Nonlinear Dynamics Perspective Of Wolfram's New Kind Of Science, A (Volume Vi)

(Volume VI)

Leon O Chua

• Publisher: WSPC
• Year: 2013
• ISBN:
  9789814460897
• Topic: Ciencias biológicas
• Subtopic: Ciencias en general

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. In particular, given any color configuration belonging to an attractor of any one of the 108 Boolean cubes from group 4, any child can predict the color configuration after τ generations, without any computation, by merely shifting each cell σ bits to the left (resp. right) if σ > 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.

Contents:

• Bernoulli σ τ -Shift Rules:
  • Introduction
  • Basin Tree Diagrams, Omega-Limit Orbits and Space-Time Patterns
  • Robust and Nonrobust ω-Limit Orbits of Rules from Group 4
  • Concluding Remarks
• More Bernoulli σ τ -Shift Rules:
  • Introduction
  • Bernoulli σ τ -Shift Rules
  • Robust and Nonrobust ω-Limit Orbits of Rules from Group 4
  • Summary of Elementary 1D Cellular Automata
  • Concluding Remarks
• 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

Readership: Students, researchers, academics as well as laymen interested in nonlinear dynamics, computer science and complexity theory.

Chapter 1

BERNOULLI σ_τ-SHIFT RULES

More than one third of the 88 globally-independent Cellular Automata rules exhibit robust simple Bernoulli-shift dynamics. Among them we find rule 170, which we proved to be chaotic in the previous episodes of our chronicle, and rule 184, the famous global majority rule. Therefore, we cannot overstate the importance of the Bernoulli σ_τ-shift rules which we will present in two parts of our continuing odyssey on the Nonlinear Dynamics Perspective of Cellular Automata. This paper covers the first 15 of the 30 Bernoulli σ_τ-shift rules. In this paper, after recalling the main concepts of Bernoulli rules — such as the role of the three Bernoulli parameters σ,τ and β — we will display the basin tree diagrams of these rules together with a convenient summary of the results extracted from them. Then, we will show that the superstring

is an excellent testing signal to find the robust behavior of a given rule. Finally, we will conclude this paper with a discussion about the difference between robust and nonrobust ω-limit orbits of the Bernoulli σ_τ-shift rules.

Keywords: Cellular automata; nonlinear dynamics; Bernoulli σ_τ-shift rules; group 4 rules; ω-limit orbits; attractors; Isles of Eden; basin tree diagrams; superstring; robust ω-limit orbits; nonrobust ω-limit orbits.

1. Introduction

We continue our analysis of one-dimensional Cellular Automata “through the eyes” of Nonlinear Dynamics, as already done in the previous twelve parts of our work: Part I [Chua et al., 2002], Part II [Chua et al., 2003], Part III [Chua et al., 2004], Part IV [Chua et al., 2005a], Part V [Chua et al., 2005b], Part VI [Chua et al., 2006], Part VII [Chua et al., 2007a], Part VIII [Chua et al., 2007b], Part IX [Chua et al., 2008], Part X [Chua et al., 2009a], Part XI [Chua et al., 2009b], and Part XII [Chua & Pazienza, 2009]. As usual, we aim to give sound scientific results about the “New Kind of Science” presented by Wolfram in his monumental tome [Wolfram, 2002], whic...

Table of contents

• Cover
• Half Title
• Series Title
• Title
• Copyright
• Dedication
• Preface
• Contents
• Volume VI
• References
• Appendices
• Index