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Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 17, No. 9 (2007) 2839–3012 c World Scientiﬁc Publishing Company

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND OF SCIENCE. PART VII: ISLES OF EDEN LEON O. CHUA, JUNBIAO GUAN∗ , VALERY I. SBITNEV and JINWOOK SHIN Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA ∗ Department of Mathematics, Shanghai University, Shanghai 200436, P. R. China Received February 5, 2007; Revised June 12, 2007 This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulli στ -shift rules and eight hyper Bernoulli στ -shift rules, the latter including such famous rules 30 and 110 . All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞. Basin tree diagrams of all ten complex Bernoulli στ -shift rules are exhibited for lengths L = 3, 4, . . . , 8. Superﬁcial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely, • Rule 90 has no Isles of Eden. • Rules 105 and 150 are composed of nothing but Isles of Eden for all string lengths L not divisible by 3. Explicit global state transition formulas are given for local rules 90 , 105 and 150 . Such formulas led to the rigorous proof of several surprising periodicity constraints for rule 90 , and to the discovery of a new global, quasi-equivalence class, deﬁned via an alternating transformation. In particular, local rules 105 and 150 are globally quasi-equivalent where corresponding spacetime patterns can be derived from each other by simply complementing every other row. Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules 90 , 105 and 150 . In particular, the period “T ” of all attractors of rules 90 , 105 and 150 , as well as of all isles of Eden of rules 105 and 150 , increases linearly with unit slope, in logarithmic scale, with the length L. Keywords: Cellular automata; nonlinear dynamics; attractors; Isles of Eden; Bernoulli shift; shift maps; basin tree diagram; Bernoulli velocity; Bernoulli return time; complex Bernoulli shifts; hyper Bernoulli shifts; rule 90; rule 105; rule 150; binomial series; scale-free phenomena.

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L. O. Chua et al.

1. Recap of Main Results from Parts I to VI A rigorous analytical theory of one-dimensional cel∆ lular automata composed of L = I + 1 identical cells, as shown in Fig. 1, has been studied in the following series of papers from a nonlinear dynamics perspective1 : Part I: Threshold of Complexity [Chua et al., 2002] Part II: Universal Neuron [Chua et al., 2003] Part III: Predicting the Unpredictable [Chua et al., 2004] Part IV: From Bernoulli shift to 1/f spectrum [Chua et al., 2005a] Part V: Fractals everywhere [Chua et al., 2005b] Part VI: From Time-reversible attractors to the arrow of time [Chua et al., 2006]

1.1. Local rules and Boolean cubes Observe that the “zeros” and “ones” in Wolfram’s truth tables [Wolfram, 2002] are symbolic variables denoting a logic “Yes” or “No” state, or a “high” or “low” state in digital electronic circuit implementations. In order to exploit powerful mathematical tools from nonlinear dynamics, it is necessary to work with real numbers. Consequently, in the papers cited above, the symbolic truth table shown in Fig. 1(c) is converted into the numeric truth table shown in Fig. 1(d). One could also redeﬁne the “0” and “1” in the symbolic truth tables as real numbers, instead of changing “0” to “−1”. There are two reasons why we opted for the latter choice. First, each of the 256 local rules can be implemented on a cellular neural network (CNN) chip [Chua & Roska, 2002] with at least three orders of magnitude faster speed than computing on standard digital computers. Such CNN implementations require that the truth tables be formulated in terms of “1” and “−1” [Chang & Muthuswamy, 2007]. The second reason is that the numeric truth table shown in Fig. 1(d) can be conveniently represented by merely coloring the eight vertices of a “unit Boolean cube” whose center is

1

located at the origin of the (ui−1 , ui , ui+1 ) — input space, as shown in Fig. 1(e). Such a representation in turn leads to simple visualizations of many rotational symmetrical transformations [Chua et al., 2003]. Each of the 256 local rules corresponds to exactly one Boolean cube in Table 1 (extracted from [Chua et al., 2003]). Observe that the number N printed under each cube corresponds to the local rule number in [Wolfram, 2002]. This number is easily obtained by adding the “vertex weights” of all red vertices in the Boolean cube, where the vertex weight for vertex kmis equal to 2k , as speciﬁed in Fig. 1(e), as well as in the lower part of Table 1.

1.2. Threshold of complexity Observe also that the identiﬁcation number N of each Boolean cube is colored in red, blue or green, depending on whether the red vertices can be segregated and separated from each other by κ = 1, 2, or 3 parallel planes, where κ is called the index of complexity of the local rule N [Chua et al., 2002]. Table 2 lists all 256 local rules along with their index of complexity. The index of complexity κ is not a deﬁnition of complexity. Rather it measures the relative number of electronic devices needed to implement each local rule. A κ = 1 local rule requires the smallest number of transistors. More transistors must be added to realize a κ = 2 local rule. Still more transistors are required to implement a κ = 3 local rule. In other words, the index of complexity κ measures the relative “cost” of hardware (Chip) implementations. While the asymptotic qualitative behaviors of all κ = 1 local rules, and all κ = 3 local rules, have been completely understood and characterized in [Chua et al., 2006], and in this paper (for Rules 105 , and 150 ), there are some κ = 2 local rules that have not yet been characterized, including rules 110 , 124 , 137 and 193 [Chua et al., 2004]. Since these four rules are universal Turing machines, they can never be completely characterized. In other words, it seems that κ = 2 can be considered as the threshold of complexity, in the sense articulated in [Wolfram, 2002].

These 6-part papers have been republished, with errors corrected, in two recent edited books [Chua, 2006] and [Chua, 2007].

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden

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Fig. 1. (a) A one-dimensional Cellular Automata (CA) made of L = 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 a local rule N , where N is a decimal number speciﬁed by a binary string {β0 , β1 , . . . , β7 }, βi ∈ {0, 1}. (c) The symbolic truth table specifying each local rule N , N = 0, 1, 2, . . . , 255. (d) By recoding “0” to “−1”, each row of the symbolic truth table in (c) can be recast into a numeric truth table, where γk ∈ {−1, 1}. (e) Each row of the numeric truth table in (d) can be represented as a vertex of a Boolean Cube whose color is red if γk = 1, and blue if γk = −1.

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L. O. Chua et al. Table 1.

Encoding 256 local rules deﬁning a binary 1D CA onto 256 corresponding “Boolean Cubes”.

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 1.

(Continued )

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L. O. Chua et al. Table 1.

(Continued )

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 1.

(Continued )

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L. O. Chua et al.

Table 2. List of 256 local rules with their complexity index coded in red (κ = 1), blue (κ = 2) and green (κ = 3), respectively.

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

κ = 1 (Red) 104 rules κ = 2 (Blue) 126 rules κ = 3 (Green) 26 rules 1.3. Only 88 local rules are independent Among the 256 local rules, only 88 are dynamically independent2 from each other in the sense that the dynamics and solutions (space-time diagrams) of any one of the remaining 168 local rules can be derived exactly from one of the 88 globally equivalent rules, listed in Table 3 [Chua et al., 2004], via one of the following three topological conjugacies: 3 Global Equivalence Transformations 2

1. left-right transformation T † 2. global complementation T 3. left-right complementation T ∗

For the reader’s convenience, each of the 256 local rules is listed in the left-most column in Table 4, along with its equivalent local rule with respect to each of the above three global equivalence transformations. Observe that due to symmetries possessed by certain rules, some rules have only two † distinct rules e.g. equivalent T ( 1 ) = 1 and ∗ † T 1 = T 1 = 127 ; T 29 = T 29 = 71 and T ∗ 29 = 29 ; T † 15 = T ∗ 15 = 85 and T 15 = 15 . Such rules are identical twins. There are altogether 72 identical twin local rules, as listed in Table 5. A few

We thank Andy Adamatzky [Adamatzky, 2007] for suggesting possible intersections of our work with [Wuensche & Lesser, 1992]. We thank Andy Wuensche for informing us that the concept of global equivalence classes was ﬁrst mentioned in [Walker, 1971]. The 88 equivalence classes of local rules were listed in [Walker & Aadryan, 1971] and [Wuensche & Lesser, 1992], using diﬀering numbering schemes. It is likely that other results published, or yet to be published, in our series of tutorial expositions on “Wolfram’s New Kind of Science” may also intersect, if not contained, in other works. We apologize to all such authors for not citing their publications, and we will appreciate their informing us of any such intersections so that future acknowledgments can be made. Being novice on the mature subject of cellular automata, the high probability of such inadvertent omissions is what prompted the authors to publish their papers as expositions for a nonspecialist audience, and not as original papers, in the Tutorial-Review section of this journal.

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 3. The ﬁrst 88 globally-independent local rules among the 256 listed in Table 2.

88 Global Equivalence Classes 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

18 19 22 23 24 25 26 27 28 29 30 32 33 34 35 36 37 38 40 41 42 43 44 45 46 50 51 54 56 57 58 60 62 72 73 74 76 77 78 90 94 104 105 106 108 110 122 126 128 130 132 134 136 138 140 142 146 150 152 154 156 160 162 164 168 170 172 178 184 200 204 232

local rules areendowed with additional symmetries such that T † N = T N = T ∗ N = N . Such rules are identical quadruplets. There are only eight identical quadruplet rules, as listed in Table 6.

1.4. Robust characterization of 70 independent local rules By virtue of the three global equivalence transformations derived in [Chua et al., 2004] it suﬃces to conduct an in-depth analysis of only the 88 local rules listed in Table 3, out of 256, a saving of nearly 70% of otherwise wasted man hours! By using random bit strings (with at least L = 400 bits) as testing signals, we have found via extensive computer simulations, and supplemented by analytical studies [Chua et al., 2006], that the robust time asymptotic dynamics of 70, out of 88, local rules can be characterized by only one of four steady-state behaviors.

1.4.1. Steady-state behavior 1: Period-1 attractors or period-1 isles of Eden Table 7 lists 26 local rules from Table 3 which exhibit a robust period-1 steady-state behavior 3

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corresponding to ﬁxed points of the time-1 characteristic function χ1N of local rule N [Chua et al., 2004]. Except for rule 204 where all orbits are period-1 isles of Eden, the generic steady-state behavior of the other 25 rules in Table 7 are all period-1 attractors. This asymptotic behavior holds for almost all initial random bit strings, and for ∆ arbitrary length L = I + 1.

1.4.2. Steady-state behavior 2: Period-2 attractors or period-2 isles of Eden Table 8 lists 13 local rules from Table 3 which exhibit a robust period-2 steady-state behavior corresponding to ﬁxed points of the time-2 characteristic function χ2N of local rule N [Chua et al., 2006]. Except for rule 51 where all orbits are period-2 isles of Eden, the generic steady-state behavior of the other 12 rules in Table 8 are all period-2 attractors. This asymptotic behavior holds for almost all initial random bit strings, and for arbitrary L.

1.4.3. Steady-state behavior 3: Period-3 attractors There is only one rule from Table 3 which exhibits a robust period-3 attractor, namely, rule 62 . As demonstrated in, Figs. 5–14 of [Chua et al., 2006], almost all initial bit strings of 62 converge to a period-3 orbit corresponding to ﬁxed points of the time-3 characteristic function χ362 of local rule 62 [Chua et al., 2006]. The other attractors of 62 have a relatively small basin of attraction. The period-3 isles of Eden of 62 have no basins of attraction and therefore require an initial bit string falling exactly on one of the three bit strings forming an isle of Eden.

1.4.4. Steady-state behavior 4: Bernoulli στ -shift attractors or isles of Eden Table 9 lists 30 local rules from Table 3 which exhibit a robust Bernoulli στ -shift steady-state behavior corresponding to a period-T attractor or a period-T isle of Eden, where T ≤ τ L. The three parameters (σ, τ , β) characterizing each Bernoulli rules are listed in Table 10 for each of the 30 robust Bernoulli rules listed in Table 9.3 We will henceforth call “σ” the Bernoulli Shift Velocity, “τ ” the Bernoulli Return Time and “β” the Bernoulli Complementation sign, or simply Bernoulli

Table 10 is constructed from Table 16 of [Chua et al., 2005, pp. 1159–1162].

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Table 4. Table of globally equivalent local rules. All local rules in each row are globally equivalent to each other. Rows with red, blue, or green background colors denote local rules with a complexity index κ = 1, 2, or 3, respectively.

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 4.

(Continued )

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L. O. Chua et al. Table 4.

(Continued )

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 4.

(Continued )

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Table 5.

0

1

4

5

250

251

254

255

15

18

19

22

233

236

237

240

29

32

33

36

219

222

223

226

37

43

50

54

201

205

212

218

55

57

71

72

183

184

198

200

73

76

85

90

165

170

179

182

91

94

95

99

156

160

161

164

104

108

109

113

142

146

147

151

122

123

126

127

128

129

132

133

Table 6.

23

51

in Table 29A of [Chua et al., 2006, pp. 1293– 1297] for rules 74 , 88 , 173 and 229 . Each of these attractors has a large enough basin of attraction that diﬀerent random initial bit strings could converge to one of these robust Bernoulli στ shift attractors. This steady-state behavior does not ∆ depend on the length L = I + 1 of the bit string. Except for local rule 15 and 170 , whose orbits are all isles of Eden, all other generic steady states converge to a Bernoulli στ -shift attractor.

List of 72 identical twin rules.

1.4.5. There are ten complex Bernoulli and eight hyper Bernoulli shift rules

List of eight identical quadruplet rules.

77

105

150

178

204

232

velocity, time, and sign, respectively. Observe that local rules 6 , 9 , 11 , 14 , 27 , 35 , 38 , 43 , 56 , 57 , 58 , 134 , 142 , and 184 have two robust Bernoulli attractors, whereas local rules 25 and 74 have three robust Bernoulli attractors. Observe from Table 10 that only ﬁve rules listed in Table 10 11 , 14 , 15 , 43 and 142 have a negative sign for β. The space-time evolution patterns of these ﬁve rules are generated by following the same procedures as the other rules (shift left by σ bits if σ > 0, or shift right by |σ| bits if σ < 0, every τ iterations), and then complementing (change color of all bits) the resulting bit string. In fact, except for rule 15 , only one of two Bernoulli attractors from the other four rules have a negative sign for β. Observe that any Bernoulli (σ, τ , β) rule with β < 0 is equivalent to iterating the rule with twice the velocity and return time without complementation, i.e. (σ, τ, β) = (2σ, 2τ, |β|),

if β < 0

(1)

For examples illustrating this equivalence, see Table 5 (pp. 2393) for 15 in [Chua et al., 2003], Fig. 29(a2 ) for 11 , Fig. 29(b2 ) for 14 , Fig. 29(d2 ) for 43 , and Fig. 29(i2 ) for 142 in [Chua et al., 2006]. ∆

In general, T = τ L if T0 = τ L/|σ| is not an integer. If T0 is an integer, then T = τ L/|σ| for |σ| ≥ 2. If each bit string in the period-T orbit consists of a concatenation of m identical substrings, then the period T is reduced further to T /m. Each Bernoulli rule listed in Table 9 can possess up to three robust Bernoulli attractors, as depicted

Together, Tables 7–9, plus the period-3 rule 62 , made up 70, out of the 88, local rules from Table 3. The robust steady-state behaviors of these 70 local rules have been completely characterized in [Chua et al., 2006]. The remaining 18 rules listed in Table 3 that have not yet been characterized are listed in Table 11, dubbed complex Bernoulli-shift rules, and Table 12, dubbed hyper Bernoulli-shift rules. It will be clear from the sequel that all of these 18 yet uncharacterized rules are also identiﬁed with Bernoulli shifts because they behave like Bernoulli στ -shifts from Table 9 except that the number of attractors is no longer bounded by 3, but increases Table 7.

List of 26 robust Period-1 local rules.

26 Topologically-Distinct Period-1 Rules

0

4

8

12 13

32 36 40 44 72 76 77 78 94 104 128 132 136 140 160 164 168 172 200 204 232

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 8.

List of 13 robust Period-2 local rules.

13 Topologically-Distinct Period-2 Rules

1 5 19 23 28 29 33 37 50 51 108 156 178

eight hyper Bernoulli-shift rules in Table 12 are nonbilateral, and correspond to those listed in column 1 of Table 18 of [Chua et al., 2006]. Table 13 gives a composition of the asymptotic behaviors of all 88 dynamically-independent local rules listed in Table 3. In this paper (Part VII) only the ten complex Bernoulli-shift rules from Table 11 will be studied. The remaining eight Hyper Bernoulli-shift rules from Table 12 will be studied in Part VIII.

2. Basin Tree Diagrams of Ten Complex Bernoulli Shift Rules For binary bit strings xn = (xn0

xn1

xn2

···

xnL−1 )

List of 30 robust Bernoulli στ -shift local rules.

30 Topologically-Distinct Bernoulli στ -shift Rules

2

3

6

7

9

10 11 14 15 24

xn → χ1N (xn ) = xn+1

(3)

under local rule N must converge to either a ﬁxed point x∗ = (xn0

∗

xn1

∗

∗

∗

xn2 · · · xnL−1 ) (4) or to a periodic orbit ΓT N of period T ≤ Tmax , at some ﬁnite time n∗ = Ttransient + T , where → (5) χ1N : is the time-1 characteristic function deﬁned in [Chua et al., 2005a], and Tmax = 2L ∆

25 27 34 35 38 42 43 46 56 57 58 74 130 134 138 142 152 162 170 184 ∆

with the length L = I + 1 of the bit strings. The ten complex Bernoulli shift rules in Table 11 are bilateral, and correspond to those listed in column 1 of Table 17 of [Chua et al., 2006, pp. 1176]. The

(2)

at time n with ﬁnite L and periodic (or ﬁxed) boundary conditions, the evolution ∆

Table 9.

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(6)

is the number of distinct binary bit strings of length L.

