Nonlinear Dynamics Perspective of Wolfram's New Kind of Science (Volume VI).
Title:
Nonlinear Dynamics Perspective of Wolfram's New Kind of Science (Volume VI).
Author:
Chua, Leon O.
ISBN:
9789814460880
Personal Author:
Physical Description:
1 online resource (579 pages)
Series:
World Scientific Series on Nonlinear Science Series A ; v.85
World Scientific Series on Nonlinear Science Series A
Contents:
CONTENTS -- Dedication -- Preface -- Volume VI -- Chapter 1. Bernoulli στ -Shift Rules -- 1. Introduction -- 1.1. Brief notes on Bernoulli στ -shift rules -- 2. Basin Tree Diagrams, Omega-Limit Orbits and Space-Time Patterns -- 2.1. Basin tree diagrams and portraits of the ω-limit orbits -- 2.2. Space-time patterns of Bernoulli rules using the super string as initial string -- 3. Robust and Nonrobust ω-Limit Orbits of Rules from Group 4 -- 3.1. Robust ω-limit orbits of rules from Group 4 -- 3.2. Nonrobust ω-limit orbits of rules from Group 4 -- 4. Concluding Remarks -- Chapter 2. More Bernoulli στ -Shift Rules -- 1. Introduction -- 2. Bernoulli στ -Shift Rules -- 2.1. General aspects of the Bernoulli στ -shift rules -- 2.2. Basin-tree diagrams and portraits of their ω-limit orbits -- 2.3. Space-time patterns of Bernoulli rules using the superstring as initial state -- 3. Robust and Nonrobust ω-Limit Orbits of Rules from Group 4 -- 3.1. Robust ω-limit orbits of rules from Group 4 -- 3.2. Non-robust ω-limit orbits of rules from Group 4 -- 3.3. Rules with multiple robust ω-limit orbits -- 4. Summary of Elementary 1D Cellular Automata -- 4.1. Boolean cubes, complexity index, and threshold of complexity -- 4.2. Globally and quasi-globally equivalent rules -- 4.3. Rotations and symmetries -- 4.4. Classification of the local rules -- 4.5. Fractality and quasi-ergodicity -- 4.6. Isles of Eden and Omega-limit orbits -- 5. Concluding Remarks -- Chapter 3. Remembrance of Things Past -- Vignettes from Volume I -- Vignettes from Volume II -- Vignettes from Volume III -- Vignettes from Volume IV -- Vignettes from Volume V -- Vignettes from Volume VI -- Vignettes of Metaphors from Biology, Cosmology, Physics, etc. -- Vignettes of 256 Boolean Cubes -- References -- Appendices.
Appendix I: Correspondence between Chapters from Edited Book and Papers from IJBC Journal -- Appendix II: Useful and Generic Tables and Figures -- Appendix III: Pages where 16 Exquisite Elementary CA Rules are Cited, Discussed, or Characterized -- Appendix IV: Contents of Volumes I-VI -- Index.
Abstract:
This invaluable volume ends the quest to uncover the secret recipes for predicting the long-term evolution of a ring of identical elementary cells where the binary state of each cell during each generation of an attractor (i.e. after the transients had disappeared) is determined uniquely by the state of its left and right neighbors in the previous generation, as decreed by one of 256 truth tables. As befitting the contents aimed at school children, it was found pedagogically appealing to code each truth table by coloring each of the 8 vertices of a cubical graph in red (for binary state 1), or blue (for binary state 0), forming a toy universe of 256 Boolean cubes, each bearing a different vertex color combination.The corresponding collection of 256 distinct Boolean cubes are then segegrated logically into 6 distinct groups where members from each group share certain common dynamics which allow the long-term evolution of the color configuration of each bit string, of arbitrary length, to be predicted painlessly, via a toy-like gaming procedure, without involving any calculation. In particular, the evolution of any bit string bearing any initial color configuration which resides in any one of the possibly many distinct attractors, can be systematically predicted, by school children who are yet to learn arithmetic, via a simple recipe, for any Boolean cube belonging to group 1, 2, 3, or 4. The simple recipe for predicting the time-asymptotic behaviors of Boolean cubes belonging to groups 1, 2, and 3 has been covered in Vols. I, II, ..., V.This final volume continues the recipe for each of the 108, out of 256, local rules, dubbed the Bernoulli rules, belonging to group 4. Here, for almost half of the toy universe, surprisingly simple recipes involving only the following three pieces of information are derived in Vol. VI; namely, a positive integer τ,
a positive, or negative, integer σ, and a sign parameter β > 0, or β 0 (resp. σ < 0), and then change the color of each cell if β < 0.As in the five prior volumes, Vol. VI also contains simple recipes which are, in fact, general and original results from the abstract theory of 1-dimensional cellular automata. Indeed, both children and experts from cellular automata will find this volume to be as deep, refreshing, and entertaining, as the previous volumes.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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Electronic Access:
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