Chaotic Dynamics and Fractals.

Front Cover -- Chaotic Dynamics and Fractals -- Copyright Page -- Table of Contents -- Contributors -- Preface -- Part I: Chaos and Fractals -- CHAPTER 1. CHAOS: SOLVING THE UNSOLVABLE, PREDICTING THE UNPREDICTABLE! -- 1. CHAOS: AN ILLUSTRATIVE EXAMPLE -- 2. ALGORITHMIC COMPLEXITY THEORY -- 3. ALGORITHMIC INTEGRABILITY -- 4. ALGORITHMIC RANDOMNESS -- 5. QUANTUM CHAOS, IF ANY? -- REFERENCES -- CHAPTER 2. MAKING CHAOTIC DYNAMICAL SYSTEMS TO ORDER -- ABSTRACT -- 1. INTRODUCTION -- 2. THE COLLAGE THEOREM -- 3. MAKING DIFFERENTIAL EQUATIONS WITH PRESCRIBED ATTRACTORS -- REFERENCES -- CHAPTER 3. ON THE EXISTENCE AND NON-EXISTENCE OF NATURAL BOUNDARIES FOR NON-INTEGRABLE DYNAMICAL SYSTEMS -- ABSTRACT -- 1. INTRODUCTION -- 2. NONLINEAR DIFFERENTIAL EQUATIONS AND ALGEBRAIC INTEGRABILITY -- 3. A CANONICAL EXAMPLE -- 4. SOME SIMPLE EXAMPLES -- ACKNOWLEDGMENT -- REFERENCES -- CHAPTER 4. THE HENON MAPPING IN THE COMPLEX DOMAIN -- 1. INTRODUCTION -- 2. HISTORY AND MOTIVATION -- 3. THE RELATION WITH THE THEORY OF POLYNOMIALS -- 4. RATES OF ESCAPE FOR THE HENON FAMILY -- 5. ANGLES OF ESCAPE -- 6. A PROGRAM FOR DESCRIBING MAPPINGS IN THE HENON FAMILY -- CHAPTER 5. DYNAMICAL COMPLEXITY OF MAPS OF THE INTERVAL -- 1. THE ŠARKOVSKII STRATIFICATION -- 2. TOPOLOGICAL ENTROPY -- 3. TURBULENCE -- 4. ENTROPY MINIMAL ORBITS -- 5. HOMOCLINIC ORBITS -- ACKNOWLEDGEMENTS -- REFERENCES -- CHAPTER 6. A USE OF CELLULAR AUTOMATA TO OBTAIN FAMILIES OF FRACTALS -- ABSTRACT -- 1. A SHORT HISTORY OF CELLULAR AUTOMATA -- 2. WHAT ARE CELLULAR AUTOMATA? -- 3. RESCALING TO OBTAIN FRACTALS IN THE LIMIT -- 4. WAYS OF OBTAINING SOME NUMBERS FROM THE LIMIT SETS -- 5. CONCLUSIONS AND DISCUSSION -- REFERENCES -- Part II: Julia Sets -- CHAPTER 7. EXPLODING JULIA SETS -- ABSTRACT -- 1. INTRODUCTION -- 2. AN EXPLOSION IN THE EXPONENTIAL FAMILY -- 3. AN EXPLOSION IN THE SINE FAMILY -- 4. CONCLUSION.