2.1. Basin of attraction and basin trees In general, many initial bit strings can converge to one of several period-T orbits, including period-1 orbits (i.e. ﬁxed points of χ1N ). Deﬁnition 1. Basin of attraction B ΓT N of ΓT N . The union of all bit strings which converge to a period-T orbit ΓT N of local rule N , including all bit strings belonging to ΓT N , is called the of ΓT N . basin of attraction B ΓT N

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L. O. Chua et al. Table 10. Bernoulli Parameters σ (Bernoulli shift velocity), τ (Bernoulli return time), and β (Bernoulli complementation sign) associated with the 30 Robust Bernoulli Rules from Table 9.

N

2 3 6 7 9 10 11 14 15 24 25

27 34 35 38

σ

τ

β

N

1 -1 2 -2 -1 -2 2 1 1 -1 1 -1 -1 -1 -1 3 2 -1 2 1 -1 1 2 2

1 2 2 2 2 2 3 1 1 1 1 1 1 1 2 3 5 2 2 1 2 1 2 2

+ + + + + + + + +

42

Table 11.

43 46 56 57 58

+ 74

+ + + + + + + + + + +

130 134 138 142 152 162 170 184

σ

τ

β

1 1 -1 1 1 -1 1 -1 1 -1 1 2 -3 1 2 -2 1 1 -1 -1 1 1 1 -1

1 1 1 1 1 1 1 1 1 2 1 2 3 1 2 2 1 1 1 1 1 1 1 1

+ +

List of ten complex Bernoulli-shift rules.

18 22 54 73 90 105 122 126 146 150

+ + + + + + + + + + + + + + + + + + + +

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 12. rules.

ΓT N , excluding ΓT N , is called the basin trees of ΓT N .

List of eight Hyper Bernoulli-shift

26 30 41 45

More precisely,

60 106 110 154

An example of a basin tree is shown in Fig. 3(g) of [Chua et al., 2006] for rule 62 with L = 9. In this case, ΓT = Γ1 62 = 0m , and (ΓT ) = Γ1 62 m m = 73 , 85 , 146 , 149 , 165 , 169 , 170 , 292 , 298 , 330 , 338 , 340 , 511 ∆ (9) B ΓT 62 = (Γ1 ) ∪ 0m

∆ (ΓT ) = B ΓT N \ΓT N

Table 13. Steady-state characterization of 88 dynamicallyindependent local rules.

Topological Classifications of 88 Equivalence Classes Topologicallydistinct Rules

Number

Period-1 Rules

26 13 1 30

Period-2 Rules Period-3 Rules Bernoulli στ -Shift Rules ComplexBernoulli-Shift Rules Hyper Bernoulli-Shift Rules

In this case, one can associate the basin tree (Γ3 ) m m m , 60m as two subtrees 40m and {23m , 1 , 35 , 22 } emerging from the period-3 orbit Γ3 62 , which is analogous to a cluster of roots. For large L, a basin tree in general is made of many topologically similar subtrees, such as Fig. 11 of [Chua et al., 2006]. In this case, we have a period-14 orbit m , 102 , 93m , 51m , 110 , 89m , 55 , Γ( 62 ) = 59m m 108 , 91m , 54 , 109 , 27m , 118 , 77m (12) and the basin tree Γ14 62 of Γ14 62 is made of seven subtrees having identical topologies.

8 88

More precisely, ∆ =∪ x∈ : ρnN (x) ∈ ΓT B ΓT N

(7)

∆

ρnN (x) = ρ1N ◦ ρ1N ◦ · · · ◦ ρ1N (x) n times

is the time-n map of N [Chua et al., 2005a] ρnN : x0 → xn , where n depends in general on x. Deﬁnition 2. Basin Trees (ΓT ). The set of all bit strings which converges to a period-T orbit 4

(8)

Observe from Fig. 3(g) that the digraph of is a directed tree from graph theory. Γ1 62 Another example of a basin tree is shown in Fig. 6 of [Chua et al., 2006]. Consider the period-3 orbit ∆

m , 38m , 61 (10) Γ3 62 = 3m in Fig. 6(a)-i. The basin tree of Γ3 62 is the set of bit strings ∆

m , 23 , 60m , 1m , 35m , 22m (11) (Γ3 ) = 40m

10

Total

where

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2.2. Garden of Eden Deﬁnition 3. Garden of Eden. A bit string

x = (x0

x1

x2

...

xL−1 )

is said to be a garden of Eden of a local rule N iﬀ its preimage is an empty set. More precisely,4 a bit string x is a garden of Eden of N iﬀ it has no predecessors in the sense

Under Deﬁnition 3, a fixed point x∗ of χ1N , i.e. a period-1 orbit, is not a garden of Eden of N because χ−1 (x∗ ) = x∗ .

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L. O. Chua et al.

that there does not exist a bit string y such that x = χ1N (y).

Proof.

Follows directly from Deﬁnitions 2 and 4.

Many examples of gardens of Eden can be found in [Chua et al., 2006]. In particular, all gardens of Eden of 62 are identiﬁed by a pink color in Figs. 3, 6, 8, 9, 11–14, in [Chua et al., 2006]. Observe that they are just the terminus of subtrees.

A bit string x is a period-n isle of Eden of N ⇔ the orbit through x is a period-n orbit Γn N where each bit string x, n−1 1 2 χ N (x), χ N (x), . . . , χ N (x) has a unique preimage.

Corollary 1.

Proof.

Follows from Eq. (14) and Proposition 1.

2.3. Isle of Eden A cursory inspection of the basin of attractions of the period-3 orbits Γ3 62 of rule 62 in Figs. 5(a)– 5(f) in [Chua et al., 2006] reveals that there are no basin trees converging to any node (i.e. bit string) belonging to these period-3 orbits! Such orbits are indeed special, and except for rules 15 , 85 , 45 , 105 , 150 , 154 , 170 , and 240 , they are isolated period-T orbits which are buried amidst neighboring bit strings belonging to basin trees of other periodic orbits. We will see in Part VIII that for large L, these isolated period-T orbits could have extremely long periods and hence are very, very hard to ﬁnd,5 like well-hidden Easter eggs! Moreover, such rare objects cannot exist in Rn in view of the Zubov– Ura–Kimura Theorem [Garay & Hofbauer, 2003], which implies that “no compact isolated invariant sets in Rn can be an isle of Eden”. These objects can be either isolated or dense, and are called Isles of Eden in [Chua et al., 2005b] and [Chua et al., 2006]. It’s time to give a formal deﬁnition. Deﬁnition 4. Isle of Eden.

A bit string x = (x0

x1

x2

···

xL−1 )

is said to be a period-n isle of Eden of a local rule N iﬀ its preimage under χnN is itself, where χnN is the time-n characteristic function of N . More precisely, x is a period-n isle of Eden of a local rule N iﬀ (x) = x χ−n N

(13)

Proposition 1. A bit string x is a period-n isle of

Eden of N ⇔ x belongs to a period-n orbit Γn N with an empty basin tree; i.e. =Ø (14) Γn N when Ø denotes the empty set.

Remarks 1. To avoid clutter, we will usually refer to all bit strings belonging to the orbit of a period-n isle of Eden also as an isle of Eden. 2. Every bit string belonging to a period-n isle of Eden has exactly one incoming and one outgoing bit string, for all n ≥ 2.

2.4. Gallery of basin tree diagrams

The collection of all period-n orbits Γn N of all possible periods n = 1, 2, . . . and their assoof an L-bit celluciated basin trees Γn N lar automata under local rule N is called a basin tree diagram of local rule N . An examination of such diagrams, even for a relatively small L, can reveal certain characteristic qualitative behaviors of the space-time patterns of many local rules. These empirical characteristics can sometimes be proved to be true in general, as will be illustrated for the complex Bernoulli shift rules 105 and 150 in this paper, and for the hyper Bernoulli shift rules 45 and 154 in Part VIII. A gallery of such basin tree diagrams for the ten complex Bernoulli shift rules listed in Table 11 is exhibited in Tables 14–23 for L = 3, 4, 5, 6, 7 and 8, respectively. Each table displays the periodic orbits and their basin trees, where each bit string is displayed in color along with its decimal identiﬁcation number, calculated from the decimal equivalent of the binary bit string as in Fig. 6 of [Chua et al., 2006]. For example, for L = 3, the two and in Gallery binary bit strings 18-1 from Table 14 would be identiﬁed by the decimal numbers6 1 • 22 + 0 • 21 + 0 • 20 = 4

5 Every isolated long-period isle of Eden is a gem worth digging for. They would provide ideal havens for cryptographic systems. Any one who discovers a long-period isle of Edens earns the right of naming it after himself for posterity reasons! 6 Each page of the basin tree diagrams listed under Tables 14–23 will be called a gallery, and identiﬁed by a Gallery number N − k, k = 1, 2, . . . , where N is the local rule number.

2857

Basin tree diagrams for rule 18 .

7 0

3

1

4 Transient phase

Gallery 18 -1

6

5

2

Transient phase

0 1 2 3 4 5

0 1 2 3 4 5

Basin tree diagrams for Rule 18 8 =1 18 , L = 3 (a) Period-1 Attractor : ρ 1 = −− 8

Table 14.

4

2

2858

11

7

2

4

5

0

10 13

8

14

15

1

12 = 0.75 ρ 1 = −− 16

(a) Period-1 Attractor :

18 , L = 4

(Continued )

0

0.5

φn

1

0

0.5

σ = +− 2, τ = 1

12

Gallery 18 - 2

3

φn - 1

6

2 = 0.25 ρ 2 = 2 16 −−

1

9

(b) Bernoulli (σ = +− 2, τ = 1) Period-2 Isles of Eden :

Table 14.

β>0

2859

18 , L = 5

(Continued )

21

22

15

23

13

26

30

= 0.375

27

29

Gallery 18 - 3

11

0

31

12 (a) Period-1 Attractor : ρ 1 = −− 32

Table 14.

(Continued )

14

28

25

17

3

6

10

20

9

4

8

16

Transient state

0.5

φn - 2

1

σ = −5

the length is equal to 2 .L = 10

0

σ = +5

0 1 2 3 4 5

0

0.5

φn

σ =+ − 5, τ = 2

Gallery 18 - 4

7

24

5

2

19

12

18

1

1

528

16

4 = 0.625 (b) Bernoulli (σ = +− 5, τ = 2) Period-2 Attractors : ρ 2 = 5 32 −−

T=2

β>0

18 , L = 5

Table 14.

τ=2 τ=2

2860

2861

38

37

0 1 2 3 4 5 6 7

44

24

52

39

3

60

18 , L = 6

Transient phase

36

30

30

26

27

(Continued )

33

22

0

45

18

9

11

48

51

50

13

6

21

25

57 41

0

63 42

Gallery 18 - 5

15

54

12

19

59

62

53

43

46 = 0.71875 (a) Period-1 Attractor : ρ 1 = −− 64

Table 14.

47

23

0

0

31

29

55

61

58

46

8

28

Transient state

0

0 1 2 3 4 5

0.5

σ = +− 3, τ = 1

σ = +3

0

0.5

φn

Gallery 18 - 6

34

1

35 20

14

17

4

10

32

49

40

7

5

16

2

56

1

σ = −3

φn - 1

6 = 0.28125 (b) Bernoulli (σ = +− 3, τ = 1) Period-2 Attractors : ρ 2 = 3 64 −−

(Continued )

τ=1

τ=1

16

1

β>0

18 , L = 6

Table 14.

T=2

2862

2863

50

79

100

23

21

81 49

113

4

77 78

75

74

88

7

96

5

103

39

8

17

72

99

27 29 31

14

10

48

120 104

2

53

35 98

20

0

73

40

68

86

18

12

56

3

33

82

9

97

83 51 114

76

105

13

11

41

57

89

108 92

116

19

15 124 32

80

112

6

121

64

26 30

84

22

38

Gallery 18 - 7

69 62 70 58 16 71

43

45

115

1

42 46 54

65

85

90

66

52 60

34

28

(Continued )

128 (a) Period-1 Attractor : ρ1 =128 −− = 1

36

67

44

106

24

101 37 102

25

18 , L = 7

Table 14.

119

123

118

59

117

122 0

0

0

111

0

125

109

93

107

61

95

63

55

47

127

126

91

110

87

94

2864

57

0 1 2 3 4 5 6 7 8

74

78

21

177

81

64

27

69

16

40

130

1

78

4 113

10

160

56

221

0

248

28

85

187

80

34

227

20

65

247

251

239

253

0

8

62 128

138 139

223

254

99

42

54 73

156

201

191

127

148

82

114

37

46

39

141

216

2

226 162

58

143 32

5

163

142

136

232

119

193

7

184 168

(Continued )

Gallery 18 - 8

0

255

112

224

238

14

170

131

209 241

17

29

68

116

124

71

92

23 31

197 199

84 108

198

228

164

41

147

146

Table 14.

Period-1 Attractor :

132 ρ1 = 256 −− = 0.515625

18 , L = 8 (a)

Transient phase

246

123

245

250

181

218

117

186

237

189

235

125

107

109

234

93

0

0

0

0

219

222

215

190

214

182

213

174

183

111

175

95

173

91

171

87

48

135

72

158 11

149

151

155

13

96

15

144

157 159 154 150

79 133 134

202

203

220

188

172

88 244

212 252 236

3

104

105

185

121

169 249 217

6

208 176 233 9 89

240

180

132

120

70

38

35

19

137

145

196

200

51 204

22 26

192

30

33

45

60

66

165

115

242

178

97 211

230

229

101

194 167

166 231 103

24

67

83 243 179

12

161

Gallery 18 - 9

102 153

52

129

36

195

210

18

225

σ = −2

0 1 2 3 4

σ = +2

τ=1

τ=1

204

25

152

140

76

0

0.5

φn

1

0

0.5

φn

1

0

0

0.5

σ =+ − 2, τ = 1

0.5

137

σ = +4

φn - 1

φn - 3

σ = +− 4, τ = 3

σ = −4

0 1 2 3 4 5 6 7 8

30 = 0.46875 ρ2 = 4 256 −−

2 = 0.015625 ρ 3 = 2 256 −−

122 44

86

94

118 126 106 90 110

61

43

47

55

53

63

59

(b) Bernoulli (σ = + − 4, τ = 3) Period-6 Attractors :

(c) Bernoulli (σ = + − 2, τ = 1) Period-2 Isles of Eden :

98

100

49

50

206 207 77 75 205

18 , L = 8 Transient phase T=6

(Continued )

τ=3

1

β>0 1

β>0

Table 14.

T=2

2865

2866

L. O. Chua et al.

and 1 • 22 + 1 • 21 + 0 • 20 = 6 respectively. These numbers are enclosed by small circles, and are represented as nodes of a digraph where a directed edge pointing from node Sm 1 to node Sm 2 means that bit string S1 maps to bit string S2 after one iteration under rule N . example, Gallery 18-1 shows the basin trees For ; 2m 5m ; 4m , 3m ; 1m , 6m con Γ1 18 = 7m verging to a period-1 (ﬁxed point) orbit Γ1 18 = 0 0 means that The self-loop attached to node {}. 0 maps into itself, ad inﬁnitum, thereby bit string 0 is a period-1 orbit. implying Each sequence of nodes along each branch of depicts successive evolutions the tree Γ1 18 2 → 5 → 0 over time. For example, the sequence translates into the space-time pattern shown in the upper right-hand corner of Table 14-1. Similarly, the 4 → 3 → 0 translates into the spacesequence time pattern shown in the lower right-hand corner. Observe that the ﬁrst two rows in both spacetime patterns on the right of Gallery 18-1 represent the transient phase of the dynamic evolution; they 7 correspond to nodes belonging to the basin tree Γ1 18 . The next four rows in these two spacetime patterns correspond to the steady state, which is a period-1 orbit in this case. Whenever a basin tree Γ1 18 is not empty, 0 in steady state is the associated periodic orbit 0 called an attractor because the period-1 bit string attracts all orbits belonging to the tree Γ1 18 . We now extend this deﬁnition to period-n orbits. Deﬁnition 5. Period-n attractor. A period-n orbit

Γn N of a local rule N is said to be a period-n attractor iﬀ it has a nonempty basin tree, i.e. = Ø (15) Γn N

It follows from Proposition 1 that every periodn orbit of a local rule N is either an attractor, or an isle of Eden. Although a period-n orbit of N contains n distinct bit strings Γn N (x), we will usually refer x, χ1N (x), χ2N (x), . . . χn−1 N k to each bit string x or χ N (x), k = 1, 2, . . . , n − 1, as a period-n attractor, or a period-n isle of Eden, respectively, to avoid clutter. In other words, a period-n attractor or isle of Eden can mean either any bit string in a “ring” orbit, or to the collection of all “n” bit strings in the “ring”.

Also listed on top of each gallery is the robustness coeﬃcient ρi =

n ∆ ni i = L 2L n

(16)

of the ith period-n orbit (n is a generic symbol denoting the actual period of each periodic orbit)

where n( L ) denotes the total number of all bit

L composed strings in the symbolic state space of all binary bit strings of length L, and where ni denotes the total number of nodes (i.e. bit strings) in the basin of attraction of the ith period-n orbit, where i = 1, 2, . . . , m, and m is the total number of period-n orbits. In Gallery 18-1, m = 1 since there is only “one” attractor when L = i = 1 in 3.Hence, Gallery 18-1. In the basin tree Γ1 18 shown in Gallery 18-1, there are all together eight nodes and hence ni = 8. Since L = 3, we have ρ1 = 8/23 = 1. The robustness coeﬃcient ρi in Eq. (16) measures the percentage of initial bit strings which converge to the ith attractor in question. In this case ρi = ρ1 = 1 because there is only one attractor in this example and hence all orbits must converge to 0 In general, 0 < ρi ≤ 1, where ρi = 1 correspond . to maximum robustness.

