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Generalized Chaotic Maps and Elementary Functions Between Analysis and Implementation GENERALIZED CHAOTIC MAPS AND ELEMENTARY FUNCTIONS BETWEEN ANALYSIS AND IMPLEMENTATION By Wafaa Saber AbdelHalim Sayed A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Mathematical Engineering FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2015 GENERALIZED CHAOTIC MAPS AND ELEMENTARY FUNCTIONS BETWEEN ANALYSIS AND IMPLEMENTATION By Wafaa Saber AbdelHalim Sayed A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Mathematical Engineering Under the Supervision of Prof. Abdel-Latif E. Hussien Prof. Hossam A. H. Fahmy Professor Professor Engineering Mathematics and Physics Department Electronics and Communication Engineering Department Faculty of Engineering, Cairo University Faculty of Engineering, Cairo University Assoc. Prof. Ahmed G. Radwan Associate Professor Engineering Mathematics and Physics Department Faculty of Engineering, Cairo University FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2015 GENERALIZED CHAOTIC MAPS AND ELEMENTARY FUNCTIONS BETWEEN ANALYSIS AND IMPLEMENTATION By Wafaa Saber AbdelHalim Sayed A Thesis Submitted to the Faculty of Engineering at Cairo University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Mathematical Engineering Approved by the Examining Committee: Prof. Abdel-Latif E. Hussien, Thesis Main Advisor Prof. Hossam A. H. Fahmy, Member Assoc. Prof. Ahmed G. Radwan, Member Prof. Salwa K. Abd-El-Hafiz, Internal Examiner Prof. Nasser H. Sweilam, External Examiner Professor, Faculty of Science, Cairo University FACULTY OF ENGINEERING, CAIRO UNIVERSITY GIZA, EGYPT 2015 Engineer’s Name: Wafaa Saber AbdelHalim Sayed Date of Birth: 20/3/1991 Nationality: Egyptian E-mail: [email protected] Registration Date: 1/10/2013 Awarding Date: –/–/2015 Degree: Master of Science Department: Engineering Mathematics and Physics Supervisors: Prof. Abdel-Latif E. Hussien Prof. Hossam A. H. Fahmy Assoc. Prof. Ahmed G. Radwan Examiners: Prof. Nasser H. Sweilam, Professor, (External examiner) Faculty of Science, Cairo University Prof. Salwa K. Abd-El-Hafiz (Internal examiner) Prof. Abdel-Latif E. Hussien (Thesis main advisor) Prof. Hossam A. H. Fahmy (Member) Assoc. Prof. Ahmed G. Radwan (Member) Title of Thesis: Generalized Chaotic Maps and Elementary Functions between Analysis and Implementation Key Words: Logistic map; Tent map; Digital implementation; Finite precision; Power function Summary: In this thesis, we aim to bridge the gap between mathematical analysis of generalized one-dimensional discrete chaotic maps and their implementation on digital platforms. We propose several variations and generalizations on the logistic and tent maps and employ the power function in a general map that could yield each of them and other new maps. We present negative control parameter maps that provide wider alternating-sign output ranges which are controllable by scaling parameters. Moreover, the proposed general powering map is mathematically analyzed and verified for boundary cases. A transition map is presented that exhibits responses in the intermediate range from tent to logistic map. Finite precision logistic map is studied explaining the impact of finitude on its properties. In addition, floating-point implementations of the power function are tested on the occurrence of special values of the operands. Acknowledgements First and foremost, I am thankful to Almighty God for His uncountable grants upon all of us. I owe sincere gratitude to Prof. Abdel-Latif E. Hussein, Prof. Hossam A. H. Fahmy, and Assoc. Prof. Ahmed G. Radwan for their valuable guidance and support. It is my pleasure to have this distinguished group of professors as thesis advisors, and I hope I managed to be as trustworthy as they expected. They gave me so much of their precious time and helped me out of many problems with their knowledge and experience. I highly appreciate the too many hours that Prof. Hossam A. H. Fahmy and Assoc. Prof. Ahmed G. Radwan dedicated to our meetings which were undoubted fruitful towards presenting this work. They have always encouraged me to explore new areas of science whenever I come across them. They provided me with invaluable guidance in my early experiences in international