules are bilateral. However,there are 16 invertible but nonbilateral period-1rules; they all yield trivial homogeneous 0 or 1patterns and are listed in Tables 5 and 6.

As an illustration, the dynamic patternsDN [x(0)] of three invertible (and bilateral ) period-1 rules selected from Table 3 are displayed in theleft column of Fig. 14; namely, N = 4 , 77 ,and 232 . Observe that since 223 in Table 3belongs to the same global equivalence class15 as that of 4 , it has the same qualitativebehaviors as 4 [Chua et al., 2004], and need

not be examined. Three additional period-1 pat-terns chosen from three noninvertible and non-bilateral rules listed in Table 4 ( 44 , 78 , and172 ) are displayed in the right column of Fig. 14.By the same principle of global equivalence, wecan predict that the three rules

{100 , 203 , 217

} 216 must have the same qualitative behavior as44 . Similarly, the three rules

{92 , 141 , 197

} 35 must have the same qualitative behaviors as78 , and the three rules

{228 , 202 , 216

} 38 must have the same qualitative behaviorsas 172 .

Except for the 12 homogeneous 0 rules inTable 5 and the 12 homogeneous 1 rules inTable 6, all other period-1 rules in Tables 3 and4 consist of clusters of period-1 points distributedover different locations on the main diagonal ofthe respective vignettes in Table 2. To demon-strate that the three time-1 map properties fromSec. 3.3 hold for all period-1 attractors, two typ-ical period-1 points are highlighted as red andblue dots in each period-1 vignette in Table 2.Observe that the red and blue dots occupy iden-tical positions in the left and the right frames ofeach vignette for all bilateral period-1 rules (e.g.4 , 36 , 72 , etc.), as predicted by the bilateral

Fig. 14. Gallery of six period-1 dynamic patterns. The patterns on the left are invertible and bilateral. Those on the rightare noninvertible and nonbilateral. Each pattern has 68 rows and 26 columns. The initial configurations (row 0) are chosenrandomly.


mapping invariance. Clusters associated with non-bilateral period-1 rules in the left frame are differ-ent from those in the right frame of correspondingvignettes, (e.g. 12 , 13 , 44 , etc.). However, the-rotation mapping symmetry implies that the leftframe of vignette N must be identical to the rightframe of vignette T [N ], modulo 180 rotation aboutthe center. Indeed, the left frame of vignette 12 andthe right frame vignette T [12] = 207 are related bya 180 rotation, as predicted.

While there are many nonhomogeneousperiod-1 attractors, each represented by a pointbelonging to some dull blue cluster in Table 2,there is only one attractor (shown in red ) in all,except eight (namely, 40 , 96 , 168 , 224 , 235 ,249 , 234 , and 248 ), homogeneous period-1 rulesin Table 2. These eight exceptions, shown in blue,are endowed with a second attractor of a morecomplicated type (called a Bernoulli 1-shift) to bediscussed in Sec. 5.

Since there are a total of 69 period-1 CA rules(45 in Table 3 and 24 in Table 4), and since 24among them (12 in Table 5 and 12 in Table 6)have only one period-1 attractors, namely, 12 homo-geneous 0 attractors and 12 homogeneous 1attractors, there are altogether 45 period-1 CA ruleshaving many distinct period-1 points clustered indisconnected groups along the main diagonal in theleft and right frames of their associated vignettesin Table 2 (printed in dull blue color). For finite I,these period-1 point are rational numbers on (0, 1).Since rational numbers are denumerable [Niven,1967], these period-1 points are sparsely distributedand almost every point on (0, 1) are not period-1.

4.2. Period-2 rules

An examination of Table 2 shows that there are17 invertible CA rules possessing period-2 attrac-tors. They are listed in Table 7, organized into 10global equivalent classes. All 17 rules in Table 7 arebilateral, i.e. N = N . In addition, there are eightnoninvertible CA rules from three global equivalentclasses possessing period-2 attractors, as exhibitedin Table 8. Observe that all of these rules are non-bilateral.

Each period-2 attractor is manifested by twoisolated points, symmetrically positioned withrespect to the main diagonal in both forward andbackward time-1 maps in Table 2. As in the period-1case, in general there are many distinct period-2attractors for each period-2 CA rule, and they tend

to be organized in various disconnected clusters;they are depicted in dull blue color in each period-2vignette in Table 2. In addition, two prototypeperiod-2 points are singled out and printed in redand blue colors, respectively, at their precise loca-tions (within the resolution of the printer).

The 17 rules in Table 7 are invertible becausecorresponding points in their forward and backwardtime-1 maps in the corresponding left and rightvignette frames in Table 2 are sy

A Nonlinear Dynamics Perspective

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