2.4.1. Highlights from Rule 18 Gallery 18-1 : L = 3, n

3

=8

There are seven basin-tree strings, all of which 0 Hence converge to the global period-1 attractor {}. 0 has maximum robustness the period-1 attractor with ρ1 = 1. Gallery 18-2 : L = 4, n

4

= 16

0 (a) There is a period-1 attractor {} with robustness coeﬃcient ρ1 = 0.75.

, 12m }, (b) There are two period-2 isles of Eden { 3m , 9m } with a combined robustness coefand { 6m ﬁcient ρ2 = 0.25. The dynamics on each isle of Eden is a Bernoulli στ -shift with σ1 = 2, τ = 1, or σ2 = −2, τ = 1, as depicted in the φn → φn−1 time-1 map in Gallery 18-2. Here, the red lines have

` ` ´´ Note that our deﬁnition of a basin tree Γn N does not include bit strings belonging to the associated period-n orbit ` ´ Γn N . 7

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden

slope equal to 2σ1 = 4, and the blue lines have slope equal to 2σ2 = 1/4. Both sets of parallel lines have a positive slope, implying that β > 0. Observe that the two period-2 “red” dots correspond to the decimal representation φ=

L−1

2−(i+1) xi

(17)

i=0

(deﬁned in Eq. (2) of [Chua et al., 2006]) of bit mof the isle of Eden { 3m , 12m } on string 3mand 12 the left; namely, 3m→ 1 • 2−3 + 1 • 2−4 = 0.1875 (left red dot) 12m→ 1 • 2−1 + 1 • 2−2 = 0.75 (right red dot) Observe that the two red dots lie at the intersection of corresponding pairs of red and blue “Bernoulli” lines, thereby conﬁrming that the dynamics on this isle of Eden can be described by a left shift of two bits (σ = 2) or, equivalently, by a right shift of two bits (σ = −2), per iteration (τ = 1), as extensively illustrated in [Chua et al., 2005a] and [Chua et al., 2006]. 5 Gallery 18-3, 18-4 : L = 5, n = 32 0 (a) There is a period-1 attractor {} with robustness coeﬃcient ρ1 = 0.375.

(b) There are ﬁve period-2 attractors with a combined robustness coeﬃcient ρ2 = 0.625. The dynamics on each attractor is a Bernoulli στ -shift with σ1 = 5, τ = 2, or σ2 = −5, τ = 2, as depicted in the φn−2 → φn time-2 map. The time-2 map φn−2 → φn consists of β = σ 1 2 = 32 parallel red Bernoulli lines with slope 2σ1 = 32, or equivalently, to β = 2|−σ2 | = 32 parallel blue Bernoulli lines with slope 2σ2 = 1/32. Observe that the two red dots now fall on the diagonal of the time-2 map, as expected of period-2 orbits. Again, β > 0 because the slope of each red (or blue) Bernoulli line is positive. For ease of visualization, we have displayed the space-time pattern using bit strings with double the length, namely, 2L = 10, which corresponds to shifting around the period-2 ring twice. Note m that the decimal code of the 5-bit basin tree 16 translates into the corresponding 10-bit string 528 shown in Gallery 18-4. Observe that all basin subtrees contain only one bit string, implying that all basin trees of rule 18 are gardens of Eden, when L = 5.

Gallery 18-5, 18-6 : L = 6, n

6

2867

= 64

0 (a) There is a period-1 attractor {} with robustness coeﬃcient ρ1 = 0.71875. Note that there are 0 at three locathree blue lines joining bit string tions in the basin tree diagram. This is done to avoid clutter. The reader should interpret all three nodes 0 as representing the same node. Observe labeled also from the basin tree diagram that the longest transient regime is four iterations, such as the one depicted in the space-time pattern originating from string 30min Gallery 18-5. The shortest transient regime is one iteration; they correspond to the 15 gardens of Eden in the three “translated” subtrees joined by blue lines.

(b) There are three period-2 attractors with a combined robustness coeﬃcient ρ2 = 0.28125. The dynamics on each attractor is a Bernoulli στ -shift with σ1 = 3, τ = 1, or σ2 = −3, τ = 1. In this case, all basin trees are gardens of Eden. Gallery 18-7 : L = 7, n

7

= 128

There are 127 basin tree strings, all of which 0 It converge to the global period-1 attractor {}. follows that we have maximum robustness with ρ1 = 1, as in Gallery 18-1. Gallery 18-8, 18-9 : L = 8, n

8

= 256

0 (a) There is a period-1 attractor {} with robustness coeﬃcient ρ1 = 0.515625. The transient regime ranges from one iteration (corresponding to subtrees composed of garden of Edens) to ﬁve iterations, as illustrated in a typical space-time diagram starting from bit string 78min Gallery 18-8.

(b) There are four period-6 attractors with a combined robustness coeﬃcient ρ2 = 0.46875. The dynamics on each attractor is a Bernoulli στ -shift with σ1 = 4, τ = 3, or σ2 = −4, τ = 3. The time-3 map φn−3 → φn shows β = 24 = 16 parallel Bernoulli “red” lines with slope 2σ1 = 16, or equivalently, 16 parallel Bernoulli “blue” lines with slope 2σ2 = 1/16. Observe that there are six red dots in the time-3 map, implying a period-6 attractor. Again, β > 0 because both red and blue lines have a positive slope.

2868

8

7

6

5

4

3

L

1 1 2 1 2 1 2 1 1 2 3

i

1 5 1 3 1 1 4

1 1

attractors

2

2

Eden

Number Period-n Period-n Isles of

1 1 2 1 2 1 2 1 1 6 2

n

0 0 2 0 5 0 3 0 0 4 2

σ1

1 1 1 1 2 1 1 1 1 3 1

τ1

+ + + + + + + + + + +

β1

-4 -2

-3

-5

-2

σ2

3 1

1

2

1

τ2

+ +

+

+

+

β2

ρ3 = 0.015625

ρ2 = 0.46875

ρ1 = 0.515625

ρ1 = 1

ρ2 = 0.28125

ρ1 = 0.71875

ρ2 = 0.625

ρ1 = 0.375

ρ2 = 0.25

ρ1 = 0.75

ρ1 = 1

ρ

coefficient

Summary of Qualitative properties of local rule 18 extracted from Gallery 18 for Rule 18 Number ID Number Bernoulli Parameters Robustness Period of of

(c) There are two period-2 isles of Eden with a combined robustness coeﬃcient ρ3 = 0.015625. The dynamics on each isle of Eden is a Bernoulli στ -shift σ1 = 2, τ = 1, or σ2 = −2, τ = 1. The qualitative properties of local rule 18 extracted from the above basin-tree Galleries 18-1 to 18-9 are summarized below:

2869

Basin tree diagrams for rule 22 .

4

1

2

7

22 , L = 3

6

3

Gallery 22 - 1

0

5

(a) Period-1 Attractor : 8

8 =1 ρ 1 = −−

Basin tree diagrams for Rule 22

Table 15.

2870

8

4

13

14 0

11

7 15

10

16

2 = 0.125 ρ 2 = −−

σ = −2

0 1 2 3 4

12

σ = +2

3

τ=1

τ=1

3

0

0.5

φn

1

0

6

0.5

σ =+ − 2, τ = 1

9

2 = 0.25 ρ3 = 2 16 −−

φn - 1

(c) Bernoulli (σ = + − 2, τ = 1) Period-2 Isles of Eden :

5

(b) Period-1 Isles of Eden :

(Continued )

Gallery 22 - 2

1

2

10 = 0.625 ρ1 = 16 −−

(a) Period-1 Attractor :

T=2

22 , L = 4

Table 15.

1

β>0

Transient phase

26

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

2

1

7

13

19

26

24

12

5

18

23

0 1 2 3 4 5 6 7

20

15

3

9

21

28

6

8

Gallery 22 - 3

21

4

14

0

27

10 17

29

30

31

Transient phase

13

(Continued )

32 = 1 (a) Period-1 Attractor : ρ 1 = 32 −−

Table 15.

Transient phase

11

25

0 1 2 3 4 5 6 7

16

Transient phase

22 , L = 5

Transient phase

2871

0 1 2 3 4 5 6 7

22

22

11

2872

8

37

26

1

12

19

44

33

57 6

28

35

48

34

20

18

15

9

21

42

2 = 0.03125 ρ 2 = 64 −−

59

3

55

0

62

60

17

47

31

7

50

52

5

10

58

2

40

45

25

4 14

27

41

53

11

56

49

13

16

38

22

32

23

46

15

29

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8

62 = 0.96875 (a) Period-1 Attractor : ρ 1 = 64 −−

(Continued )

Gallery 22 - 4

54

36

61

63

24

39

(b) Period-1 Isles of Eden :

43

29

51

30

22 , L = 6

Table 15.

Transient phase Transient phase

2873

86

15

82

41

30

38

80

32

33

64

46

51

78

115

12

97

6

9

58

36

126

0

1

90

72

116

25

103

125

65

34

62

123

119

39

73

57

7

8

2

48

17

28

54

20

85

122

77 5

74

79

114

4

27

31

42

105

14

96

83

101

107

99

(Continued )

53

120

50

117

10

100

21

113

106

0 1 2 3 4 5 6 7 8

127

118

59 0

109

93

91

110 55

15

114 ρ 1 =128 −− = 0.890625

(a) Period-1 Attractor :

Gallery 22 - 5

60

66

102

67 24

63

75

111

68

56

95

61 37

92

3

16

87

84

108

18

40

124

71

112

19

43

23

76

29

89

121

45

94

47

22 , L = 7

Table 15.

Transient phase

98

44

52

35

22

69

70

49

88

104

81

11

13

0

0.5

φn

1

0

0.5

φn

1

0

0

φn - 1

φn - 1 1

1

0

0.5

φn

1

0

0.5

φn

1

0

0

0.5

φn - 2

β>0

φn - 2

β>0

σ = −1, τ = 2

0.5

σ = +1, τ = 2

1

1

σ = −1

0 1 2 3 4 5 6 7

σ = −3

0 1 2 3 4 5 6 7

σ = +3

σ = +1

104

7 = 0.109375 Period-7 Isles of Eden : ρ 3 = 2 −− 128

Gallery 22 - 6

0.5

σ = +3, τ = 1

0.5

σ = −3, τ = 1

β>0

26

β>0

(b) Bernoulli (σ = −3, σ = +3, τ = 1), (σ = +1, σ = −1, τ = 2)

(Continued )

T=7

T=7

22 , L = 7

Table 15.

88

τ=1 τ=2

τ=1 τ=2

2874

2875

89

202

77

71

58

106

67

113

95

212

11

23

152

140

107

158

47

149

116

176

4

34

223

239

17

164

191

119

187

247

148

190

128

182

61

193

162

200

94

28

134

137

229

211

8

82

41

14

233

245

0

167

56

25 208

19

154

97

109

125

250

218

35

194

53

145

226

178

22

188

122

1

2

38

197

244

142

49

44

121

37

98

146 131

7

203

84

214

16

215

73

68

254

253

136

112

151

50

161

29

101

92

13

46

69

138

26

(Continued )

169

232

209

83

43

86

0 1 2 3 4 5 6 7 8 9 10

246

123 0

219

222

255

237

189

183

111

149

154 ρ1 = 256 −− = 0.6015625

(a) Period-1 Attractor :

Gallery 22 - 7

127

238

221

251

74

79

173

32

175

168

22 , L = 8 181

21

242

70

224

52

76

163

133

64

196

184

100

91

88

235

104

42

81

139

166

172

Table 15.

Transient phase

15

96

105

155

157

72

144

159

135

48

180

205

206

207

249

9

3

6

75

236

220

217

185

150 240

252

(Continued )

60

66

126

33

231

36

12

24

195

243

18

90

103

230

179

115

0

0.5

φn

1

0 1 2 3 4 5 6 7 8

0.5

σ = +− 4, τ = 3

σ = +4

0

Gallery 22 - 8

165

110

129

30

192

210

118

55

59

63

225

45

σ = −4

φn - 3

●

14 = 0.21875 (b) Bernoulli : (σ = + −− − 4, τ = 3) Period-6 Attractors : ρ 2 = 4 256

120

132

22 , L = 8

Table 15.

Transient phase T=6

165

1

β>0 τ=3

2876

31

174

143

160

80

177

27

216

10

5

234

241

117

248

147

78

201

39

124

186

62

93

130

65

198

108

99

54

40

20

171

199

213

227

57

156

0

Transient phase

0

0.5

φn

1

51

153 51 102

85

σ = +2

0 1 2 3 4

102 204 153

σ = −2

Gallery 22 - 9

170 85

2 (e) Period-1 Isles of Eden : ρ5 = −− 256 = 0.0078125

204

τ=1

204 τ=1

0

0.5

φn

1

0

0.5

σ = +− 2, τ = 1

σ = +4

0 1 2 3 4 5 6

0.5

σ =+ − 4, τ = 2

φn - 1

σ = −4

φn - 2

10 = 0.15625 (c) Bernoulli : (σ = + −− − 4, τ = 2) Period-4 Attractors : ρ 3 = 4 256

(Continued )

T=2

2 = 0.015625 (d) Bernoulli : (σ = + −− − 2, τ = 1) Period-2 Isles of Eden : ρ4 = 2 256

228

114

87

141

22 , L = 8

Table 15.

T=4

57

1

β>0 τ=2τ=2 1

β>0

2877

2878

8

7

5 6

4

3

L

1 1 2 3 1 1 2 1 2 1 2 3 4 5

i

1 4 4

1

1 1

1 1

attractors

2 2

2

2

2 2

Eden

Number Period-n Period-n Isles of

σ1

1 0 1 0 1 0 2 2 1 0 1 0 1 0 1 _0 7 +3 1 0 6 4 4 4 2 2 1 0

n

1 1 1 1 1 1 1 1 1 1 3 2 1 1

τ1

+ + + + + + + + + + + + + +

β1

-4 -4 -2

_1 +

-2

σ2

3 2 1

2

1

τ2

+ + +

+

+

β2

ρ1 = ρ1 = ρ2 = ρ3 = ρ1 = ρ1 = ρ2 = ρ1 = ρ2 = ρ1 = ρ2 = ρ3 = ρ4 = ρ5 =

1 0.625 0.125 0.25 1 0.96875 0.03125 0.890625 0.109375 0.601563 5625 0.21875 0.15625 0.015625 0.0078125

ρ

coefficient

Summary of Qualitative properties of local rule 22 extracted from Gallery 22 for Rule 22 Number ID Number Bernoulli Parameters Robustness Period of of

The basin tree diagrams of Rule 22 for L = 3, 4, . . . , 8 are exhibited in Table 15. Following a detailed analysis of these diagrams, the qualitative properties of local rule 22 extracted from basin-tree Galleries 22-1 to 22-9 of Table 15 are summarized below:

2.4.2. Highlights from Rule 22

2879

Basin tree diagrams for rule 54 .

54 , L = 3

5

2

4 7

1

Gallery 54 - 1

0

3

(a) Period-1 Attractor : ρ1 = −− =1 8

6

Basin tree diagrams for Rule 54 8

Table 16.

1

2

(a) Bernoulli (σ = + − 2, τ = 2)

11

14

7

13

4

8

0

0.5

φn

1

0

σ = +2

0 1 2 3 4 5 6 7 8 σ = −2

φn - 2

8

1

6

9

0

0.5

φn

1

0

0.5

σ = +− 2, τ = 1

φn - 1

1

0

15

5

10

4 = 0.25 (c) Period-1 Attractor : ρ3 = 16 −−

12

3

Gallery 54 - 2

0.5

σ = +− 2, τ = 2

2 Period-2 Isles of Eden : ρ2 = 2 −− 16 = 0.25

(b) Bernoulli (σ = + − 2, τ = 1)

(Continued )

β>0

4 Period-4 Isles of Eden : ρ 1 = 2 −− 16 = 0.5

54 , L = 4

T=4

β>0

Table 16.

τ=2 τ=2

2880

2881

23

21

27

8

4

0

11

28

14

54 , L = 5

3

17

20

10 31

26

9

5

(Continued )

25 16

18

15

12

22

19

13

1

30

Gallery 54 - 3

6

24

7

2

29

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

32 (a) Period-1 Attractor : ρ1 = −− 32 = 1

Table 16.

Transient phase Transient phase

22

30

2882

19

11

37

41

60

0

30

(Continued )

27

3

33

26

39

24

36

18

12

51

13

25

63

44

42

48

6

15

57

38

50

52

54

22

Gallery 54 - 4

9

21

34 (a) Period-1 Attractor : ρ 1 = −− 64 = 0.53125

45

54 , L = 6

Table 16.

Transient phase Transient phase

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

52

44

49

56

28

46

23

43 8

55

35

1

62

61

59

0 Transient phase

0

0.5

φn

1

Gallery 54 - 5

29

20

34

7 2

58

16

47

40

14

5

53 4

31

17

32

10

0 1 2 3 4 5 6 7 8 9

σ=0

0.5

σ = 0, τ = 4

5 = 0.46875 (b) 6 Period-4 Attractors : ρ 2 = 6 64 −−

(Continued )

T=4

54 , L = 6

Table 16.

φn - 4

β>0

46

1

τ=4

2883

2884

54 , L = 7

0

37

74

(Continued )

127

41

42

Gallery 54 - 6

82

21

84

9 = 0.0703125 (a) Period-1 Attractor : ρ1 = 128 −−

Table 16.

97

94

0 1 2 3 4 5 6 7 8 9 10

σ=0

64

18

12

115

63

70

112

47

118

38

33

76

109

89

30

51

45

114

105

101

0 1 2 3 4 5 6 7 8 9 10

32

9

6

121

80

15

116

35

σ=0

95

86

44

56

87

59

19

108

57

22

τ=4

44

Transient phase

16

68

3

124

40

71

58

81

0 1 2 3 4 5 6 7 8 9 10

111

43

σ=0

28

107

93

73

54

92

11

108

8

34

65

62

20

99

29

104

Transient phase

119

0 1 2 3 4 5 6 7 8 9 10

14

117

110

85 100

27

46

69

σ=0

4

17

96

31

10

113

78

52

Gallery 54 - 7

T=4

70

T = 4 Transient phase

85

123

106

7

122

55

50

77

23

98

17 = 0.9296875 (b) 7 Period-4 Attractors : ρ2 = 7128 −−

τ=4

88

τ=4

(Continued )

T=4

54 , L = 7

T = 4 Transient phase

Table 16.

