Barnsley / Demko Chaotic Dynamics and Fractals

1. Auflage 2014
ISBN: 978-1-4832-6908-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 304 Seiten, Web PDF

Chaotic Dynamics and Fractals covers the proceedings of the 1985 Conference on Chaotic Dynamics, held at the Georgia Institute of Technology. This conference deals with the research area of chaos, dynamical systems, and fractal geometry. This text is organized into three parts encompassing 16 chapters. The first part describes the nature of chaos and fractals, the geometric tool for some strange attractors, and other complicated sets of data associated with chaotic systems. This part also considers the Henon-Hiles Hamiltonian with complex time, a Henon family of maps from C2 into itself, and the idea of turbulent maps in the course of presenting results on iteration of continuous maps from the unit interval to itself. The second part discusses complex analytic dynamics and associated fractal geometry, specifically the bursts into chaos, algorithms for obtaining geometrical and combinatorial information, and the parameter space for iterated cubic polynomials. This part also examines the differentiation of Julia sets with respects to a parameter in the associated rational map, permitting the formulation of Taylor series expansion for the sets. The third part highlights the applications of chaotic dynamics and fractals. This book will prove useful to mathematicians, physicists, and other scientists working in, or introducing themselves to, the field.

Autoren/Hrsg.

Barnsley, Michael F.

Demko, Stephen G.