REFERENCES -- CHAPTER 8. ALGORITHMS FOR COMPUTING ANGLES IN THE MANDELBROT SET -- 1. NOTATIONS -- 2. POTENTIAL AND EXTERNAL ARGUMENTS -- 3. HOW TO COMPUTE Argc(z) FOR c ϵ D0 u D2, z PREPERIODIC -- 4. EXTERNAL ARGUMENTS IN M -- 5. USE OF EXTERNAL ARGUMENTS -- 6. INTERNAL AND EXTERNAL ARGUMENTS IN ∂W0 -- 7. TUNING -- 8. FEIGENBAUM POINT AND MORSE NUMBER -- 9. SPIRALING ANGLE -- 10. HOW TO DETERMINE πw0 (c) KNOWING 1 PAIR (t,t') SUCH THAT t tc t' , t ≠ t' -- 11. ACKNOWLEDGEMENTS -- CHAPTER 9. THE PARAMETER SPACE FOR COMPLEX CUBIC POLYNOMIALS -- 1. INTRODUCTION -- 2. ABOUT POLYNOMIALS OF DEGREE d ≥ 2 IN GENERAL -- 3. DICHOTOMY FOR DYNAMICAL BEHAVIOR -- 4. TRICHOTOMY FOR DYNAMICAL BEHAVIOR OF CUBIC POLYNOMIALS -- 5. A TOPOLOGICAL DESCRIPTION OF Sr -- 6. FINE STRUCTURE FOR y+r(α) -- 7. FINAL REMARKS -- CHAPTER 10. DISCONNECTED JULIA SETS -- INTRODUCTION -- 1. NOTATION AND BACKGROUND MATERIAL -- 2. SYMBOLIC CODINGS FOR CUBICS -- 3. SUMMARY AND OPEN PROBLEMS -- REFERENCES -- CHAPTER 11. CALCULATION OF TAYLOR SERIES FOR JULIA SETS IN POWERS OF A PARAMETER -- ABSTRACT -- 1. INTRODUCTION -- 2. DERIVATIVES OF JULIA SETS WITH RESPECT TO THE PARAMETER -- 3. RESULTS OF CALCULATIONS -- REFERENCES -- CHAPTER 12. DIOPHANTINE PROPERTIES OF JULIA SETS -- ABSTRACT -- 1. INTRODUCTION -- 2. FEKETE'S THEOREM, KRONECKER'S THEOREM -- 3. JULIA SETS ARE NATURAL SETS FOR LOCALIZATION OF ALGEBRAIC INTEGERS -- 4. GENERALIZATION TO POLYNOMIALS WITH ALGEBRAIC INTEGERS COEFFICIENTS -- 5. QUANTITATIVE FORMULATION AND GENERALIZATION OF LEHMER'S PROBLEM -- 6. EXTENSION TO THE MANDELBROT M SET -- 7. CONCLUSION -- REFERENCES -- Part III. Applications -- CHAPTER 13. REAL SPACE RENORMALIZATION AND JULIA SETS IN STATISTICAL MECHANICS -- 1. INTRODUCTION -- 2. RENORMALIZATION GROUP -- 3. HIERARCHICAL LATTICES -- 4. THE CRITICAL BEHAVIOUR OF f.

5. ZEROS OF THE PARTITION FUNCTION [6] - JULIA SET -- 6. DISORDER ON HIERARCHICAL LATTICES -- 7. CONCLUSION -- ACKNOWLEDGEMENTS -- REFERENCES -- CHAPTER 14. REGULAR AND CHAOTIC CYCLING IN MODELS FROM POPULATION AND ECOLOGICAL GENETICS -- INTRODUCTION -- 1. POPULATION GENETICS -- 2. CLASSICAL SELECTION MODEL -- 3. ECOLOGICAL GENETICS AND DENSITY REGULATED SELECTION -- 4. OSCILLATORY BEHAVIOR IN OTHER GENETIC SYSTEMS -- 5. CONCLUSIONS -- REFERENCES -- ACKNOLWEDGMENTS -- CHAPTER 15. A BIFURCATION GAP FOR A SINGULARLY PERTURBED DELAY EQUATION -- ABSTRACT -- 1. DELAY EQUATIONS AND INTERVAL MAPS -- 2. THREE QUESTIONS -- 3. A BIFURCATION GAP -- 4. THE COUNTEREXAMPLES -- 5. A NEW DYNAMICAL SYSTEM -- REFERENCES -- CHAPTER 16. TRAVELLING WAVES FOR FORCED FISHER'S EQUATION -- 1. INTRODUCTION -- 2. LINEARIZATION ANALYSIS -- 3. BIFURCATION -- REFERENCES.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.