τ=4

2885

1

0

0.5

φn

2

72

48

79

5

120

39

26

0

67

61

91

25

0.5

σ = 0, τ = 4

125

53

102

75

49

φn - 4

β>0

1

36

24

103

66

60

83

13

1

126

90

2886

54 , L = 8

0

141

27

(Continued )

41

73

85

255

148

146

99

74

37

108

Gallery 54 - 8

198

82

164

54

170

177

216

20 = 0.078125 (a) Period-1 Attractor : ρ 1 = 256 −−

Table 16.

251

84

254

125

245

81

1

4

131

187 238

14

54 , L = 8

17 68

68 17

(Continued )

56

238 187

224

16

64

95

239

69

215

191

21 247

2

8

7

119 221

28

34 136

136 34

112

221 119

193

32

128

42

190

127

138

175

223

Gallery 54 - 9

168

253

250

235

162

0

0.5

φn

1

0 1 2 3 4 5 6 7 8 9 10 11 12

σ = −2

0

0.5

σ = +− 2, τ = 2

σ = +2

φn - 2

24 = 0.1875 (b) Bernoulli (σ = + −− − 2, τ = 2) 2 Period-4 Attractors : ρ 2 = 2 256

Table 16.

Transient phase T=4

1

138

β>0 τ=2 τ=2

2887

133

174

10

234

88

181

78

241

31

228

91

52

67

160

54 , L = 8

(Continued )

194

87

5

117

44

218

39

248

143

114

173

26

161

97

109

147

124

199

57

13

208

40

176

213

65

93

11

182

201

62

227

156

107

134

104

Gallery 54 - 10

80

171

130

186

22

214

0

σ = −4

0.5

φn - 2

σ = +4

σ = +− 4, τ = 2

Transient 0 1 phase 2 3 20 4 5 6 7 8 9 10 11 12

0

0.5

φn

1

14 = 0.21875 (c) Bernoulli (σ = + −− − 4, τ = 2) 4 Period-4 Attractors : ρ3 = 4 256

Table 16.

T=4

1

109

β>0 τ=2 τ=2

2888

2889

185

70

233

222

172

45

77

30

243

33

12

103

152

229

94

137

118

192

18 63

220

212

210

(Continued )

111

244

86

150

166

15

249 144

6

96

9 159

230

106

105

101

246

240

79

53

167

207

120

123

178

180

25

50

132

48

157

98

151

121

3

72 252

43

115

189

211

89

90

154

110

140

145

75 83

183

135

122

Gallery 54 - 11

179

76

242

47

196

59

205

38

217

(d) Bernoulli (σ = + − 4, τ = 3) 4 Period-6 Attractors : 155

35

100

202

237

225

158

54 , L = 8

Table 16.

60

231 66

24

206

49

203

188

19

236

129

36 126

169

165

149

219

195

61

200

55

28 = 0.4375 ρ 4 = 4 256 −−

0

σ = +4

0.5

σ =+ − 4, τ = 3

0 Transient 1 2 phase 3 4 5 6 7 8 9 10 11 12

0

0.5

φn

1

σ = −4

φn - 3

(d) Bernoulli (σ = + − 4, τ = 3) 4 Period-6 Attractors (continued) :

155

1

46 197

92

29

184

226

232

58

23

71

0

0.5

φn

1

0

0.5

φn

1

0

0

153

102

1

1

2 = 0.015625 ρ 6 = 2 256 −−

Gallery 54 - 12

51

204

φn - 1

φn - 1

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

0

0.5

φn

1

0

0.5

σ = +− 2, τ = 1

σ = +3

φn - 1

σ = −3

8 = 0.0625 ρ 5 = 2 256 −−

2 Period-2 Isles of Eden :

0.5

σ = +3, τ = 1

0.5

σ = −3, τ = 1

(f) Bernoulli (σ = + − 2, τ = 1)

113

139

163

116

142

209

(e) Bernoulli (σ = −3, σ = +3, τ = 1) 2 Period-8 Isles of Eden :

(Continued )

β>0 β>0

Table 16.

T=8

T=8

54 , L = 8

T=6

β>0 τ=3 τ=3

232

184

τ=1 τ=1 1

β>0

2890

2891

8

7

5 6

4

3

L

1 1 2 3 1 1 2 1 2 1 2 3 4 5 6

i

1 1 1 6 1 7 1 2 4 4

1

attractors

2 2

2 2

Eden

Number Period-n Period-n Isles of

1 4 2 1 1 1 4 1 4 1 4 4 6 8 2

n

0 2 2 0 0 0 0 0 0 0 2 4 4 3 2

σ1

1 2 1 1 1 1 4 1 4 1 2 2 3 1 1

τ1

+ + + + + + + + + + + + + + +

β1

-2 -4 -4 -3 -2

-2 -2

σ2

2 2 3 1 1

2 1

τ2

+ + + + +

+ +

β2

ρ1 = ρ1 = ρ2 = ρ3 = ρ1 = ρ1 = ρ2 = ρ1 = ρ2 = ρ1 = ρ2 = ρ3 = ρ4 = ρ5 = ρ6 =

1 0.5 0.25 0.25 1 0.53125 0.46875 0.0703125 0.929688 6875 0.078125 0.1875 0.21875 0.4375 0.0625 0.015625

ρ

coefficient

Summary of Qualitative properties of local rule 54 extracted from Gallery 54 for Rule 54 Number Period Bernoulli Parameters Robustness ID Number of of

The basin tree diagrams of Rule 54 for L = 3, 4, . . . , 8 are exhibited in Table 16. Following a detailed analysis of these diagrams, the qualitative properties of local rule 54 extracted from basin-tree Galleries 54-1 to 54-12 of Table 16 are summarized below:

2.4.3. Highlights from Rule 54

2892

Basin tree diagrams for rule 73 .

7

73 , L = 3

0

4

2

3

6

5

3 = 0.375 ρ2 = −− 8

(b) Period-1 Isles of Eden :

Gallery 73 - 1

1

5 = 0.625 ρ1 = −− 8

(a) Period-2 Attractor :

Basin tree diagrams for Rule 73

Table 17.

15

0

13

5

10

14

8 = 0.5 ρ 1 = 16 −−

(a) Period-2 Attractor :

73 , L = 4

12

6

3

Gallery 73 - 2

7

11

9

4 = 0.25 ρ 2 = 16 −−

(b) Period-1 Isles of Eden : (σ = + − 2, τ = 1)

(c) Bernoulli

1

0

8

1

0 1 2 3

φn - 1

σ = −2

0.5

σ =+ −2, τ = 1

σ = +2

0

0.5

φn

2

2 = 0.25 ρ 3 = 2−− 16

8

4

Period-2 Isles of Eden :

T=2

(Continued )

τ=1

Table 17.

1

β>0 τ=1

2893

30

25

19

31

15

18

9

29

0

5

28

20

10

23

14

27

22

11

21

26

13

6

3

17

24

12

16

8

4

2

1

0 1 2 3 4 5 6

Transient phase

0 1 2 3 4 5 6

Transient phase

0 1 2 3 4 5 6

Transient phase

16

4

2

3 = 0.46875 (b) Period-2 Attractors : ρ2 = 5 32 −−

(Continued )

Gallery 73 - 3

7

17 = 0.53125 (a) Period-2 Attractor : ρ 1 = −− 32

73 , L = 5

Table 17.

T=2

T=2 T=2

2894

2895

61

23

31

5

53

16

17

55

73 , L = 6

(Continued )

4

7

1 28

15

9

21

49

20

57

29

18 51

0

63

30

42

39

36

35

56

60

8

59

34

2

43

40

62

47

58

45

37 33

27

11

12

41

3

54

22

24

6

44

19

13

25

48

38

26

52 50

7 = 0.328125 ρ 2 = 3 64 −−

(b) Period-1 Attractors :

Gallery 73 - 4

14

32

10

46

43 = 0.671875 (a) Period-2 Attractor : ρ 1 = −− 64

Table 17.

2896

117

62

31

107

17

34

127

4

8

73 , L = 7

(Continued )

122

113

99

124

72

0

74

79

120 2

21

42

84

71

16

68

87

60

82

1

30

61

36

64

103

18

47

94

115

Gallery 73 - 5

37

41

15

32

9

121

44 = 0.34375 (a) Period-2 Attractor : ρ 1 = 128 −−

Table 17.

98

49

88

44

22

11

69

104

52

26

13

70

35

81

2 = 0.109375 ρ 2 = 7128 −−

(b) Period-2 Isles of Eden :

2897

14

67

112

28

110

106

91

90

118

86

93

85

96

24

6

65

73 , L = 7

100

10

116

78

46

83

75

123

25

66

29

114

126

38

80

95

73

20

92

119

(Continued )

55

53

109

45

59

43

48

12

3

50

5 39

23

105

101

125

76

33

58

57

63

19

40

Gallery 73 - 6

7

97

56

111

9 = 0.4921875 ρ3 = 7128 −−

(c) Period-3 Attractors :

Table 17.

108

54

27

77

102

51

89

Isles of Eden : 7 = 0.0546875 ρ4 =128 −−

(d) Period-1

1

113

124

84

29

2

226

248

168

58

23

16

209

69

199

46

32

163

138

143

4

197

8

227

162

232

92

64

139

71

21

31

184

128

142

42

62

0

0.5

φn

1

Gallery 73 - 7

241

81

116

0

1

92

σ = −4

φn - 3 Transient phase 0 1 2 3 4 5 6 7 8 9 10 11 12 σ = +4

0.5

σ = +−4, τ = 3

10 = 0.15625 (a) Bernoulli (σ = +− 4, τ = 3) Period-6 Attractors : ρ 1 = 4 256 −−

(Continued )

T=6

73 , L = 8

Table 17.

τ=3

β>0 τ=3

2898

117

7

93

193

87

112

213

28

80

20

5

65

114

223

156

247

39

253

201

127

160

40

10

130

228

191

57

239

78

251

147

254

0

σ=0

0.5

σ = 0, τ = 3

φn - 3

5 = 0.15625 ρ 2 = 8 256 −−

Transient phase 0 1 2 3 4 5 6 7 8 9 10 11 12

0

0.5

φn

Gallery 73 - 8

234

14

186

131

174

224

171

56

1

(b) Bernoulli (σ = 0, τ = 3) Period-3 Attractors :

(Continued )

T=3

73 , L = 8

Table 17.

1

251

β>0 τ=3

2899

2900

15

105

111

195

90

219

240

150

246

60

165

189

96

24

6

129

(Continued )

118

106

9

101

154

66

89

110

249

103

79

47

203 126 211 155

217

242 144 159 244 166 230

86

169

36

188 231 61 185

157

192

48

12

3

212

18

202

53

132

178

77

33

172

83

72

233

229

94 243 158 220

206

151 252 167 55

179

205

63

236

121 207 122 115

59

Gallery 73 - 9

30

210

222

135

180

183

225

45

237

120

75

123

43

12 = 0.375 (c) Period-3 Attractors : ρ 3 = 8 256 −−

149

73 , L = 8

Table 17.

T=3

T=3

Transient phase 0 1 2 3 4 5 6 7 8 9 10 11 12

Transient phase 0 1 2 3 4 5 6 7 8 9 10 11 12

252

121

2901

107

218

182

173

99

216

54

141

104

11

26

194

134

176

161

44

73 , L = 8

98

196

Gallery 73 - 10

208

22

70

52

145

35 198

177

133

76

49

214

181

13

152

200

38

97

100

67

88

50

108

27

25

6 = 0.1875 ρ 4 = 8256 −−

19

109

91

Period-2 Attractors :

(Continued )

137

140

(d)

Table 17.

204

102

51

153

4 = 0.015625 ρ 5 = 256 −−

(e) Period-1 Isles of Eden :

146

37

85

221

73

0

119

164

0

0.5

φn

1

0

0.5

σ =+ −2, τ = 1

245

95

175

250

φn - 1

17

136

0 1 2 3 4 5 6 7 8 σ = +2

1

68

34

125

215

190

235

17

6 = 0.046875 ρ 7 = 2 256 −−

(g) Bernoulli (σ = +− 2, τ = 1) Period-2 Attractors :

(Continued )

Gallery 73 - 11

82

41

238

148

170

187

16 = 0.0625 ρ 6 = 256 −−

(f) Period-2 Attractor :

74

255

73 , L = 8

Table 17.

T=3

β>0

τ=1 σ = −2

τ=1

2902

2903

8

7

5 6

4

3

L

i 1 2 1 2 3 1 2 1 2 1 2 3 4 1 2 3 4 5 6 7

1 2

4 8 8 8

7

1 5 1 3 1

1

1

attractors

4

7

7

4 2

3

Eden

Number Period-n Period-n Isles of

2 1 2 1 2 2 2 2 1 1 2 3 1 6 3 3 2 1 2 2

n 0 0 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 2

σ1 1 1 1 1 1 1 2 1 1 1 2 3 1 3 3 3 2 1 1 1

τ1 + + + + + + + + + + + + + +

β1

-2

-4

-2

σ2

1

3

1

τ2

+

+

+

β2

ρ1 = ρ2 = ρ1 = ρ2 = ρ3 = ρ1 = ρ2 = ρ1 = ρ2 = ρ1 = ρ2 = ρ3 = ρ4 = ρ1 = ρ2 = ρ3 = ρ4 = ρ5 = ρ6 = ρ7 =

0.625 0.375 0.5 0.25 0.25 0.53125 0.46875 0.671875 0.328115 0.34375 0.109375 0.492188 1875 0.0546875 0.15625 0.15625 0.375 0.1875 0.015625 0.0625 0.046875

ρ

coefficient

Summary of Qualitative properties of local rule 73 extracted from Gallery 73 for Rule 73 Number Bernoulli Parameters Robustness Period ID Number of of

The basin tree diagrams of Rule 73 for L = 3, 4, . . . , 8 are exhibited in Table 17. Following a detailed analysis of these diagrams, the qualitative properties of local rule 73 extracted from basin-tree Galleries 73-1 to 73-11 of Table 17 are summarized below:

2.4.4. Highlights from Rule 73

2904

Basin tree diagrams for rule 90 .

3

0

90 , L = 3

4

7

2

Gallery 90 - 1

5

2 =1 (a) Period-1 Attractors : ρ 1 = 4 −− 8

6

Basin tree diagrams for Rule 90

Table 18.

1

2905

12

11

6

14

90 , L = 4

(Continued )

9

1

15

10

3

4

0

8

7

0 1 2 3

2

13

0 1 2 3

Gallery 90 - 2

Transient phase

5

Transient phase

16 = 1 (a) Period-1 Attractor : ρ 1 = −− 16

Table 18.

6

Transient phase

4

0 1 2 3

7

6

9

Transient phase

22

16

15

0 1 2 3 4 5 6 7 8

25

90 , L = 5

T=3

2906

(Continued )

11

8

3

20

22

23

21

17

10 27

26

24

5 29

7

13

1

12

18 30

0

31

2 (b) Period-1 Attractor : ρ 2 = −− 32 = 0.0625

14

2

Gallery 90 - 3

28

4

6 = 0.9375 (a) Period-3 Attractors : ρ 1 = 5 −− 32

Table 18.

19

2907

45

0

7

56

21

63

90 , L = 6

(Continued )

18

42

9

35

14

28

36

Gallery 90 - 4

54

27

49

4 = 0.25 (a) Period-1 Attractors : ρ 1 = 4−− 64

Table 18.

58

48

52

σ = 3, τ = 1 OR σ = −3, τ = 1

33

11

σ = 3, τ = 1 OR σ = −3, τ = 1

37

26

OR σ = 3, τ = 1

σ = −3, τ = 1

47

16

51

57

40

90 , L = 6

(Continued )

22

30

43

15

32

5

41

3

62

1

53

31

12

38

6

44

2

61

39

25

34

19

17

23

60

20

10

Transient phase

Transient phase

Transient phase

σ = −3

0 1 2 3 4 5 6

σ = −3

0 1 2 3 4 5 6

σ = −3

0 1 2 3 4 5 6

σ = +3

σ = +3

σ = +3

τ=1 τ=1

33

τ=1 τ=1

3

τ=1 τ=1

47

Gallery 90 - 5

13

24

σ = 3, τ = 1 OR σ = −3, τ = 1

29

OR σ = 3, τ = 1

50

8

46

σ = −3, τ = 1

55

4

59

σ = 3, τ = 1 OR σ = −3, τ = 1

0

0.5

φn

1

0

0.5

σ =+ − 3, τ = 1

8 = 0.75 (b) Bernoulli (σ = + − 3, τ = 1) Period-2 Attractors : ρ2 = 6 −− 64

Table 18.

T=2

T=2

T=2

2908

φn - 1

1

β>0

103

37

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

60

24

67

102

90

91

(Continued )

1

79

74

126

36

120

48

25

66

61

7

77

53

55

1

50

5

122

113

96

2

14

27

106

62

42

123

17

99

65

100

10

117

Gallery 90 - 6

125

72

31

21

110

28

54

85

93

4

119

34

107

73

20

0 1 2 3 4 5 6 7 8

14 = 0.765625 (a) Bernoulli (σ = 0, τ = 7) Period-7 Attractors : ρ 1 = 7128 −−

Table 18.

Transient phase

T=7

90 , L = 7

Transient phase

T=7

2909

8

62

124

84

0 1 2 3 4 5 6 7 8

71

3

56

108

43

59

(Continued )

121

56

41

111

68

6

15

19

40

87

112

89

86

118

16

38

80

47

30

12

32

97

51

45

63

18

0

0.5

φn

1

76

33

94

Gallery 90 - 7

95

9

115

82

109

0

0 1 2 3 4 5 6 7 8

0.5

σ = 0, τ = 7

64

(a) Bernoulli (σ = 0, τ = 7) Period-7 Attractors (continued)

Table 18.