Inhaltsverzeichnis

1;Front Cover;1
2;Chaotic Dynamics and Fractals;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;8
6;Preface;10
7;Part I: Chaos and Fractals;14
7.1;CHAPTER 1. CHAOS: SOLVING THE UNSOLVABLE, PREDICTING THE UNPREDICTABLE!;14
7.1.1;1. CHAOS: AN ILLUSTRATIVE EXAMPLE;17
7.1.2;2. ALGORITHMIC COMPLEXITY THEORY;21
7.1.3;3. ALGORITHMIC INTEGRABILITY;37
7.1.4;4. ALGORITHMIC RANDOMNESS;49
7.1.5;5. QUANTUM CHAOS, IF ANY?;61
7.1.6;REFERENCES;65
7.2;CHAPTER 2. MAKING CHAOTIC DYNAMICAL SYSTEMS TO ORDER;66
7.2.1;ABSTRACT;66
7.2.2;1. INTRODUCTION;66
7.2.3;2. THE COLLAGE THEOREM;69
7.2.4;3. MAKING DIFFERENTIAL EQUATIONS WITH PRESCRIBED ATTRACTORS;76
7.2.5;REFERENCES;80
7.3;CHAPTER 3. ON THE EXISTENCE AND NON-EXISTENCE OF NATURAL BOUNDARIES FOR NON-INTEGRABLE DYNAMICAL SYSTEMS;82
7.3.1;ABSTRACT;82
7.3.2;1. INTRODUCTION;82
7.3.3;2. NONLINEAR DIFFERENTIAL EQUATIONS AND ALGEBRAIC INTEGRABILITY;86
7.3.4;3. A CANONICAL EXAMPLE;102
7.3.5;4. SOME SIMPLE EXAMPLES;106
7.3.6;ACKNOWLEDGMENT;111
7.3.7;REFERENCES;112
7.4;CHAPTER 4. THE HENON MAPPING IN THE COMPLEX DOMAIN;114
7.4.1;1. INTRODUCTION;114
7.4.2;2. HISTORY AND MOTIVATION;115
7.4.3;3. THE RELATION WITH THE THEORY OF POLYNOMIALS;116
7.4.4;4. RATES OF ESCAPE FOR THE HENON FAMILY;118
7.4.5;5. ANGLES OF ESCAPE;120
7.4.6;6. A PROGRAM FOR DESCRIBING MAPPINGS IN THE HENON FAMILY;123
7.5;CHAPTER 5. DYNAMICAL COMPLEXITY OF MAPS OF THE INTERVAL;126
7.5.1;1. THE ŠARKOVSKII STRATIFICATION;126
7.5.2;2. TOPOLOGICAL ENTROPY;127
7.5.3;3. TURBULENCE;128
7.5.4;4. ENTROPY MINIMAL ORBITS;129
7.5.5;5. HOMOCLINIC ORBITS;132
7.5.6;ACKNOWLEDGEMENTS;133
7.5.7;REFERENCES;134
7.6;CHAPTER 6. A USE OF CELLULAR AUTOMATA TO OBTAIN FAMILIES OF FRACTALS;136
7.6.1;ABSTRACT;136
7.6.2;1. A SHORT HISTORY OF CELLULAR AUTOMATA;136
7.6.3;2. WHAT ARE CELLULAR AUTOMATA?;138
7.6.4;3. RESCALING TO OBTAIN FRACTALS IN THE LIMIT;142
7.6.5;4. WAYS OF OBTAINING SOME NUMBERS FROM THE LIMIT SETS;146
7.6.6;5. CONCLUSIONS AND DISCUSSION;151
7.6.7;REFERENCES;152
8;Part II: Julia Sets;154
8.1;CHAPTER 7. EXPLODING JULIA SETS;154
8.1.1;ABSTRACT;154
8.1.2;1. INTRODUCTION;154
8.1.3;2. AN EXPLOSION IN THE EXPONENTIAL FAMILY;157
8.1.4;3. AN EXPLOSION IN THE SINE FAMILY;162
8.1.5;4. CONCLUSION;166
8.1.6;REFERENCES;166
8.2;CHAPTER 8. ALGORITHMS FOR COMPUTING ANGLES IN THE MANDELBROT SET;168
8.2.1;1. NOTATIONS;168
8.2.2;2. POTENTIAL AND EXTERNAL ARGUMENTS;169
8.2.3;3. HOW TO COMPUTE Argc(z) FOR c . D0 u D2, z PREPERIODIC;169
8.2.4;4. EXTERNAL ARGUMENTS IN M;170
8.2.5;5. USE OF EXTERNAL ARGUMENTS;171
8.2.6;6. INTERNAL AND EXTERNAL ARGUMENTS IN .W0;172
8.2.7;7. TUNING;173
8.2.8;8. FEIGENBAUM POINT AND MORSE NUMBER;177
8.2.9;9. SPIRALING ANGLE;178
8.2.10;10. HOW TO DETERMINE pw0 (c) KNOWING 1 PAIR (t,t') SUCH THAT t tc t' , t . t';180
8.2.11;11. ACKNOWLEDGEMENTS;181
8.3;CHAPTER 9. THE PARAMETER SPACE FOR COMPLEX CUBIC POLYNOMIALS;182
8.3.1;1. INTRODUCTION;182
8.3.2;2. ABOUT POLYNOMIALS OF DEGREE d = 2 IN GENERAL;182
8.3.3;3. DICHOTOMY FOR DYNAMICAL BEHAVIOR;185
8.3.4;4. TRICHOTOMY FOR DYNAMICAL BEHAVIOR OF CUBIC POLYNOMIALS;187
8.3.5;5. A TOPOLOGICAL DESCRIPTION OF Sr;188
8.3.6;6. FINE STRUCTURE FOR y+r(a);190
8.3.7;7. FINAL REMARKS;192
8.4;CHAPTER 10. DISCONNECTED JULIA SETS;194
8.4.1;INTRODUCTION;194
8.4.2;1. NOTATION AND BACKGROUND MATERIAL;195
8.4.3;2. SYMBOLIC CODINGS FOR CUBICS;201
8.4.4;3. SUMMARY AND OPEN PROBLEMS;212
8.4.5;REFERENCES;213
8.5;CHAPTER 11. CALCULATION OF TAYLOR SERIES FOR JULIA SETS IN POWERS OF A PARAMETER;216
8.5.1;ABSTRACT;216
8.5.2;1. INTRODUCTION;216
8.5.3;2. DERIVATIVES OF JULIA SETS WITH RESPECT TO THE PARAMETER;217
8.5.4;3. RESULTS OF CALCULATIONS;220
8.5.5;REFERENCES;226
8.6;CHAPTER 12. DIOPHANTINE PROPERTIES OF JULIA SETS;228
8.6.1;ABSTRACT;228
8.6.2;1. INTRODUCTION;228
8.6.3;2. FEKETE'S THEOREM, KRONECKER'S THEOREM;229
8.6.4;3. JULIA SETS ARE NATURAL SETS FOR LOCALIZATION OF ALGEBRAIC INTEGERS;230
8.6.5;4. GENERALIZATION TO POLYNOMIALS WITH ALGEBRAIC INTEGERS COEFFICIENTS;233
8.6.6;5. QUANTITATIVE FORMULATION AND GENERALIZATION OF LEHMER'S PROBLEM;234
8.6.7;6. EXTENSION TO THE MANDELBROT M SET;238
8.6.8;7. CONCLUSION;239
8.6.9;REFERENCES;239
9;Part III. Applications;242
9.1;CHAPTER 13. REAL SPACE RENORMALIZATION AND JULIA SETS IN STATISTICAL MECHANICS;242
9.1.1;1. INTRODUCTION;242
9.1.2;2. RENORMALIZATION GROUP;244
9.1.3;3. HIERARCHICAL LATTICES;244
9.1.4;4. THE CRITICAL BEHAVIOUR OF f;247
9.1.5;5. ZEROS OF THE PARTITION FUNCTION [6] - JULIA SET;249
9.1.6;6. DISORDER ON HIERARCHICAL LATTICES;251
9.1.7;7. CONCLUSION;253
9.1.8;ACKNOWLEDGEMENTS;254
9.1.9;REFERENCES;254
9.2;CHAPTER 14. REGULAR AND CHAOTIC CYCLING IN MODELS FROM POPULATION AND ECOLOGICAL GENETICS;256
9.2.1;INTRODUCTION;256
9.2.2;1. POPULATION GENETICS;257
9.2.3;2. CLASSICAL SELECTION MODEL;258
9.2.4;3. ECOLOGICAL GENETICS AND DENSITY REGULATED SELECTION;260
9.2.5;4. OSCILLATORY BEHAVIOR IN OTHER GENETIC SYSTEMS;270
9.2.6;5. CONCLUSIONS;273
9.2.7;REFERENCES;274
9.2.8;ACKNOLWEDGMENTS;275
9.3;CHAPTER 15. A BIFURCATION GAP FOR A SINGULARLY PERTURBED DELAY EQUATION;276
9.3.1;ABSTRACT;276
9.3.2;1. DELAY EQUATIONS AND INTERVAL MAPS;277
9.3.3;2. THREE QUESTIONS;282
9.3.4;3. A BIFURCATION GAP;283
9.3.5;4. THE COUNTEREXAMPLES;286
9.3.6;5. A NEW DYNAMICAL SYSTEM;290
9.3.7;REFERENCES;297
9.4;CHAPTER 16. TRAVELLING WAVES FOR FORCED FISHER'S EQUATION;300
9.4.1;1. INTRODUCTION;300
9.4.2;2. LINEARIZATION ANALYSIS;302
9.4.3;3. BIFURCATION;304
9.4.4;REFERENCES;305