Gallery 137-1

1372 3

0 1

4 5

6 7

vertex

0

Basin Tree Diagrams of Rule 13722 = 4 23 = 8

26 = 6427 = 1137

20 = 1

25 = 32

21 = 2

1234567

0

1

24 = 16

L=3 L=4

Table B1.

S

Gallery 137-2

L=5137

Gallery 137-3

L=6137

Gallery 137-4

L=7137

Gallery 137-5

L=8137

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρiβ σ τ

3 1 1 0 1 x0( ) + 0 1 3 ρ1 = 1

4 1 2 0 2

x0( )

+ 2 1 2x1( )

x0( )

2 1 0 1 x0( ) + 0 1 2 ρ2 = 0.25

5 1 1 0 1 x0( ) + 0 1 5 ρ1 = 1

6 1 2 0 9

x0( )

+ -2 3 3x8( )

x0( )

2 1 0 1 x0( ) + 0 1 3 ρ2 = 0.1562

137

nMaximum TransientLength

δmax

Table B2. Portraits of ω-Limit Orbits of rule

7

1

4

15

31

1

37

63

7

1

4

15

31

1

37

63

2

8

2

11

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρiβ σ τ

7 1 1 0 14x0( )

+ -3 2 7x13( )

2 1 0 1 x0( ) + 0 1 2 ρ2 = 0.0703

8

1 2 0 16

x0( )

+ 2 4 7

x15( )

x0( )

2 0 1 8x0( )

+ 3 1 0x7( )

3 2 0 2

x0( )

+ 2 1 2x1( )

x0( )

4 1 0 1 x0( ) + 0 1 3 ρ4 = 0.0781

nMaximum TransientLength

δmax3

83

127

5

145

17

68

255

13

161

3

83

127

5

145

17

68

255

10

35

34

136

13

161

Gallery 166-1

1662 3

0 1

4 5

6 7

vertex

0

Basin Tree Diagrams of Rule 16622 = 4 23 = 8

26 = 6427 = 128

20 = 1

25 = 32

21 = 2

1234567

0

1

24 = 16

Table B3.

L=3 L=4

S

Gallery 166-2

166 L=5

Gallery 166-3

L=6166

Gallery 166-4

L=7166

ρ2

ρ3

ρ4

ρ5

ρ6ρ7

Gallery 166-5

L=8166ρ4

ρ5

ρ6

ρ7

ρ8ρ9

ρ10

ρ11

Table B4. Portraits of ω-Limit Orbits of rule

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρi

β σ τ

31 0 2 1

x0( )+ 0 1 0

x0( )

2 0 1 6x0( ) + 2 2 0x5( )

4

1 0 1 1 x0( ) + 0 1 0 ρ1 = 0.0625

2 0 2 4

x0( )

+ 2 2 0x3( )

x0( )

3 1 0 1 x0( ) + 0 1 2 ρ3 = 0.4375

5

1 0 1 1 x0( ) + 0 1 0 ρ1 = 0.0312

2 0 1 10x0( ) + 2 2 0x9( )

3 0 1 20x0( ) + -1 4 0x19( )

4 0 1 1 x0( ) + 0 1 0 ρ4 = 0.0312

166

nMaximum TransientLength

δmax0

7

1

6

0

1

12

0

1

24

5

19

31

15

0

7

1

6

0

1

12

0

1

24

5

19

31

2

9

15

!1 = 2!!

18

"

=0.25!2 = 1!

!68

"

=0.75

!2 = 2!!

416

"

=0.5

!2 = 1!!

1032

"

=0.3125!3 = 1!

!2032

"

=0.625

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρi

β σ τ

6

1 0 1 1 x0( ) + 0 1 0 ρ1 = 0.0156

2 0 3 6

x0( )

+ 2 2 0

x5( )

x0( )

x5( )

3 0 2 12

x0( )

+ -2 4 0x11( )

x0( )

4 0 2 3

x0( )

+ -2 1 0x2( )

x0( )

nMaximum TransientLength

δmax0

1

48

5

35

13

52

0

1

48

2

33

9

54

9

54

5

35

13

52

10

7

26

41

!2 = 3!!

664

"

=0.2812

!3 = 2!!

1264

"

=0.375

!4 = 2!!

364

"

=0.0938

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρi

β σ τ

6 5 3 0 2

x0( )

+ 3 1 1x1( )

x0( )

6 1 0 1 x0( ) + 0 1 1 ρ6 = 0.0469

7

1 0 1 1 x0( ) + 0 1 0 ρ1 = 0.0078

2 0 3 14

x0( )

+ 2 2 0x13( )

x0( )

3 0 3 28

x0( )

+ 4 4 0

x27( )

x0( )

4 0 1 1 x0( ) + 0 1 0 ρ4 = 0.0078

nMaximum TransientLength

δmax

0

1

96

5

67

127

23

58

63

0

1

96

5

67

127

23

58

63

9

102

23

79

13

100

37

91

4619

5343

!5 = 3!!

464

"

=0.1875

!2 = 3!!

14128

"

=0.3281

!3 = 3!!

28128

"

=0.6562

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρi

β σ τ

8

1 0 1 1 x0( ) + 0 1 0 ρ1 = 0.0039

2 0 2 4

x0( )

+ 2 2 0x3( )

x0( )

3 0 1 8x0( ) + -3 1 0x7( )

4 0 2 8

x0( )

+ 2 2 0x7( )

x0( )

nMaximum TransientLength

δmax0

17

204

25

200

1

192

0

17

204

25

200

1

192

34

153

2

129

!2 = 2!!

4256

"

=0.0312

!3 = 1!!

8256

"

=0.0312

!4 = 2!!

8256

"

=0.0625

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρi

β σ τ

8

5 0 2 8

x0( )

+ 2 2 0x7( )

x0( )

6 0 2 8

x0( )

+ 2 2 0x7( )

x0( )

7 0 4 8

x0( )

+ 4 4 0x7( )

x0( )

nMaximum TransientLength

δmax9

198

13

196

5

131

9

198

18

141

13

196

26

137

5

131

10

7

20

14

40

28

!5 = 2!!

8256

"

=0.0625

!6 = 2!!

8256

"

=0.0625

!7 = 4!!

8256

"

=0.125

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρi

β σ τ

8

8 0 4 8

x0( )

+ 4 4 0x7( )

x0( )

9 0 4 8

x0( )

+ 4 4 0x7( )

x0( )

10 0 4 8

x0( )

+ 4 4 0x7( )

x0( )

nMaximum TransientLength

δmax21

206

31

202

37

155

37

155

74

55

148

110

41

220

31

202

62

149

124

43

248

113

21

206

42

157

84

59

168

118

!8 = 4!!

8256

"

=0.125

!9 = 4!!

8256

"

!10 = 4!!

8256

"

=0.125

=0.125

LDistinct Orbit

ID number

iNumber of Period-Tattractors

Number of Period-TIsles ofEden

Period

T

Bit-strings (coded by integer )

for Period-T Orbitof Length L

BernoulliParameters

RobustnessCoefficient ρi

β σ τ

8 11 0 4 8

x0( )

+ 4 4 0x7( )

x0( )

12 1 0 1 x0( ) + 0 1 4 ρ12 = 0.1211

nMaximum TransientLength

δmax53

147

53

147

106

39

212

78

169

156

255 255

!11 = 4!!

8256

"

=0.125