Transient phase

T=7

90 , L = 7

Transient phase

T=7

2910

φn - 7

1

64

49

44

σ = −5

0 1 2 3 4 5 6 7 8

58

78

σ = +2

69

τ=1

1

0

0.5

φn

τ=1

88

105

83

11

Table 18.

(Continued )

0

22

29

88

0.5

φn - 1 1

81

70

σ = +5

0 1 2 3 4 5 6 7 8

75

46

σ = −2

τ=1

τ=1

52

52

114

57

26

0

0.5

φn

1

0

13

23

35

0.5

φn - 1

σ = −2, σ = +5, τ = 1

92

101

104

Gallery 90 - 8

σ = +2, σ = −5, τ = 1

39

116

98

14 = 0.21875 ρ2 = 2128 −−

(b) Bernoulli (σ = + −2 , σ = −+5 , τ = 1) Period-7 Attractors :

β>0

90 , L = 7

Transient phase

T=7

Transient phase

T=7

2911

1

β>0

0

127

= 0.015625

2 ρ3 = 128 −−

(c) Period-1 Attractor :

167

(a) Period-1 Attractor : ρ = 256 −− = 1 256

(Continued )

148 1 88 193 107 133 122 242 227 13 182 62 168 208 47 145 73 156 99 87 196 110 28 138 117 253 54 201 2 230 179 59 223 32 76 119 235 70 19 80 5 190 25 213 185 128 236 65 175 127 49 34 136 42 100 169 154 20 250 247 86 207 155 252 206 101 8 93 162 139 116 3 237 184 85 67 22 48 222 33 159 188 71 135 120 202 96 233 39 45 18 210 232 189 52 221 53 97 114 66 204 240 158 203 7 141 165 23 219 216 142 82 90 113 37 153 36 248 173 15 112 218 249 143 6 83 172 255 0 174 212 81 129 140 115 251 195 126 4 102 43 217 38 150 246 64 21 95 10 105 9 163 191 92 60 51 234 160 197 58 55 17 24 98 111 144 225 245 12 30 231 75 180 157 77 89 200 46 63 192 178 170 29 166 243 69 16 106 149 186 123 209 214 226 131 72 239 40 198 132 183 124 50 68 238 41 103 125 147 244 152 205 1 130 108 11 161 94 215 57 187 0 84 35 56 199 1 26 171 254 146 228 109 2 118 220 14 27 78 177 79 229 3 91 137 194 176 44 61 4 31 164 241 104 151 121 211 5 224 74 6 134 181

90 , L = 8

Table 18.

173

Transient phase

Gallery 90 - 9

Transient phase

0 1 2 3 4 5 6

Transient phase

0 1 2 3 4 5 6

Transient phase Transient phase

2912

0 1 2 3 4 5 6

0 1 2 3 4 5 6

109

227

129

189

2913

8

7

6

5

3 4

L

1 1 1 2 1 2 1 2 3 1

i

4 1 5 1 4 6 7 2 1 1

attractors

Eden

Number Period-n Period-n Isles of

1 1 3 1 1 2 7 7 1 1

n

0 0 0 0 0 3 0 _2 + 0 0

σ1

1 1 3 1 1 1 1 1 1 1

τ1

+ + + + + + + + + +

β1

1 1

_ +5

τ2

-3

σ2

+

+

β2

ρ1 = 1

ρ3 = 0.015625

ρ2 = 0.21875

ρ1 = 0.765625

ρ2 = 0.75

ρ1 = 0.25

ρ2 = 0.0625

ρ1 = 0.9375

ρ1 = 1

ρ1 = 1

ρ

coefficient

Summary of Qualitative properties of local rule 90 extracted from Gallery 90 for Rule 90 Number Bernoulli Parameters Robustness ID Number Period of of

The basin tree diagrams of Rule 90 for L = 3, 4, . . . , 8 are exhibited in Table 18. Following a detailed analysis of these diagrams, the qualitative properties of local rule 90 extracted from basin-tree Galleries 90-1 to 90-9 of Table 18 are summarized below:

2.4.5. Highlights from Rule 90

2914

Basin tree diagrams for rule 105 .

105 , L = 3

2

4

1 7

Gallery 105 - 1

0

8 =1 (a) Period-2 Attractor : ρ 1 = −− 8

6

3 5

Basin tree diagrams for Rule 105

Table 19.

3 6

0 1 2 3 4

σ = −2

0 1 2 3 4

σ = −2

8 4

σ = +2

14 τ=1 τ=1

τ=1 τ=1

1

σ = +2

2 1

0

0.5

φn

1

0

9

φn - 1

11

13

1

7

0

15

(b) Period-2 Isles of Eden : 2 = 0.25 ρ2 = 2 16 −−

0

0.5

φn

1

0

5

0.5

τ=1

τ=1

10

φn - 1

σ = +1

σ = +− 1, τ = 1

σ = −1

0 1 2 3 4

10

Bernoulli (σ = + − 1, τ = 1) shift :

Gallery 105 - 2

0.5

σ = +− 2, τ = 1

14

2 = 0.5 ρ 3 = 4 16 −−

β>0

(c) Period-2 Isles of Eden: Bernoulli (σ = + − 2, τ = 1) shifts :

12

1 (a) Period-1 Isles of Eden : ρ 1 = 4 −− 16 = 0.25

(Continued )

T=2

Table 19.

1

β>0

105 , L = 4

T=2

T=2

2915

28

29

30

8

11

23

20

5

7

3

25

0

0.5

φn

1

0

16

0.5

σ = 0, τ = 3

6

4

φn - 3

1

16

0

31

2 = 0.0625 ρ2 = 32 −−

(b) Period-2 Isle of Eden :

σ=0 β0

T=4 β>0

T=4 β>0

σ = −4

σ = −4

σ = −4

160

240

τ=2 τ=2

216

τ=2 τ=2 τ=2 τ=2

2925

2926

0 1 2 3 4

σ = −2

221

34

119

136

σ = +2

238

17

τ=1

τ=1

221

187

68

0

0.5

σ = +− 2, τ = 1

φn - 1

σ = +2

τ=1

τ=1

34

1

0 1 2 3 4

170

0.5

φn - 1

τ=1

τ=1

170

1

255

0

(f) Period-2 Isle of Eden 2 = 0.0078125 ρ6 = 256 −−

0

0.5

0

σ = +1 σ = −1 σ =+ − 1, τ = 1 1 φn

Gallery 105 - 13

0

0.5

φn

1

σ = −2

0 1 2 3 4

85

Period-2 Isle of Eden 2 = 0.0078125 ρ5 = 256 −−

(d) 2 Bernoulli (σ = + − 2, τ = 1) Period-2 Isles of Eden 2 = 0.03125 ρ 4 = 4 256 −−

T=2

T=2

153

51

102

β>0

204

(e) Bernoulli (σ = + − 1, τ = 1)

4 = 0.015625 Period-1 Isles of Eden ρ 3 = 256 −−

(Continued )

T=2

105 , L = 8 (c)

Table 19.

β>0

2927

8

7

6

5

4

3

L

1 1 2 3 1 2 1 2 1 2 3 1 2 3 4 5 6

i

1 1

1

attractors

1 7 2 48 12 4 4 1 1

4 2 4 5 1

Eden

Number Period-n Period-n Isles of

2 1 2 2 6 2 2 2 2 14 14 4 4 1 2 2 2

n

0 0 1 2 0 0 0 1 0 0 1 0 4 0 2 1 0

σ1

1 1 1 1 3 1 1 1 1 7 2 1 2 1 1 1 1

τ1

_ + + + _ _ _ + _ + + + + + + + _

β1

1 1

2

-4 -2 -1

2

1

1 1

τ2

-1

-1

-1 -2

σ2

+ +

+

+

+

+ +

β2

ρ1 = ρ1 = ρ2 = ρ3 = ρ1 = ρ2 = ρ1 = ρ2 = ρ1 = ρ2 = ρ3 = ρ1 = ρ2 = ρ3 = ρ4 = ρ5 = ρ6 =

1 0.25 0.25 0.5 0.9375 0.0625 0.5 0.5 0.015625 0.765625 0.21875 0.75 0.1875 0.015625 0.03125 0.0078125 0.0078125

ρ

coefficient

Summary of Qualitative properties of local rule 105 extracted from Gallery 105 for Rule 105 Number Bernoulli Parameters Robustness Period ID Number of of

The basin tree diagrams of Rule 105 for L = 3, 4, . . . , 8 are exhibited in Table 19. Following a detailed analysis of these diagrams, the qualitative properties of local rule 105 extracted from basin-tree Galleries 105-1 to 105-13 of Table 19 are summarized below:

2.4.6. Highlights from Rule 105

2928

Basin tree diagrams for rule 122 .

122 , L = 3

0

1

3

2

=1 ρ 1 = −− 8

Gallery 122 - 1

6

7

5

(a) Period-1 Attractor :

4

Basin tree diagrams for Rule 122 8

Table 20.

2929

0

0

0.5

β>0

φn - 1 1

0

2

0.5

1

4

1

β>0

φn - 1

10

σ = +−1, τ = 1

8

5

6 = 0.375 ρ 2 = 16 −−

Period-2 Attractor :

Gallery 122 - 2

0

0.5

0.5

1

φn

σ =+ −2, τ = 1

11

14

(Continued )

(b) Bernoulli (σ = + − 1, τ = 1)

φn

1

7

13

2 = 0.25 ρ1 = 2 16 −−

(a) Bernoulli (σ = + − 2, τ = 1) Period-2 Isles of Eden :

122 , L = 4

Table 20.

3

6

0

15

9

12

6 = 0.375 ρ 3 = 16 −−

(c) Period-1 Attractor :

2930

10

5

18

2

1

20

8

4

9

12

13

24

26

17

21

3

11

6

22

30

29

27

23

15

19

7

14

28

25

Transient phase

0

0.5

σ = 0, τ = 2

σ=0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0

0.5

φn

1

(Continued )

φn - 2

Gallery 122 - 3

T=2

16

(a) Bernoulli (σ = 0, τ = 2) 6 = 0.9375 Period-2 Attractors : ρ 1 = 5 32 −−

122 , L = 5

Table 20.

1

8

τ=2

0

31

2 = 0.0625 ρ 2 = −− 32

(b) Period-1 Attractor :

0

0.5

φn

1

0

0.5

σ = +−3, τ = 1

30

60

24

12

57

48

φn - 1

β>0 1

33

3

39

51

6

15

4 = 0.1875 ρ 1 = 3 64 −−

Period-2 Attractors :

σ = −3

0 1 2 3 4 5 6 7

σ = +1

11

29

58

τ=1 τ=1

32

41

19

55

0

47

22

25

28

9

56

32

Gallery 122 - 4

σ = +3

48

38 = 0.59375 ρ 3 = −− 64

Attractor :

(c) Period-1

σ = −1

0 1 2 3 4

Transient phase

14 = 0.21875 ρ 2 = −− 64

Transient phase

(a) Bernoulli (σ = +− 3, τ = 1)

(Continued )

(b) Bernoulli (σ = +− 1, τ = 1) Period-2 Attractor :

T=2

122 , L = 6

T=2

Table 20.

τ=1

τ=1

2931

17

13

62

18

50

63

5

42

20

35

54

45

7

61

26

2

8

1

43

36

27

23

34

21

40

16

53

49

14

46

10

44

31

59

4

38

37

52

2932

0

65

39

62

20

83

57

10

119

100

4

99

93

73 114

8

111 56 43

84 68

40

16

113

14

106

96 31

110

123

21

17

104

98

49

27

53

50

92

125

7

29

26

61

51

48

120

105

55

88

102

89

47

79

36

90 37

32

1

46

91

60

23

58

116

24

103

64

76

12

109

33

63

30

115

0 1 2 3 4 5 6 7 8

ρ1 = 1

(a) Period-1 Attractor :

25

18

82

126

13

45

97

66

67

44

94

9

80

38

70

41

22

86

112

95

75

Gallery 122 - 5

2

5

77 122

72 74

52

117

127

6

(Continued )

15 118

121

11

35

34 81 87 69 42 85 108 107 28 54

59

78 101 19

124 71

3

122 , L = 7

Table 20.

Transient phase

1

2933

251

14

253

7

216 175 19 143 83 70 86

55 208 233

88 188 103

113 116 218

37 117

(Continued )

87 82

94 179

54 196 235 227 212 145 149

127

193

73 93

182

47 217

29

92

205 52 122

22

254 108 137 215 199 169 35 43

155 104 244

44

146 186 184 58 109 131

Gallery 122 - 6

23

71

100 53 248 49 250 141 13 115 158 223 118 203 133 112 173

101

174 164

142 46

200 106 241 98 245 27 26 230 61 191 236 151 11 224 91

202

239

134 185 79

50 154 125 152

247

190

77 76

67 220 167

25

157 242 97 28 107 209 197 213 148

99

62

89

59 229 194 56 214 163 139 171 41

198

124

178

40 = 0.625 (a) Bernoulli (σ = + −− − 4, τ = 3) Period-6 Attractors : ρ1 = 4 256

177 95 38 31 166 140 172

110 161 211

176 121 206

226 232 181

74 234

122 , L = 8

Table 20.

0

σ = −4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.5

σ = +−4, τ = 3

Transient phase

0

0.5

φn

1

1

133

σ = +4

φn - 3

66

72

(Continued )

165

180

219

123

90

36

231

60

102

204

135

252

189

24

195

0

120

129

126

207

48

3

183

75

132

255

(b) Period-1 Attractor :

Table 20.

Gallery 122 - 7

(a) Bernoulli (σ = +− 4, τ = 3) Period-6 Attractors (continued) :

122 , L = 8

T=6

β>0 τ=3

τ=3

2934

192

30

15

96 6

243

249

12

33

210

237

63

225

51

153

240

159

246

105

144

222

111

45

150

54 = 0.2109375 ρ2 = 256 −−

18

9

2935

17

1

21

81

0 1 2 3 4 T=2 5 σ = −1

10

160

Transient phase

4

64

68

40

σ = +1

69

170

84

130

16

138

85

168

2

0

0.5

φn

1

0

0.5

σ =+ −1, τ = 1

128

8

1

β>0

φn - 1

T=2

Transient phase

114

39

228

78

0

0.5

φn

1

0

0.5

φn - 1

1

β>0

τ=1 τ=1

57

201

156

147

57

σ = +2

119

238

σ =+ −2, τ = 1

0 1 2 3 σ = −2

221

187

(d) Bernoulli (σ = +− 2, τ = 1) 6 = 0.046875 Period-2 Attractors : ρ4 = 2 256 −−

(Continued )

Gallery 122 - 8

34

65

20

136

162

42

τ=1 τ=1

1

5

80

32

(c) Bernoulli (σ = + − 1, τ = 1) 30 = 0.1171875 Period-2 Attractor ρ 3 = 256 −−

122 , L = 8

Table 20.

2936

8

7

6

5

4

3

L

1 1 2 3 1 2 1 2 3 1 1 2 3 4

i

1 1 5 1 3 1 1 1 4 1 1 2

1

attractors

2

Eden

Number Period-n Period-n Isles of

1 2 2 1 2 1 2 2 1 1 6 1 2 2

n

0 2 1 0 0 0 3 1 0 0 4 0 1 2

σ1

1 1 1 1 2 1 1 1 1 1 3 1 1 1

τ1

+ + + + + + + + + + + + + +

β1

-1 -2

-4

-3 -1

-2 -1

σ2

1 1

3

1 1

1 1

τ2

+ +

+

+ +

+ +

β2

ρ1 = ρ1 = ρ2 = ρ3 = ρ1 = ρ2 = ρ1 = ρ2 = ρ3 = ρ1 = ρ1 = ρ2 = ρ3 = ρ4 =

1 0.25 0.375 0.375 0.9375 0.0625 0.1875 0.21875 0.59375 1 0.625 0.210938 9375 0.117188 1875 0.046875

ρ

coefficient

Summary of Qualitative properties of local rule 122 extracted from Gallery 122 for Rule 122 Number ID Number Bernoulli Parameters Robustness Period of of

The basin tree diagrams of Rule 122 for L = 3, 4, . . . , 8 are exhibited in Table 20. Following a detailed analysis of these diagrams, the qualitative properties of local rule 122 extracted from basin-tree Galleries 122-1 to 122-8 of Table 20 are summarized below:

2.4.7. Highlights from Rule 122

2937

Basin tree diagrams for rule 126 .

126 , L = 3

0

1

7

Gallery 126 - 1

5

4

6

3

2

8 =1 (a) Period-1 Attractor : ρ 1 = −− 8

Basin tree diagrams for Rule 126

Table 21.

2938

0

5

15 3

9

T=2

Transient phase

6

12

10

8 = 0.5 ρ 1 = 16 −−

(a) Period-1 Attractor :

126 , L = 4

σ = +2

τ=1

τ=1

4

1

0

0.5

φn

8

4

0

7

11

0.5

σ =+ −2, τ = 1

13

14

φn - 1

1

β>0

2

1

(b) Bernoulli (σ = +− 2, τ = 1) 4 = 0.5 Period-2 Attractors ρ 2 = 2 16 −−

(Continued )

Gallery 126 - 2

σ = −2

0 1 2 3 4

Table 21.

10

T=2

Transient phase

0

11

13

20

0 1 2 3 4

5

31

26

12 = 0.375 ρ 1 = −− 32

(a) Period-1 Attractor :

126 , L = 5

21

9

12

(Continued )

25

15

23

28 8

16

27

29

30

14

7

19

4

2

1

6

3

17

24

12

4 = 0.625 Period-2 Attractors ρ2 = 5 32 −−

(b) Bernoulli (σ = 0, τ = 2)

Table 21.

