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Fractals and Chaos. an Understanding of These Basic Concepts Simplifies This Analysis

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Fractals and Chaos. an Understanding of These Basic Concepts Simplifies This Analysis Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1991-06 Fractals and chaos Beaver, Philip Frederick Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/28232 NAVAL POSTGRADUATE SCHOOL Monterey, California FRACTALS AND CHAOS by Philip Frederick Beaver June 1991 Thesis Advisor: Maurice D. Weir Co-Advisor: Ismor Fischer Approved for public release; distribution is unlimited. T256243 fnclassificd ecurity Qassification of this page REPORT DOCUMENTATION PAGE a Report Security Classification Unclassified lb Restrictive Markings a Security Classification Authority 3 Distribution Availability of Report b Declassification/Downgrading Schedule Approved for public release; distribution is unlimited. Performing Organization Report Number(s) 5 Monitoring Organization Report Number(s) a Name of Performing Organization 6b Office Symbol 7 a Name of Monitoring Organization Javal Postgraduate School (If Applicable) MA Naval Postgraduate School c Address (city, state, and ZIP code) 7 b Address (city, state, and ZIP code) fonterey, CA 93943-5000 Monterey, CA 93943-5000 a Name of Funding/Sponsoring Organization 8b Office Symbol 9 Procurement Instrument Identification Number (If Applicable) c Address (city, state, and ZIP code) 1 Source of Funding Numbers Program Element Number | Project No | Talk No | Work Unit Accession No 1 Tide (Include Security Classification) FRACTALS AND CHAOS 2 Personal Authors) Beaver, Philip F. 3 a Type of Report 13b Time Covered 1 4 Date of Report (year, month/lay) 1 5 Page Count laster's Thesis I From To June 1991 179 6 Supplementary Notation The views expressed in this thesis are those of the author and do not reflect the official olicy or position of the Department of Defense or the U.S. Government 7 Cosati Codes 1 8 Subject Terms (continue on reverse if necessary and identify by block number) ield Group Subgroup Fractal geometry and chaotic dynamical systems 9 Abstract (continue on reverse if necessary and identify by block number The study of fractal geometry and chaotic dynamical systems has received considerable attention in the past ecade. Motivated by the interesting computer graphics produced by these fields, mathematicians have attempted ) formalize the theoretical structure of the results, physicists have attempted to apply the theory to real world henomena, and laymen have enjoyed much of the popular literature and television programs the field has astered. Unfortunately, the mathematics associated with these subjects has made them inaccessible to most ndergraduates, even if they have a strong background in mathematics. This thesis presents the basic ideas of ractal geometry and chaotic dynamical systems in a setting that can be understood by undergraduate students who ave had a course in advanced calculus. We hope it will allow them to gain an appreciation of the fields and loti vate them to pursue further study. Distribution/Availability of Abstract 21 Abstract Security Classification unclassified/unlimited |X| | tame at report DTIC users Unclassified J 2a Name of Responsible Individual 22b Telephone (Include Area code) 22c Office Symbol Maurice D. Weir (408) 646- 2608 MA/Wc D FORM 1473, 84 MAR 83 APR edition may be used until exhausted security classification of this page All other editions are obsolete Unclassified Approved for public release; distribution is unlimited. Fractals and Chaos by Philip Frederick Beaver Captain, United States Army B.S., United States Military Academy, 1983 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN APPLIED MATHEMATICS from the NAVAL POSTGRADUATE SCIHnni ABSTRACT The study of fractal geometry and chaotic dynamical systems has received considerable attention in the past decade. Motivated by the interesting computer graphics produced by these fields, mathematicians have attempted to formalize the theoretical structure of the results, physicists have attempted to apply the theory to real world phenomena, and laymen have enjoyed much of the popular literature and television programs that the field has fostered. Unfortunately, the mathematics associated with these subjects has made them inaccessible to most undergraduates, even if they have a strong background in mathematics. This thesis presents the basic ideas of fractal geometry and chaotic dynamical systems in a setting that can be understood by undergraduate students who have had a course in advanced calculus . We hope it will allow them to gain an appreciation of the fields and motivate them to pursue further study. in &./ TABLE OF CONTENTS I. INTRODUCTION 1 IL BASIC CONCEPTS 4 A. INTRODUCTION 4 B. METRIC SPACES 4 C ITERATED FUNCTION SYSTEMS 6 