(1+1)-NONLNR UNIVERSE PARABOLIC MAP & COMBINATORICS, THE

by James D Louck, Myron L Stein

Overview

This monograph develops chaos theory from properties of the graphs inverse to the parabolic map of the interval [0, 2], where the height at the midpoint x = 1 may be viewed as a time-like parameter, which together with the x-coordinate, provide the two parameters that uniquely characterize the parabola, and which are used throughout the monograph. There is only one basic mathematical operation used: function composition. The functions studied are the n-fold composition of the basic parabola with itself. However, it is the properties of the graph inverse to this n-fold composition that are the objects whose properties are developed. The reflection symmetry of the basic parabola through the vertical line x = 1 gives rise to two symmetry classes of inverse graphs: the inverse graphs and their conjugates. Quite remarkably, it turns out that there exists, among all the inverse graphs and their conjugates, a completely deterministic class of inverse graphs and their conjugates. Deterministic in the sense that this class is uniquely determined for all values of the time-like parameter and the x-coordinate, the entire theory, of course, being highly nonlinear — it is polynomial in the time-like parameter and in the x-coordinate. The deterministic property and its implementation are key to the argument that the system is a complex adaptive system in the sense that a few axioms lead to structures of unexpected richness.This monograph is about working out the many details that advance the notion that deterministic chaos theory, as realized by a complex adaptive system, is indeed a new body of mathematics that enriches our understanding of the world around us. But now the imagination is also opened to the possibility that the real universe is a complex adaptive system.* deceased

Product Details

ISBN-13:	9789814632430
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/30/2014
Sold by:	Barnes & Noble
Format:	eBook
Pages:	192
File size:	44 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface vii

1 Introduction and Point of View 1

1.1 Function Composition and Graphs 1

1.1.1 Words on Two Letters 10

1.1.2 A New Notation for Branches 10

1.1.3 A Total-Order Relation on τ Sequences 10

1.1.4 Reality of the Inverse Graph 11

1.2 Inverse Graphs Created at ς = 1 12

1.2.1 The Order Rule for General Branches 14

1.2.2 Fixed Points 16

1.2.3 The Primordial p-Curve and Its Evolution 17

1.2.4 MSS Polynomials and Roots 19

1.2.5 Lexical Sequences 21

1.3 Preview of the Full ς-Evolution 22

1.4 The Baseline 23

1.5 Vocabulary, Symbol Definitions, and Explanations 26

2 Recursive Construction 27

2.1 Construction of the Baseline B_n 28

2.1.1 Properties of MSS Roots 28

2.2 Reducible and Irreducible Sequences 31

3 Description of Events in the Inverse Graph 39

3.1 Domains of Definition of Branches and Curves 39

3.2 Concatenation, Harmonics, and Antiharmonics 41

3.3 Fixed Points as Dynamical Objects 42

3.4 The Fabric of Bifurcation Events 43

3.5 The Anatomy of Period-Doubling Bifurcations 44

3.5.1 The Complete Description for n. = 4 44

3.6 Signature Properties of Fixed Points 46

3.6.1 More Vocabulary and Associated Events 46

3.6.2 r-Cycles, Permutation Cycle Classes, and Words 50

3.7 Young Hook Tableaux and Gelfand-Tsetlin Patterns 54

4 The (1+1)-Dimensional Nonlinear Universe 59

4.1 The Parabolic Map 59

4.2 Complex Adaptive Systems 63

5 The Creation Table 65

5.1 The Creation Intervals 68

5.2 Properties of the Creation Table 69

5.2.1 The Table of Creation Sequences for Prime n 75

6 Graphical Presentation of MSS Roots 85

7 Graphical Presentation of Inverse Graphs 93

Bibliography 171

Epilogue 175

Index 177