Gallery 126 - 3

σ=0

18

22

τ=2

2939

0

0.5

φn

1

0

0.5

σ =+ −3, τ = 1

30

60

24

12

57

48

φn - 1 1

β>0

51

39

15

σ = +3

0 1 2 3 4 5 6 7 σ = −3

21

42

τ=1

τ=1

6

63

(Continued )

0

43

1

2

20

40

62

9

61

23

19

34

55

0 Transient 1 2 phase 3 4 5 6

29

63

38 14

27

52 49

8

37 36 11 13 28

41 54 35

25

50

7

58 16

56 18 26 22 44 45

47

5

4

17

53

10

29

46

59

31

32

52 = 0.8125 (b) Period-1 Attractor : ρ 2 = 64 −−

Table 21.

Gallery 126 - 4

Transient phase

33

3

6

4 = 0.1875 ρ 1 = 3 64 −−

Period-2 Attractors :

(a) Bernoulli (σ = + − 3, τ = 1)

126 , L = 6

T=2

2940

2941

59

(Continued )

Gallery 126 - 5

101

128 ρ1 = 128 −− = 1

(a) Period-1 Attractor :

69 70 68 58 57 107 90 16 20 111 35 56 40 119 28 62 65 98 87 11 38 45 118 124 13 117 53 99 19 9 42 71 3 10 96 21 95 116 46 31 54 76 108 4 113 32 114 14 78 112 27 84 74 17 123 73 127 89 15 121 80 110 81 6 47 77 86 48 120 49 51 41 25 106 22 122 37 102 79 26 5 0 30 63 82 7 100 60 97 18 109 115 50 12 24 2 64 105 125 103 33 23 39 101 67 43 94 72 85 126 0 61 66 104 1 88 44 2 55 52 1 3 36 4 83 5 91 75 6

92 29 8 93 34

126 , L = 7

Table 21.

Transient phase

2942

55

103

87

110

206

174

10

78

5

39

168

200

152

81

145

49

250

143

253

2

245

31

251

4

126 , L = 8

(Continued )

216

7

177

14

112

141

224

27

32

223 137

140

138

114

80

19

25

21

228

160

117

118

115

234

236

230

205

217

213

155

179

171

65

201

42

50

38

130

147

84

100

76

190

227

127

128

125

199

254

1

Gallery 126 - 6

248

175

64

191

241

95

54

193

108

131

28

99

56

198

8

247

62

235

16

239

124

215

70

69

98

35

162

156

20

196

57

40

93

157

220

186

59

185

28 = 0.4375 (a) Bernoulli (σ = + −− − 4, τ = 3) Period-6 Attractors : ρ 1 = 4 256

Table 21.

0

σ = −4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.5

σ = +−4, τ = 3

Transient phase

0

0.5

φn

1

1

110

σ = +4

φn - 3

(Continued )

0

0.5

φn

1

0

0.5

φn - 1

1

β>0

238

68

σ = +−2, τ = 1

221

136

T=2

σ = −2

τ=1

τ=1

17

σ = +2

17

187

Transient phase

34

119

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

4 = 0.03125 ρ 2 = 2 256 −−

(b) Bernoulli (σ = +− 2, τ = 1) Period-2 Attractors :

Table 21.

Gallery 126 - 7

Period-6 Attractors (continued) :

(a) Bernoulli (σ = +− 4, τ = 3)

126 , L = 8

T=6

β>0 τ=3 τ=3

2943

2944

0

113

142

211

122

209

46

44

88

194

36

48

3

66

232

67

233

26

229

192

30

15

96

71

237

63

225

51

153

240

159

246

163

18

12

6

9

22

97

222

161

208

111

176

184

33

243

94

47

249

144

11

92

79

139

116

158

58

197

173

91

182

202

149

43

214

109

101

86

107

53

74

75

150

45

77

154

181

218

178

41 148

89

172

136 ρ3 = 256 −− = 0.53125

Gallery 126 - 8

85

13

244

242

255

188

61

60

102

231 189

24

195

204

(Continued )

(c) Period-1 Attractor :

170

151

104

167

135

252

183

29

120

72

129

126

203

121

207

132

23

219

52

134

123

133

226

126 , L = 8

Table 21.

165

90

166

106

37

82

210

105

180

83

169

212

146

73

164

2945

8

7

6

5

4

3

L attractors

1 1 2 1 5 3 1 1 4 2 1

i

1 1 2 1 2 1 2 1 1 2 3

Eden

Number Period-n Period-n Isles of

1 1 2 1 2 2 1 1 6 2 1

n

0 0 2 0 0 3 0 0 4 2 0

σ1

1 1 1 1 2 1 1 1 3 1 1

τ1

+ + + + + + + + + + +

β1

-4 -2

-3

-2

σ2

3 1

1

1

τ2

+ +

+

+

β2

ρ3 = 0.53125

ρ2 = 0.03125

ρ1 = 0.4375

ρ1 = 1

ρ2 = 0.8125

ρ1 = 0.1875

ρ2 = 0.625

ρ1 = 0.375

ρ2 = 0.5

ρ1 = 0.5

ρ1 = 1

ρ

coefficient

Summary of Qualitative properties of local rule 126 extracted from Gallery 126 for Rule 126 Number ID Number Bernoulli Parameters Robustness Period of of

The basin tree diagrams of Rule 126 for L = 3, 4, . . . , 8 are exhibited in Table 21. Following a detailed analysis of these diagrams, the qualitative properties of local rule 126 extracted from basin-tree Galleries 126-1 to 126-8 of Table 21 are summarized below:

2.4.8. Highlights from Rule 126

2946

Basin tree diagrams for rule 146 .

7

1 = 0.125 ρ 1 = −− 8

(a) Period-1 Isle of Eden :

146 , L = 3

Gallery 146 - 1

6

0

5

1

3

2

4

7 = 0.875 (b) Period-1 Attractor : ρ 2 = −− 8

Basin tree diagrams for Rule 146

Table 22.

2947

10

1

8

4

2

11

13

14

7

1 = 0.0625 ρ2 = −− 16

15

(b) Period-1 Isle of Eden :

0

5

11 = 0.6875 ρ1 = −− 16

(a) Period-1 Attractor :

146 , L = 4

(Continued )

0

0.5

σ =+ −2, τ = 1

6

3

T=2

1

β>0

φn - 1

Gallery 146 - 2

0

0.5

φn

1

9

12

σ = −2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 = 0.25 ρ3 = 2 −− 16

σ = +2

τ=1

τ=1

12

(c) Bernoulli (σ = +− 2, τ = 1) Period-2 Isles of Eden :

Table 22.

2948

21

11

7

26

0

22

13

25

1 = 0.03125 ρ2 = 32 −−

31

(b) Period-1 Isle of Eden :

14

28

11 = 0.34375 ρ1 = −− 32

(a) Period-1 Attractor :

146 , L = 5

16

8

4

2

1

9

20

10

5

18

6

3

17

24

12

15

23

27

29

30

4 = 0.625 Period-2 Attractors ρ3 = 5 32 −−

Gallery 146 - 3

19

(Continued )

(c) Bernoulli (σ = 0, τ = 2)

Table 22.

0

0.5

φn

1

0

0.5

34

5

10

1

β>0

φn - 1

20

40

17

σ =+ −3, τ = 1

8

16

58

29

32

53

43

23

46

σ = +3

0 1 2 3 4 5 6 7 σ = −3

τ=1

τ=1

58

41

25

49

7

28

6

42

12

9

57

51

18

15

0

30

13 11

48

54

45

33

26 22

1 = 0.015625 ρ3 = −− 64

55

14

39

27

60

21

35

(c) Period-1 Isle of Eden :

59

47

62

50

19

45 = 0.703125 ρ 2 = 64 −−

Gallery 146 - 4

Transient phase

1

2

4

(Continued )

(b) Period-1 Attractor :

Table 22.

6 = 0.28125 Period-2 Attractors : ρ 1 = 3 64 −−

(a) Bernoulli (σ = + − 3, τ = 1)

146 , L = 6

T=2

2949

3

61

44

37

24

63

36

56

31

38

52

2950

50

29

91

77

55

114

7

103

79

78

1

44

74

104

2

101

100

25

67

52

24

36

57 109

64

33

97

94

63

86

95

32

6

92 76

11

80

15

3

93

23

62

83

19

46

59

38

124

0 1 2 3 4 5 6 7 8

124

127

1 = 0.0078125 ρ 2 = 128 −−

(b) Period-1 Isle of Eden :

Transient phase

127 ρ 1 = 128 −− = 0.9921875

(a) Period-1 Attractor :

105

56

112

89

16

69

121

41

118

13

47

40

70

84

98 108

87

8

68

111 71

51

9

43

34

20

116

(Continued )

Gallery 146 - 5

39

65

35

28 42

85

82

30 12

26

45

99

119

0

54

73

107

49

75

106

17

10

126 18

60

90

53

115

22

61

120

72

102

37

66

96

81

4

110

31

123 113

117

21

58

122 125

48

5

88

27

14

146 , L = 7

Table 22.

2951

111

6

150

132

75

159

89

135

38

169

9

(Continued )

98

180

48

123

120

72

77

49

240

202

252

105

144

200

185 22

103 26

145

115 44

206 52

196

53

219

129

165

237

192

243

43

33

210

231

86

225

137

106

66

195

28 = 0.4375 ρ 1 = 4 256 −−

Gallery 146 - 6

134 217

133 110

50

13 179

11 220

149 100 154 96

246

249

(a) Bernoulli (σ = + − 4, τ = 3) Period-6 Attractors :

3 212 35 172 183 19 207

230 88

157 104

70

205 176

59 208

15

146 , L = 8

Table 22.

152

45

12

222

30

18

83

76

60

178

63

90

36

161 118

97 155

140

67 236

194 55

101 25 166 24

189

126

0

Transient phase

0

0.5

φn

1

σ = −4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.5

σ =+ −4, τ = 3

1

252

σ = +4

φn - 3

(a) Bernoulli (σ = + − 4, τ = 3) Period-6 Attractors (continued) :

146 , L = 8

T=6

β>0 τ=3 τ=3

2952

(Continued )

0

0.5

σ =+ −2, τ = 1

153

51

φn - 1

1

β>0

T=2

σ = −2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

255

σ = +2

τ=1

τ=1

204

1 = 0.00390625 ρ3 = 256 −−

(c) Period-1 Isle of Eden :

Gallery 146 - 7

0

0.5

φn

1

102

204

2 = 0.015625 ρ 2 = 2 256 −−

(b) Bernoulli (σ = +− 2, τ = 1) Period-2 Isles of Eden :

Table 22.

2953

223

143

190

227

247

250

168

5 37

54

65

244

253

93 8

117

61

85

34

20

95

175

92

94

211

40

68

122

251

232

69

1

254

215

171

142

56

226

174

245

242

58

84

46

47

131

233

4 158 146 130

108

10

74

81

203 139 224

124

186

41

198

238

187

177

16

170

17

160

64

234

241

(Continued )

125

239

199

31

191

156

214

78

181

107

57

0 1 2 3 4 5 6 7 8 9 10

39

218

0

173

114

109 147

182

91

228

201

127

139 ρ4 = 256 −− = 0.54296875

(d) Period-1 Attractor :

Gallery 146 - 8

138 7

0

27

164

141 82

167

21

184 14 163 188

209

80

136

32

216

221

119

99

148

248

112 197 229

2

62

162

116

73 79

128

42

29

121

87

113

28

71

213

193

151

23

235

127

146 , L = 8

Table 22.

Transient phase

2954

8

7

6

5

4

3

L

1 2 1 2 3 1 2 3 1 2 3 1 2 1 2 3 4

i

1

4

1

5 3 1

1

1 1

attractors

2 1

1

1

1

1 2

1

Eden

Number Period-n Period-n Isles of

1 1 1 1 2 1 1 2 2 1 1 1 1 6 2 1 1

n

0 0 0 0 2 0 0 0 3 0 0 0 0 4 2 0 0

σ1

1 1 1 1 1 1 1 2 1 1 1 1 1 3 1 1 1

τ1

+ + + + + + + + + + + + + + + + +

β1

-4 -2

-3

-2

σ2

3 1

1

1

τ2

+ +

+

+

β2

ρ1=0.125 ρ2=0.875 ρ1=0.6875 ρ2=0.0625 ρ3=0.25 ρ1=0.34375 ρ2=0.03125 ρ3=0.625 ρ1=0.28125 ρ2=0.703125 ρ3=0.015625 ρ1=0.992188 875 ρ2=0.0078125 ρ1=0.4375 ρ2=0.015625 ρ3=0.00390625 ρ4=0.542969 6875

ρ

coefficient

Summary of Qualitative properties of local rule 146 extracted from Gallery 146 for Rule 146 Number Period ID Number Bernoulli Parameters Robustness of of

The basin tree diagrams of Rule 146 for L = 3, 4, . . . , 8 are exhibited in Table 22. Following a detailed analysis of these diagrams, the qualitative properties of local rule 146 extracted from basin-tree Galleries 146-1 to 146-8 of Table 22 are summarized below:

2.4.9. Highlights from Rule 146

2955

Basin tree diagrams for rule 150 .

2

150 , L = 3

4

1 7

0

Gallery 150 - 1

4 (a) Period-1 Attractors : ρ 1 = 2 −− 8 =1

6

3 5

Basin tree diagrams for Rule 150

Table 23.

0

0.5

φn

1

0

σ = −1

0 1 2 3 4

0.5

τ=1

τ=1

12

φn - 1

σ = +1

σ =+ − 2, τ = 1

3

1

β>0

12

(Continued )

(c)

13

4

4

14

Gallery 150 - 2

0 1 2 3 4

8

9

Period-2 Isles of Eden : 2 = 0.5 ρ 3 = 4 16 −−

6

2 = 0.25 Period-2 Isles of Eden : ρ 1 = 2 16 −−

(a) Bernoulli (σ = + − 2, τ = 1)

T=2

150 , L = 4

T=2

Table 23.

2

T=2

2956

0 1 2 3 4

7

10

0

1

1

5

15

11

(b) Period-1 Isles of Eden : 4 = 0.25 ρ 2 = 16 −−

2957

20

23

7

26

19

13

3

2

1

0 1 2 3 4

0 1 2 3 4

9

15

14

21

150 , L = 5

T=3

T=3

(Continued )

6

4

8

28

5

29

18

30

11

24

12

16

25

10

27

22

17

1

0

0.5

σ = 0, τ = 3

φn - 3

0

31

2 ρ 2 = −− 32 = 0.0625

(b) Period-1 Isles of Eden :

0

0.5

φn

Gallery 150 - 3

12

4

3 (a) Bernoulli (σ = 0, τ = 3) Period-3 Isles of Eden : ρ 1 = 10 −− 32 = 0.9375

Table 23.

1

β>0

2958

5

51

30

40

20

39

150 , L = 6

34

45

17

54

27

15

0

60

24

63

43

36

9

46

29

53

18

6

3

48

12

58

33

23

Gallery 150 - 4

57

10

(Continued )

Transient phase

Transient phase

16 (a) Period-1 Attractors : ρ 1 = 2 −− 64 = 0.5

Table 23.

0 1 2 3 4 5 6

0 1 2 3 4 5 6

33

30

2959

47

25

2

52

8

59

150 , L = 6

62

13

7

28

49

19

42

32

4

21

55

14

35

50

1

41

56

26

31

44

16

38

61

11

Gallery 150 - 5

37

22

(Continued )

Transient phase

Transient phase

16 = 0.5 (b) Period-1 Attractors : ρ 2 = 2 −− 64

Table 23.

0 1 2 3 4 5 6

0 1 2 3 4 5 6

52

26

45

30

76

94

33

63

115

82

51

18

109

97

12

64

86

15

38

47

89

9

118

112

6

32

43

71

19

87

40

111

124

84

108

68

59

56

Gallery 150 - 6

80

95

121

41

7 = 0.765625 ρ 2 = 14128 −−

3

16

1

0

0.5

φn

0

0

0 1 2 3 4 5 6 7

0.5

σ = 0, τ = 7

φn - 7

1

β>0 12

127

2 = 0.015625 (a) Period-1 Isles of Eden : ρ 1 =128 −−

(Continued )

(b) 14 Bernouli (σ = 0, τ = 7) Period-7 Isles of Eden :

150 , L = 7

Table 23.

T=7

2960

2961

85

99

73

107

20

119

62

42

54

34

93

28 8

65

150 , L = 7

(Continued )

106

113

100

117

27

17

110

14

96

4

53

120

50

122

5

125

79

74 7

77

72

55

Gallery 150 - 7

10

123

31

21

48

2

90

60

25

61

66

126

103

37

102

36

91

67

24

1

(b) 14 Bernoulli (σ = 0, τ = 7) Period-7 Isles of Eden (continued) :

Table 23.

22

98

35

26

49

44

52

81

11

69

70

13

150 , L = 7

(Continued )

88

104

105

58

92

78

39

75

23

116

57

0

0.5

φn

1

0

0.5

φn

1

0

0

0.5

σ = −1, τ = 2

0.5

σ = +1, τ = 2

Gallery 150 - 8

29

83

101

46

114

φn - 2

φn - 2

7 = 0.21875 ρ 3 = 4128 −−

1

1

σ = −1

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

σ = +1

(c) 4 Bernoulli (σ = +1, σ = −1, τ = 2) Period-7 Isles of Eden :

Table 23.

T=7 β>0

T=7 β>0

104

88

τ=2 τ=2

2962

2963

252

207

84

69

123

48 249

246

144

72

159 111

3

9

173

183

132

214

16 168

112

56

138 218

1

7

109

131

150 , L = 8

(Continued )

63

81

21

237

33

222

18

91

224

181

42

192 231

12 126

64 162

4

219

66

189

36

182

193

107

28

129

24

128

8

Gallery 150 - 9

96 243

6

32

2

14

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

4 = 0.75 (a) 48 Period-4 Isles of Eden : ρ 1 = 48 256 −−

Table 23.