D. CODE SPACE 13 E. THE CANTOR SET 15 m. FRACTALS 19 A. THE SETTING FOR FRACTALS 19 B. CONTRACTION MAPPINGS 22 C AFFINE TRANSFORMATIONS OF THE PLANE 25 D. CONTRACTION MAPPINGS OF THE SPACE Ji(X) 29 E. CREATING FRACTALS THROUGH ITERATED FUNCTION SYSTEMS 33 1. The Cantor Set 33 2. Fractals in Two Dimensions 36 3. Condensation Sets 42 4. A Fractal Tree 43 F. APPLICATIONS OF FRACTALS TO COMPUTER GRAPHICS 46 G. THE ADDRESSES OF POINTS ON FRACTALS 50 H. FRACTAL DIMENSION 60 I. EXPERIMENTAL DETERMINATION OF FRACTAL DIMENSION 69 J. THE KOCH SNOWFLAKE 73 K. APPLICATIONS OF FRACTAL GEOMETRY 75 IV IV. CHAOS 77 A. INTRODUCTION 77 B. GRAPHICAL ANALYSIS OF FIXED POINTS OF MAPS 79 C MAPS OF THE CIRCLE 82 D. CHAOTIC DYNAMICAL SYSTEMS 85 E. TOPOLOGICAL CONJUGACY 89 F. CHAOTIC DYNAMICS ON CODE SPACE 90 G. NEWTON'S METHOD FOR X2 = -1 92 H. THE QUADRATIC FAMILY OF MAPS 96 I. BIFURCATIONS 108 J. SARKOVSKITS THEOREM 118 K. THE QUADRATIC FAMILY REVISITED 123 L. JULIA SETS 131 M. THE MANDELBROT SET. 137 N. THE SMALE HORSESHOE 144 O. THE HENON MAP 149 P. THE LORENZ EQUATIONS 155 LIST OF REFERENCES 168 XlNl 1 1/\1_> L'lO X IvlD \J X XvJIN Lid X .......................<>..... ........................................ .......... X / J. ACKNOWLEDGMENT I would like to express my appreciation to a number of individuals who have played a role in the development of this thesis. I am particularly grateful to the following: Professor Maurice D. Weir, who first suggested the subject of this thesis, and who developed the curriculum that provided the necessary background for me to write it. He continued to guide me throughout the preparation of the thesis, and spent countless hours encouraging, proofreading, and mentoring. Professor Ismor Fischer, who led me through a course on chaotic dynamical systems, provided technical guidance and expertise during the preparation of this thesis, and who remained patient throughout the entire process. Professor Aaron Schusteff, who started me on these subjects by teaching a course on fractal geometry and topology, and who remained an essential source of technical information during my further study of chaotic dynamical systems and throughout the preparation of this thesis. Finally, Mr. David Beaver of The Automation Group, who introduced me to the idea of chaos by sending me James Gleick's book on the subject, and who volunteered numerous hours of word processing advice, without which I could not have prepared this document. As usual, however, I retain sole responsibility for all the errors, mistakes, and other deviations from the truth that may be contained within this thesis. Philip Beaver Monterey, California June 1991 VI I. INTRODUCTION The subjects of fractals and chaos have attracted considerable attention in the last decade. This interest ranges from a "cult following" of laymen who are intrigued by the intricate computer graphics associated with the fields, to a rigorous mathematical treatment of the subjects by topologists and experts in dynamical systems, and to applications of these results to the real world by engineers and physicists. However, these subjects have been almost wholly inaccessible to undergraduates because of the level of mathematics required to study them. This thesis presents the subjects of fractals and chaos in a setting that can be understood by a typical undergraduate student with a solid background in mathematics through advanced calculus. The subjects of fractals and chaos are not new. The German mathematician Georg Cantor (1845-1918) knew about fractals, and the French mathematician Jules Henri Poincare (1854-1912) knew about chaos in dynamical systems in the late nineteenth century. Additionally, the French mathematicians Pierre Fatou (1878-1929) and Gaston Julia (1893-1978) knew about Julia sets in the 1920s. However, it was not until the 1970s that high- speed computers allowed others to see what these men had discovered and to recognize the true potential of these fields. The growth of these fields has resulted in significant scientific advances in the past decade. While the discovery of quantum mechanics and relativity had a profound impact on very specialized areas of science, fractals and chaos have had a universal effect on the whole scientific community. Until recently, science had become so specialized that, not only were physicists and mathematicians not communicating with each other, but molecular biologists were not communicating with population biologists either. The science of chaos has served to bring together the entire scientific community, including physicists, electrical engineers, mechanical engineers, biologists, economists, astronomers, meteorologists, medical scientists, and of course, mathematicians. To explain briefly, fractals and chaos help to describe the universe. Natural objects, from crystals, plants and geological
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