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E-grāmata: Nonlinear Dynamics Perspective of Wolfram's New Kind of Science, v. 4 [World Scientific e-book]

Edited by (Univ Of California, Berkeley, Usa)
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Nonlinear Dynamics Perspective of Wolfram's New Kind of Science, v. 4Palielināt
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Volume IV continues the author's odyssey on l-D cellular automata as chronicled in Volumes I, II and III, by uncovering a novel quasi-ergodicity phenomenon involving orbits meandering among omega-limit orbits of complex (group 5) and hyper (group 6) Bernoulli rules. This discovery is embellished with analytical formulas characterizing the fractal properties of characteristic functions, as well as explicit formulas for generating colorful and pedagogically revealing isomorphic basin tree diagrams. Many new results were derived and proved by uncovering subtle symmetries endowed by various subsets of the 256 Boolean cubes. For the first time, rigorous analyses were used to identify 67, out off 256 , local rules whose asymptotic behaviors consist of robust period-l orbits. The highlight of this continuing odyssey is the discovery of an isolated period-3240 Isle of Eden hidden among the dense omega-limit orbits of Wolfram's remarkable “random number generating” rule 30. This is the largest gem known to-date and readers are challenged to uncover even larger ones.
An Ode to the Unknowable xi
Volume IV
Chapter 1 Quasi-Ergodicity
1(156)
1 Remembrance of Things Past
2(57)
1.1 Boolean cube representation
3(1)
1.2 Index of complexity
3(8)
1.3 One formula specifies all 256 rules
11(1)
1.4 Space-time pattern and time-τ return maps
12(17)
1.5 We only need to study 88 rules!
29(1)
1.6 The "Magic" rule spaces
29(3)
1.7 Symmetries among Boolean cubes
32(1)
1.7.1 Local complementation Tc
32(1)
1.7.2 Three equivalence transformations T†, T, and T
32(4)
1.7.3 Perfect complementary rules
36(1)
1.7.4 Permutive rules
36(1)
1.7.5 Superposition of local rules
37(1)
1.7.6 Rules with explicit period-1 and/or period-2 orbits
37(1)
1.7.7 Most rules harbor at least one Isle of Eden
38(21)
2 Quasi-Ergodicity
59(20)
2.1 Only complex and hyper Bernoulli- shift rules are quasi-ergodie
59(17)
2.2 Quasi-ergodicity and Gardens of Eden
76(3)
3 Fractals in 1D Cellular Automata
79(18)
3.1 All time-1 characteristic functions are fractals
79(4)
3.2 Fractals in CA additive rules
83(1)
3.2.1 Rule 60
84(2)
3.2.2 Rule 90
86(2)
3.2.3 Rule 105
88(2)
3.2.4 Rule 150
90(3)
3.3 From the time-1 characteristic function to the rule number
93(2)
3.4 Number of fractal patterns
95(2)
4 New Results about Isles of Eden
97(9)
4.1 Definitions and basic lemmas
98(1)
4.2 Alternate proof that rules 45 and 154 are conservative for odd lengths
98(1)
4.3 There are exactly 28 strictly-dissipative local rules
99(6)
4.4 Isles of Eden for rules of group 1
105(1)
5 How to Find Analytically the Basin-Tree Diagrams for Bernoulli Attractors
106(5)
5.1 Bernoulli-στ basin-tree generation formula
106(3)
5.2 A practical application of the Bernoulli στ-shift basin tree generation formula
109(2)
6 Old Theorems and New Results for Additive Cellular Automata
111(41)
6.1 Theorems on the maximum period of attractors and Isles of Eden
113(3)
6.2 Scale-free property for additive rules
116(36)
7 Concluding Remarks
152(1)
Appendix A
152(1)
Appendix B
153(1)
Appendix C
154(3)
Chapter 2 Period-1 Rules
157(230)
1 The Second Time Around
157(6)
1.1 Recap of Period-1 Rules
163(1)
2 Basin Tree Diagrams
163(141)
2.1 Each bit string has a decimal and a fractional code
285(1)
2.2 Attractor, basin of attraction, and basin tree
285(17)
2.3 Explicit formula for generating isomorphic basin trees
302(1)
2.4 Robustness coefficient ρ
303(1)
2.5 Genotype and phenotype
303(1)
3 Omega-Limit Orbits
304(67)
3.1 General observations on ω-limit orbits from Table 6
304(1)
3.2 Portraits of ω-limit orbits from the basin tree diagrams exhibited in Table 6
304(63)
3.3 Concatenated ω-limit orbit generation algorithm
367(4)
4 Rules of Group 1 have Robust Period-1 ω-Limit Orbits
371(14)
5 Concluding Remarks
385(2)
References 387(2)
Index 389
Leon Chua is a foreign member of the Academia Europea and a recipient of eight USA patents and 12 docteur honoris causa. He has received numerous international awards, including the first IEEE Kirchhoff Award, the Neural Networks Pioneer Award, and the Top 15 Cited AuthorsA" Award based on the ISI Citation Index in Engineering from 1991 to 2001. When not immersed in science, he relaxes by searching for Wagner's leitmotifs, musing over Kandinsky's chaos, and contemplating Wittgenstein's inner thoughts.
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