T=4

T=4

192

128

2964

167

122

94

229

59

79 118

50

25

208

103

13 244

35

155

179

145

205

19

137

176 188

185

11 203

49

220

152

150 , L = 8

(Continued )

236

100

206

70

55

38

115

47

67

61

52 211

194 242

44

217

200

157

140

110

76

230

196

134

104

133

88

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

Gallery 150 - 10

161 158

26 233

97 121

22 151

98

(a) 48 Period-4 Isles of Eden (continued) :

Table 23.

T=4 T=4

208

176

2965

253

223

232

142

248

241

74

37

117 251

31

87 191

164

154

143

82

77

113 209

86

43

29 169

23

101

212

178

150 , L = 8

(Continued )

58

227

148

62

73

53

172

83

71

213 239

93 254

197

92 116

199

41

124

146

106

89

166

149

171

186

139

184

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

Gallery 150 - 11

234 247

174 127

226 163

46

202

(a) 48 Period-4 Isles of Eden (continued) :

Table 23.

T=4

T=4

223

232

114

240

80

175 95

245

250

39 228

210

105

45

150

15 225

177

160

216

5

27

150 , L = 8 (b)

141

(Continued )

65

190

235

165

90

99

156 147

60 135

20 130

125

215

75

180

198

108

57

0

0.5

φn

1

0

0.5

φn

1

0

0.5

120

40

φn

0

0

0

0.5

σ = +− 4, τ = 2

0.5

σ = +− 4, τ = 2

0.5

σ =+ − 4, τ = 2

Gallery 150 - 12

78 201

30 195

10

54

12 Bernoulli (σ = + − 4, τ = 2) Period-4 4 Isles of Eden ρ 2 = 12 −− 256 = 0.1875 1

Table 23.

φn - 2

φn - 2

φn - 2 1

1

1

0 1 2 3 4 5 6 7 8

σ = +4

0 1 2 3 4 5 6 7 8

σ = +4

0 1 2 3 4 5 6 7 8

σ = +4

T=4 β>0

T=4 β>0

T=4 β>0

σ = −4

σ = −4

σ = −4

160

240

τ=2 τ=2

228

τ=2 τ=2 τ=2 τ=2

2966

2967

0 1 2 3 4

σ = −2

34

136

119

221

σ = +2

17

68

0

0.5

φn

1

0

0.5

σ =+ − 2, τ = 1

σ = −2

φn - 1

σ = +2

τ=1

τ=1

136

1

Gallery 150 - 13

τ=1

τ=1

34

187

238

1 2 3 4

(d) 4 Bernoulli (σ = + −2, τ = 1, β < 0) Period-2 Isles of Eden 2 = 0.03125 ρ 4 = 4 256 −− 0

T=2

T=2

85

170

0

β0

2968

8

7

6

5

4

3

L

1 1 2 3 1 2 1 2 1 2 3 1 2 3 4 5

i

2 2

2

attractors

2 14 4 48 12 4 4 2

2 4 4 10 2

Eden

Number Period-n Period-n Isles of

1 2 1 2 3 1 1 1 1 7 7 4 4 1 2 2

n

0 2 0 2 0 0 0 0 0 0 1 0 4 0 2 2

σ1

1 1 1 1 3 1 1 1 1 7 2 4 2 1 1 1

τ1

+ + + _ + + + + + + + + + + _ +

β1

1 1

2

-4 -2 -2

2

1

1

τ2

-1

-2

-2

σ2

+ _ +

+

+ _

β2

ρ1 = ρ1 = ρ2 = ρ3 = ρ1 = ρ2 = ρ1 = ρ2 = ρ1 = ρ2 = ρ3 = ρ1 = ρ2 = ρ3 = ρ4 = ρ5 =

1 0.25 0.25 0.5 0.9375 0.0625 0.5 0.5 0.015625 0.765625 0.21875 0.75 0.1875 0.015625 0.03125 0.015625

ρ

coefficient

Summary of Qualitative properties of local rule 150 extracted from Gallery 150 for Rule 150 Number Bernoulli Parameters Robustness ID Number Period of of

The basin tree diagrams of Rule 150 for L = 3, 4, . . . , 8 are exhibited in Table 23. Following a detailed analysis of these diagrams, the qualitative properties of local rule 150 extracted from basin-tree Galleries 150-1 to 150-13 of Table 23 are summarized below:

2.4.10. Highlights from Rule 150

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden

Fig. 2.

Truth table, Boolean cube, Diﬀerence Equation, and space-time pattern of local rule 90 .

∆

Table 24. Table deﬁning8 xi ⊕ xj = xi XOR xj .

3. Global Analysis of Local Rule 90 The truth table, Boolean cube, and “Diﬀerence Equation” deﬁning the local rule 90 along with a space-time pattern (with a single red-pixel initial state) exhibited in Table 5 of [Chua et al., 2003] is reproduced in Fig. 2 for the reader’s convenience. For this paper, it is more instructive to recast the Diﬀerence Equation deﬁning 90 into an equivalent diﬀerence equation involving only a mod 2 addition ⊕ (deﬁned in Table 24). Substituting ui = 2xi − 1 from Eq. (4) of [Chua et al., 2005a] for ui in the Diﬀerence Equation for 90 , we obtain − 1 = sgn[1 − |2xti−1 + 2xti+1 − 2|] 2xt+1 i = sgn[1 − 2|xti−1 + xti+1 − 1|] 8

2969

(18)

0 1

0 0 1

1 1 0

Simplifying Eq. (18) using Table 24, we obtain the following equivalent Diﬀerence Equation: Rule 90

xt+1 = (xti−1 + xti+1 ) mod (2) i = xti−1 ⊕ xti+1

(19)

The mod 2 operation xi ⊕ xj between two binary variables is also called an exclusive OR operation in mathematical logic, and ∆

denoted by xi ⊕ xj = xi XOR xj .

2970

L. O. Chua et al.

3.1. Ru1e 90 has no Isle of Eden A cursory glimpse at the basin-tree diagram of rule 90 in Table 18 reveals that all bit strings converge to an attractor for 3 ≤ L ≤ 8. We now prove this property is true for all L. Theorem 1. Rule 90 does not have any isle of

Eden. It follows from Eq. (19) that an arbitrary bit string

Proof.

t

x =

(xt0

xt1

xt2

···

xtL−1 )

(20)

at time “t” is linearly related (mod 2) to its image xt+1 = (x0t+1

xt+1 1

xt+1 2

···

t+1 xL−1 )

(21)

at time “t + 1”via an L × L circulant matrix [Davis, 1979] M 90 , henceforth called the local time-1 state transition matrix:   · · ·  t+1     xt  x0   0 · · ·  t   t+1     x1   x1     t+1   ···   xt  x    2   2   ···  t   t+1    x3   x3  =   ..     ..  ..   ..   . ..  . .   .   .        ···   xt   t+1    L−2  xL−2     t ···   xL−1 xt+1 L−1 ··· xt xt+1 M 90 (22) + where addition is mod (2) sum . Note the diagonal elements of the circulant matrix M 90 are all equal to zero. Observe also the elements below (resp. above) the diag directly onal of M 90 are all equal to one. All other elements are zero, except for the top rightmost element, and the bottom leftmost elements, which are equal to one, respectively. It follows from this spe- cial structure that the leftmost column of M 90 is equal to the mod 2 sum of the L-1 remaining columns. Since the columns of M 90 are not linearly independent, mod 2, it follows that M does not have an inverse. Since the bit string xt+1 on the left side of Eq. (19) does not have a unique preimage, it 9

follows that the bit string xt is not an isle of Eden of 90 . Since xt is an arbitrary bit string, it follows that 90 cannot possess an isle of Eden for any L.

3.2. Period of Rule 90 grows with L Since rule 90 does not have isles of Eden, all bit strings of 90 must converge to period-T attractors whose period “T ” is bounded by 1 ≤ T ≤ Tmax

(23)

where Tmax = 2L as deﬁned in Eq. (6). As an example, the period T of an attractor of 90 is listed in Table 25 for 3 ≤ L ≤ 100. Observe that the period T for some L (e.g. L = 47, 49, 53, etc.) is not listed in Table 25 because it is so large that it had exceeded the maximum simulation time allocated. A bit string belonging to one of the many period-T attractors for 3 ≤ L ≤ 25 is given in Table 26. For example, the bit string listed for L = 3 corresponds to the third period-1 attractor (out of 4) listed in Gallery 90-1 of Table 18. The bit string listed for L = 5 corresponds to node 6mof Gallery 90-3 of Table 18, out of ﬁve period-3 attractors. The bit string listed for L = 6 corresponds to node 30min the ﬁfth attractor shown on the left of Gallery 90-5. The bit string listed for L = 7 corresponds to node 68min the top left attractor of 90 shown on the top left of Gallery 90-7. As examination of Table 25 shows that unlike the period-1 and period-2 local rules listed in Tables 7 and 8, and the period-3 local role 62 , which have a relatively small period, and independent of L, the period T of rule 90 can increase at an exponential rate as a function of L, as depicted in Fig. 3. Such exponential growth of T as a function of L is a signature of all complex Bernoulli rules in Table 11, and hyper-Bernoulli rules in Table 12. In spite of the very large values T of some period-T attractors of 90 , these periods are usually many orders of magnitude smaller than the upper bound Tmax listed in Table 27 for 3 ≤ L ≤ 85.9 There exists, however, period T attractors whose period T approaches the upper bound Tmax . For example, Table 28 shows a period-504 bit string of an isle of Eden of rule 45 for L = 9, which is almost as large as Tmax = 29 = 512! This example suggests that some of the empty slots in Tables 25 may never be ﬁlled.10

It would take at least 105, 104, 783, 572 years for a 1 GHz PC to simulate all Tmax = 2L distinct bit strings! Rule 45 will be studied in Part VIII where it is proved that all bit strings are isles of Eden if, and only if L is an odd number.

10

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 25.

L

Attractor

L

Period “T ” of attractors of local rule 90 for 3 ≤ L ≤ 100.

Attractor

L

Attractor

Attractor

L

21

T = 63

41

T = 1023

61

2

22

T = 62

42

T = 126

62

T = 62

82

Attractor

81 T = 2046

3

T=1

23

T = 2047

43

T = 127

63

T = 63

83

4

T=1

24

T=8

44

T = 124

64

T=1

84

T = 252

5

T=3

25

T = 1023

45

T = 4095

65

T = 63

85

T = 255

6

T=2

26

T = 126

46

T = 4094

66

T = 62

86

T = 254

7

T=7

27

T = 511

47

8

T=1

28

T = 28

48

9

T=7

29

T = 16383

49

10

T=6

30

T = 30

50

T = 2046

70

11

T = 31

31

T = 31

51

T = 255

71

12

T=4

32

T=1

52

T = 252

72

13

T = 63

33

T = 31

53

14

T = 14

34

T = 30

54

15

T = 15

35

T = 4095

55

16

T=1

36

T = 28

56

T = 56

76

17

T = 15

37

T = 87381

57

T = 511

77

18

T = 14

38

T = 1022

58

T = 32766

78

19

T = 511

39

T = 4095

59

20

T = 12

40

T = 24

60

67 T = 16

68

87 T = 60

88

T = 248

89

T = 2047

90

T = 8190

91

T = 4095

T = 56

92

T = 8188

73

T = 511

93

T = 1023

74

T = 174762

94

69

T = 1022

T = 8190

75

95 T = 2044

T = 60

The state transition formula given in Fig. 2 and Eq. (19) for rule 90 is local in time in the sense that it generates from a bit string xt = (xt0 xt1 xt2 · · · xtL−1 ) at time “t” the next bit string xt+1 xt+1 · · · xt+1 xt+1 = (xt+1 0 1 2 L−1 ) at time “t + 1”. Our next theorem gives an explicit formula which is global in time in the sense that it generates a bit string xn0 = (xn0 xn1 xn2 · · · xnL−1 ) at any future time n > t. Theorem 2. Global State-Transition Formula for

80

∆

T = 32

97 T = 8190

98 99

T = 48

T = 32767

100 T = 4092

Each pixel xni at time n > t is determined from “n + 1” initial pixels x0i−n , x0i−n+2 , . . . , x0i+n−2 , x0i+n at t = 0 via the binomial formula. xni =

n k=0

Proof.

n! • x0i−n+2k k!(n − k)!

mod (2)

(24)

Apply mathematical induction as follow:

(a) n = 1 Applying n = 1 in Eq. (24), we obtain11 x1i = x0i−1 + x0i+1 which is Eq. (19) for t = 0.

Recall the factorial notation 0! = 1.

96

79

3.3. Global state-transition formula for rule 90

11

L

1

90 .

2971

mod (2)

(25)

2972

L. O. Chua et al. Table 26.

L T

Bit strings for generating a period-T attractor of Rule 90 .

A bit string on a Period-T attractor

3 1 4 1 5 3 6 2 7 7 8 1 9 7 10 6 11 31 12 4 13 63 14 14 15 15 16 1 17 15 18 14 19 511 20 12 21 63 22 62 23 2047 24 8 25 1023

10

6

Legend :

T 10

10

10

10

- attractor period-T - attractors with T > 10 6

5

4

3

2

10

1 1 Fig. 3.

10

L=I+1

10

2

Dependence of the period “T ” of attractor of rule 90 as a function of L (in logarithmic scale).

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 27. The upper bound Tmax of the period “T ” as function of L for 3 ≤ L ≤ 85.

Tmax = 2 L

L 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536

85

38685626227668133590597632

. . .

k=0

+

k=0

m! • x0i−m+2k k!(m − k)!

k=0

mod (2)

(26)

m! x0 k!(m − k)! (i+1)−m+2k

mod (2)

Changing symbol “m” on the right-hand side of Eq. (27) to m − 1 gives

k=0

(m − 1)! x0 k!(m − 1 − k)! i−m +2k +

−1 m

k=0

(m − 1)! x0 k!(m − 1 − k)! i−m +2k+2

The terms inside the bracket can be simpliﬁed by observing for k = 1 to m − 1, we have (m − 1)! (m − 1)! + k!(m − 1 − k)! (k − 1)!(m − k)! 1 1 (m − 1)! + = (k − 1)!(m − 1 − k)! k (m − k) (m − k) + k (m − 1)! = (k − 1)!(m − 1 − k)! k(m − k) =

m ! k!(m − k)!

(31)

Moreover, when k = 0 and k = m , Eq. (31) gives the same value as the ﬁrst term on the left of Eq. (30), and the last term on the right of Eq. (30), respectively. Substituting back m = m −1 in Eq. (31), and making use of Eqs. (27)–(31), we obtain m+1 (m + 1)! • x0i−(m+1)+2k = xm+1 i k!(m + 1 − k)! mod (2) (32)

(27)

−1 m

(30)

k=0

k=0

k=0

k=1

• x0i−m +2k

m = xm xm+1 i−1 + xi+1 mod (2) i m m! x0 = k!(m − k)! (i−1)−m+2k

+

(m − 1)! x0 (29) (k − 1)!(m − k )! i−m +2k

Changing the dummy index k in Eq. (29) back to k, we obtain m −1 m (m − 1)! (m − 1)! + k!(m − 1 − k)! (k − 1)!(m − k)!

We must show that incrementing “m” to “m + 1” in Eq. (26) gives Eq. (24) with n = m + 1. Substituting Eq. (26) to Eq. (19), we obtain

m

m k =1

. . .

m

(m − 1)! x0 k!(m − 1 − k)! i−m +2k

(b) Assume Eq. (24) is true for n = m (induction hypothesis); namely, xm i =

Changing symbol k in the second summation terms in Eq. (28) to k − 1 gives −1 m

3 4 5 6 7 8 9 10 11 12 13 14 15 16

2973

mod (2) (28)

which is identical to incrementing m in the induction hypothesis (26) to m + 1. Table 29 gives the global state-transition formula (24) of rule 90 for n = 1, 2, 3, 4 and 5. n Observe that the coeﬃcients k for each time n ≥ 1 is identical to the binomial coeﬃcients in the expansion of (x + y)n , as listed in Table 30 for n = 1, 2, . . . , 11. These binomial coeﬃcients are repackaged in Table 31 into the form of a Pascal’s triangle where each coeﬃcient under the pyramid is obtained by adding adjacent left and right coeﬃcients above it.

2974

0

0.5

φn

1

0

0.5

σ = −2, τ = 56

φ n - 56

β>0 1

504 ρ1 = 512 −− = 0.984375

Isle of Eden:

Period-504

(σ = −2, τ = 56)

(a) Bernoulli

45 , L = 9

Table 28.

508

2

499

8

463

32

319

128

254

1

505

4

487

16

415

64

127

256

340

506

338

491

330

431

298

191

170

253

169

501

165

471

149

351

85

382

70

263

280

30

98

120

392

480

35

387

140

15

49

60

196

240

273

449

123

116

492

464

435

323

207

270

317

58

246

232

473

417

359

429

333

183

310

221

218

373

361

470

422

347

155

366

109

442

436

52

203

208

301

321

182

262

217

26

357

104

406

416

91

131

364

13

235

414

135

434

211

29

391

398

31

59

124

236

496

433

451

199

271

285

62

118

248

472

481

355

229

297

405

166

87

153

348

101

370

404

458

83

299

332

174

306

185

202

164

57

145

228

69

401

276

71

82

284

328

114

290

456

156

39

293

493

150

439

294

27

154

108

105

89

113

133

432

420

195

147

269

77

54

194

345

265

358

38

410

152

107

97

428

304

371

241

462

453

315

279

238

94

441

376

476

352

23

386

92

11

368

44

450

22

225

88

111

55

444

220

243

369

461

454

311

283

411

215

177

349

197

374

277

474

86

237

43

437

325

485

278

407

90

95

360

380

180

190

209

100

7

400

28

67

112

268

448

50

259

134

224

25

412

272

342

509

3

250

115

375

460

478

307

477

66

80

264

320

34

379

20

346

287

362

126

426

258

206

494

233

99

93

204

372

305

466

198

331

281

151

72

81

65

137

260

37

18

144

316

484

167

403

157

79

117

121

158

234

288

468

242

339

457

334

295

314

425

130

324

162

302

102

186

274

9

74

36

296

148

421

396

408

51

313

443

230

239

409

445

103

247

455

503

490

479

427

383

504

136

10

33

40

132

160

17

129

68

5

378

12

163

48

141

192

175

171

510

257

53

483

173

399

181

63

213

252

341

497

189

507

245

495

469

447

343

255

350

212

6

337

24

326

96

282

384

106

424

489

61

423

244

159

465

125

327

500

286

467

122

335

488

318

419

143

200

14

289

56

385

418

498

139

459

45

303

249

363

344

430

354

187

394

172

222

110

377

440

486

227

397

176

267

193

46

261

184

389

231

482

413

395

119

47

188

388

179

19

205

76

309

214

308

216

210

353

329

390

223

356

381

402

502

75

475

300

367

178

446

201

251

21

452

84

275

336

78

322

312

266

226

42

393

168

161

138

291

41

142

The period of the following isle of Eden for rule 45 is T = 504, which is equal almost to Tmax = 512.

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 29.

Table 30.

n

2975

Global state-transition formula for rule 90 for 1 ≤ n ≤ 5.

` ´∆ Table of n k = n!/k! (n − k)!, n = 1, 2, . . . , 11, k = 0, 1, 2, . . . , 11.

k 0

1

2

3

4

5

6

7

8

9

10

1

1

1

2

1

2

1

3

1

3

3

1

4

1

4

6

4

1

5

1

5

10

10

5

1

6

1

6

15

20

15

6

1

7

1

7

21

35

35

21

7

1

8

1

8

28

56

70

56

28

8

1

9

1

9

36

84

126

126

84

36

9

1

10

1

10

45

120

210

252

210

120

45

10

1

11

1

11

55

165

330

462

462

330

165

55

11

11

1

2976

L. O. Chua et al.

Table 31. Binomial coeﬃcients cal’s triangle.

`n´ k

repackaged into a Pas-

Pascal’s Triangle 1 1 1 1 1

1 2

3

1 3

1

6 4 1 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1

4

5

.. .

Taking the “mod 2 equivalent” of each coeﬃcient in Table 29, we obtain the more compact but equivalent expansion in Table 32 where all nonzero terms correspond to those in Table 29 with “odd number ” coeﬃcients. The equivalent mod 2 coefﬁcients are repackaged in Table 33. Observe that Table 33 can be obtained from Table 31 by replacing each odd (respectively, even) coeﬃcient in Table 31 by a one (respectively, a zero). If we ﬁll in the missing slot in each row of the mod 2 Pascal’s triangle, we would obtain the pyramidal “fractal” space-time pattern of rule 90 in Table 34, which is identical

Table 32.

to that shown in the bottom of Fig. 2, where the initial conﬁguration consists of a single red bit at the center, as in [Wolfram, 2002]. Example 1. Table 35 shows the space-time pattern obtained from the global state-transition formula of rule 90 in (a) when the initial conﬁguration consists of a single red bit at the center. The corresponding pattern obtained from the local statetransition formula is shown in (b). They are identical, as expected. The minor diﬀerences in the graphics and color are due to the diﬀerences in the softwares used to generate these patterns. Example 2. Table 36 shows the corresponding results when the initial conﬁguration consists of a string of random bits.

3.4. Periodicity constraints of Rule 90 Theorem 1 implies that all bit strings of rule 90 must converge to a period-T attractor, where T ≤ Tmax ≤ 2L . We will prove in this subsection that ∆ for ﬁnite length L = I + 1, the period T must satisfy certain constraints. Such periodicity constraints are useful on many occasions, such as verifying whether certain periodic orbit can exist, or to generate new periodic orbits, etc. The proof of many of these results depend on the following easily

Compact global state-transition formula for rule 90 for 1 ≤ n ≤ 5.

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden Table 33. Mod 2 binomial coeﬃcients a mod 2 Pascal’s triangle.

`n´ k

repackaged into

Mod 2 Pascal’s Triangle

(iii)

where

1 1 1 1 1 1

1 0

1 0

1 1

0

1 0

1

1

0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 .. .

(ii)

0 mod (2), = 1 mod (2),

Table 34.

0 mod (2), 1 mod (2),

for k = 2, 3, . . . , n − 1 for k = 0, 1, n, n + 1 (35)

n ∆ n! = k!(n − k)! k

(36)

Theorem 3. Periodicity Condition: L = 2m . For L = 2m , m = 2, 3, 4, . . . , rule 90 has a global period-1 attractor Γ; namely, x(Γ) = (0

0 0 ··· L=2m

0)

(37)

Let n = 2m−1 in the global state-transition formula (24). It follows from Eqs. (33) and (36) that Proof.

Binomial Coeﬃcient Lemma. If n = 2m , where m ≥ 2, then the following identities hold: 1 = 1 mod (2), for k = 0, 1, 2, . . . , n − 1 (i) n − k

n k

=

All bit strings not belonging to the attractor Γ converge to Γ in at most 2m−1 iterations.

veriﬁable identities:

n+1 k

2977

(33) for k = 1, 2, . . . , n − 1 for k = 0, n (34)

n! mod (2) k!(n − k)! 0, for k = 1, 2, . . . , n − 1 = 1, for k = 0, n = 2m−1

(38)

It follows from Eq. (38) and the global statetransition formula (24) that xni contains only two nonzero terms; namely, the leftmost and the

Space-time pattern of the Pascal triangle fractal generated by rule 90 .

2978

L. O. Chua et al. Table 35. Space-time pattern of the rule 90 with red central bit initial conﬁguration: (a) from global state-transition formula; (b) from local state-transition formula.

90 mod(2)

90

x in + 1 = xin- 1 + x in+ 1 mod(2)

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden

2979

Table 36. Space-time pattern of the rule 90 obtained from random initial state generated by: (a) global state-transition formula; (b) local state-transition formula.

90

mod(2)

90

x in + 1 = xin- 1 + x in+ 1 mod(2)

Corollary to Theorem 2. A bit string

rightmost terms. Hence, xni = x0i−n + x0i+n

mod (2)

where n = 2m−1 . Substituting i = n = 2m−1 in Eq. (39), we obtain xni = x0n−n + x0n+n

mod (2)

= x00 + x02m

mod (2)

= x00 + x0L = 2x0 =0

x01

x02

···

x0L−1 )

(41)

of length L = I + 1 (under periodic boundary condition) is a period-n attractor of local rule 90 if, and only if, the periodicity condition xnimod(L) = x0i

mod (2)

= x00 + x02n

x0 = (x00

(39)

=

n k=0

n! k!(n − k)!

• x0((i−n+2k) mod(L))

mod (2)

(42) mod (2)

mod (2) (40)

because x00 = xL . Since x0i is arbitrary, it follows that all bit strings must converge to Eq. (7) in at most 2m−1 iterations.

is satisﬁed for all i. Follows directly from Theorem 2 and the periodic boundary condition.

Proof.

The periodicity constraint equation (42) is applicable to any period-n attractor of rule 90 .

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L. O. Chua et al.

The “mod (L)” operation attached to the subscript index of x0 is just a mathematically precise algorithm for implementing the periodic boundary conditions. It is also mathematically equivalent to concatenating replicas of the L-bit string x0 x1 x2 · · · xI ad inﬁnitum; namely, · · · x0

x1

x2 · · ·

xI

L bits

x0

x0

x2 · · ·

x1

L bits

x1

x2 · · ·

L bits

xI

xI · · ·

(44)

all binomial coeﬃcients in Eq. (42) are equal to unity, in view of the Binomial Coeﬃcient Lemma; namely, k = 0, 1, 2, . . . , n

(45)

Equation (45) is obtained by substituting n + 1 = 2m from Eq. (44) in place of n in Eq. (33): (n + 1) − 1 = 1 mod (2), k = 0, 1, 2, . . . , n k (46) Substituting Eq. (46) into Eq. (42), we obtain the following simpliﬁed periodicity constraint: xnimod(L) = x0i =

k=0

x0((i−n+2k) mod(L))

mod (2)

(47) If we impose the additional constraint L = n = then we obtain the following simple method for ﬁnding period-(2m − 1) attractors:

2m − 1,

Theorem 4. Periodicity Condition: L = 2m − 1. Rule 90 has a period-n attractor where n = 2m − 1 and L = 2m − 1 if, and only if, L−1 i=0

x0i mod (2) = 0

(49)

Since “i” is an arbitrary index in Eq. (49), let it be “n”. Substituting i = n in Eq. (49), we obtain x00 = x00 + x02 + · · · +

x0 n−1

+x0((n+1) mod(L))

+ · · · + x0((2n−2) mod(L)) + x0((2n) mod(L))

(48)

2m

(50)

− 1, we have

(2n) mod (L) = 0, ((2n − 2) mod (L)) = L − 2, ((n + 1) mod (L)) = 1. Observe also that L = 2m − 1 implies that L − 2, L − 4, etc. are odd numbers. Substituting these mod (L) equivalent indices into Eq. (50), we obtain x00 = x00 + x02 + · · · + x0L−1 + x01 + x03 + · · · + x0L−2 + x00 mod (2)

(51)

Observe that whereas the ﬁrst x00 on the right-hand side of Eq. (51) comes from the corresponding ﬁrst term of Eq. (50), the last x00 of Eq. (51) comes from the last bit x0((2n) mod(L)) = x00 of Eq. (50). Rearranging the terms in increasing subscript order in Eq. (51), we obtain x00 = x00 + x00 + x01 + x02 + · · · + x0L−2 + x0L−1 mod (2) (x00

for all i

Valid for n = 2m − 1 L = 2m − 1

(i−n) mod(L) (i−n+2) mod(L) 0 + · · · + x((i+n−2) mod(L)) + x0((i+n) mod(L)) mod (2)

Observe next that for n = L =

n = 2m − 1

m −1 2

x0i mod(L)

mod (2)

where L = I + 1. In the special case where

Valid if n = 2m − 1

Let us list all terms from Eq. (47) as follows: + x0 = x0

n−1=2m −2 0, we can

generalize our deﬁnition of “Bernoulli στ -shift” to include σ = 0 for all such period-T orbits. In this case, the return map φn−τ → φn will consist of points lying on the diagonal line φn = φn−τ . We usually include such a graph whenever space permits. We note also that the period “T ” of rules 90 , 150 and 105 exhibit a scale free property as L → ∞. For example, for L = 2m , the period of · · · 0 as rule 90 is always equal T = 1 with 0 0 L bits

its global ﬁxed point attractor. To its immediate left (L = 2m − 1) and immediate right (L = 2m + 1), the period-T orbits have equal period T = 2m −1, at any scale L → ∞. To illustrate the scale-free distribution of the period “T ” of rule 90 , Fig. 6 shows a plot of log T as a function of log L of the data listed in Table 25. Observe the six period-1 red stars on the horizontal axis (T = 1) are located at L = 2m , m = 2, 3, 4, 5; namely, L = 4, 8, 16, 32, 64, as predicted by Theorem 3. Observe that all data points from Table 25 lie along straight lines with a slope equal to “one”. The distributions of the period T of rules 150 and 105 are plotted in Figs. 7 and 8, respectively, as a function of the string length L = I + 1, in base-10 logarithmic scales. The data are extracted from Table 37 for rule 150 , and from Table 38 for rule 105 , respectively. Data points corresponding to isles of Eden are shown as blue dots. Those

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L. O. Chua et al.

10

6

T 10

slope =

5

= 10

10

10

4

∆ log(T) ∆ log(L)

log(8) - log(2) =1 log(24)-log(6)

3

2

10 ∆ log(T) ∆ log(L)

1 1 Fig. 6. scales.

10

L=I+1

10

2

Relationship between the period T and the length L = I + 1 of attractors of rule 90 plotted in base-10 logarithmic

Fig. 7. Relationship between the period T and the length L = I + 1 of isles of Eden (plotted as blue dots), and attractors (plotted as red stars) of rule 150 .

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden

3011

Fig. 8. Relationship between the period T and the length L = I + 1 of isles of Eden (plotted as blue dots), and attractors (plotted as red stars) of rule 105 .

corresponding to attractors are shown as red stars. Again, the scale-free distributions are clearly seen from the parallel straight lines where these data points are located.

References Adamatzky, A. [2007] “Book review on a nonlinear dynamics perspective of Wolfram’s new kind of science,” J. Cellular Automata, in press. Chua, L. O. & Roska, T. [2002] Cellular Neural Networks and Visual Computing (Cambridge University Press, Cambridge). Chua, L. O., Yoon, S. & Dogaru, R. [2002] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part I: Threshold of complexity,” Int. J. Bifurcation and Chaos 12, 2655–2766. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2003] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part II: Universal neuron,” Int. J. Bifurcation and Chaos 13, 2377–2491. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2004] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part III: Predicting the unpredictable,” Int. J. Bifurcation and Chaos 14, 3689–3820. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2005a] “A nonlinear dynamics perspective of Wolfram’s new kind of

science. Part IV: From Bernoulli shift to 1/f spectrum,” Int. J. Bifurcation and Chaos 15, 1045–1183. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2005b] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part V: Fractals everywhere,” Int. J. Bifurcation and Chaos 15, 3701–3849. Chua, L. O. [2006] A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science, Vol. I (World Scientiﬁc, Singapore). Chua, L. O., Sbitnev, V. I. & Yoon, S. [2006] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part VI: From time-reversible attractors to the arrow of time,” Int. J. Bifurcation and Chaos 16, 1097–1373. Chua, L. O. [2007] A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science, Vol. II (World Scientiﬁc, Singapore). Chang, J. & Muthuswamy, B. [2007] “Extracting optimal CNN templates for linearly-separable onedimensional cellular automata,” Int. J. Bifurcation and Chaos 17, 749–779. Davis, P. J. [1979] Circulant Matrices (WileyInterscience, NY). Garay, B. M. & Hofbauer, J. [2003] “Robust permanence for ecological diﬀerential equations, minimax, and discretizations,” SIAM J. Math. Anal. 34, 1007–1039.

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Walker, C. [1971] “Behavior of a class of complex systems: The eﬀect of system size on properties of terminal cycles,” J. Cybern. 1, 55–67. Walker, C. C. & Aadryan, A. A. [1971] “Amount of computation preceding externally detectable steady state behavior in a class of complex systems,” J. Bio-Med. Comput. 2, 85–94. Wolfram, S. [2002] A New Kind of Science (Wolfram Media, Inc., Champaign, IL). Wuensche, A. & Lesser, M. [1992] The Global Dynamics of Cellular Automata (Addison-Wesley Publishing Company, Reading, MA).

Appendix

The bit strings {xn0 , xn1 , xn2 , . . . , xnI } and {y0n , y1n , y2n , . . . , yIn } generated respectively by rules 150 and 105 from the same initial state {z00 , z10 , z20 , . . . , zI0 } obey the following alternating symmetry relations: (−1)n xni

(A.1)

xni = αn + (−1)n yin

(A.2)

= αn +

(a) Assume n is even in Eq. (A.5). In this case αn = 0 and αn+1 = 1. Equation (A.5) reduces to: = 1 − (xni−1 + xni + xni+1 ) mod (2) 1 − xn+1 i (A.6) Hence, = (xni−1 + xni + xni+1 ) mod (2) xn+1 i

Following the same procedure let us change the symbol “x” in Eq. (A.3) into “y” by applying Eq. (A.2) to obtain

Proof.

= xni−1 + xni + xni+1 xn+1 i

mod (2)

(A.3)

Bit string {y0n , y1n , y2n , . . . , yIn } evolves under rule 105 via the formula n n + yin + yi+1 ) mod (2) yin+1 = 1 − (yi−1

(A.4)

Changing the symbol “y” in Eq. (A.4) into “x” by applying Eq. (A.1) and invoking the identity (3αn ) mod (2) = αn mod (2) we obtain αn+1 + (−1)n+1 xn+1 i = 1 − (3αn + (−1)n xni−1 + (−1)n xni + (−1)n xni+1 ) mod (2) = (1 − αn ) − (−1)n (xni−1 + xni + xni+1 )

(A.8)

Hence, both Eqs. (A.7) and (A.8) are identical to Eq. (A.3).

αn+1 + (−1)n+1 yin+1 n = (3αn + (−1)n yi−1 + (−1)n yin n + (−1)n yi+1 ) mod (2) n n n ) = αn + (−1) (yi−1 + yin + yi+1

Bit string {xn0 , xn1 , xn2 , . . . , xnI } evolves under rule 150 via the formula

(A.7)

(b) Assume n is odd in Eq. (A.5). In this case αn = 1 and αn+1 = 0. Equation (A.5) reduces to: = (xni−1 + xni + xni+1 ) mod (2) xn+1 i

105 150 Alternating Symmetry Duality

yin

Consider the following two cases:

mod (2) (A.9)

Again, we must consider two cases: (a) Assume n is even in Eq. (A.9). In this case αn = 0 and αn+1 = 1. Equation (A.9) reduces to: n n + yin + yi+1 ) mod (2) 1 − yin+1 = (yi−1 (A.10)

Hence, n n + yin + yi+1 ) mod (2) yin+1 = 1 − (yi−1 (A.11)

(b) Assume n is odd in Eq. (A.9). In this case αn = 1 and αn+1 = 0. Equation (A.9) reduces to: n n + yin + yi+1 ) mod (2) 0 + yin+1 = 1 − (yi−1 (A.12)

mod (2) (A.5)

Hence, both Eqs. (A.11) and (A.12) are identical to Eq. (A.4).

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