publication. They answered my questions patiently and paid attention to every fine detail. Moreover, I would like to thank Assoc. Prof. Ahmed G. Radwan for being confident in my capabilities, answering all my inquiries on the spot in spite of his busy schedule, and motivating me to exert relentless efforts in my work. I am deeply indebted to Prof. Hossam A. H. Fahmy for his keenness on keeping my morale high. He has played a major role in the development of my personality and mind. In fact, he spares no effort to offer moral and scientific support to all his students. All words of gratitude fail to express how much I owe him for he always guides, helps, and encourages me to acquire more skills and experiences. I thank my beloved mother for her invaluable support during my life. I could not have accomplished any success without her continual efforts to let me focus on my studying. Not to mention my beloved younger brother who is a source of inspiration and a motivation for success. I am grateful to every teacher and professor who contributed to my knowledge and my decisions throughout the route towards my career. Finally, I would like to thank my dear friends and colleagues; specifically those who are patiently always a source of support and kindness and those who have helped me since day one as a teaching assistant and postgraduate student until now. May Allah bless and reward all those who have supported me. i Table of Contents Acknowledgements i Table of Contents ii List of Tables vii List of Figures viii List of Symbols and Abbreviations xii Abstract xiii 1 INTRODUCTION 1 1.1 Motivation . 1 1.1.1 Chaotic Maps and Numerical Computation . 2 1.1.2 Power Function and Floating-Point Representation . 4 1.2 Main Contributions . 5 1.3 Publications out of This Work . 6 1.4 Organization of the Thesis . 6 2 REVIEW ABOUT THE STUDIED PROBLEMS 8 2.1 Chaos: Motivation and Mathematical Analysis . 8 2.1.1 Historical Background . 9 2.1.2 Conventional Logistic Map xn+1 = λxn(1 − xn) . 12 2.1.2.1 Exponential Versus Logistic Growth Models . 12 2.1.2.2 Operands’ Ranges . 13 2.1.2.3 Main Definitions: Fixed Points and Periodic Points . 14 2.1.2.4 Bifurcation Diagram . 14 2.1.2.4.1 Fixed and Periodic Points and Their Stability . 16 2.1.2.4.2 Different Phases on the Diagram . 16 2.1.2.4.3 Key-points of the Bifurcation Diagram . 17 2.1.3 Conventional Tent Map xn+1 = µmin(xn;1 − xn) . 17 2.1.3.1 Operands’ Ranges . 18 2.1.3.2 Bifurcation Diagram . 18 2.1.4 Maximum Lyapunov Exponent . 18 2.2 Negative Operands: How and Why? . 19 2.2.1 The Negative Parameter Case First Visited . 19 2.2.2 Enhancement of Randomness Measures . 20 2.2.3 Wider Domain of Modeling With Improved Efficiency . 21 2.3 Generalized Chaotic Maps . 23 2.4 Digital Chaotic Maps . 25 2.4.1 Multiple Precision Versus Fixed Precision . 26 2.4.2 Previous Work . 26 2.5 Elementary Functions: The Power Operation . 29 ii 2.5.1 IEEE Standard Definitions . 30 2.5.1.1 Special Values . 30 2.5.1.1.1 Signed Zero . 30 2.5.1.1.2 Infinities . 30 2.5.1.1.3 NaN . 31 2.5.1.2 Operations . 31 2.5.1.3 Operations Generating NaN (QNaN) . 31 2.5.1.4 Exceptions and Pre-Substitutions . 32 2.5.1.5 Rounding . 32 2.5.2 Previous Implementations . 33 3 SCALED POSITIVE, NEGATIVE, AND ALTERNATING SIGN MAPS 35 3.1 Generalizations of One-Dimensional Maps . 35 3.1.1 The Proposed Maps . 35 3.1.1.1 Variations on Logistic Map . 36 3.1.1.2 Variations on Tent Map . 37 3.1.2 Alternating Sign Maps . 38 3.1.2.1 Mostly Positive Logistic Map . 38 3.1.2.1.1 Parameters’ Ranges . 38 3.1.2.1.2 Fixed Points and Their Stability . 41 3.1.2.2 Mostly Positive Tent Map . 42 3.1.2.2.1 Parameters’ Ranges . 42 3.1.2.2.2 Fixed Points and Their Stability . 42 3.1.2.2.3 Periodic Points and Their Stability . 43 3.2 Scaled Logistic Map: Analysis and Results . 43 3.2.1 Independent Scaling xn+1 = ±λxn(a ± bxn) . 43 3.2.1.1 Positive Logistic Map f (x;λ,a;b) = λx(a − bx) . 43 3.2.1.1.1 Range of λ ................... 43 3.2.1.1.2 Fixed Points and Stability Condition . 44 3.2.1.2 Mostly Positive Logistic Map f (x;λ,a;b) = −λx(a − bx) 44 3.2.1.2.1 Range of λ ................... 44 3.2.1.2.2 Fixed Points and Stability Condition . 45 3.2.1.3 Maximum Lyapunov Exponent . 45 3.2.2 Vertical Scaling xn+1 = ±λxn(1 ± bxn) . 46 3.2.2.1 Positive Logistic Map f (x;λ,b) = λx(1 − bx) . 46 3.2.2.1.1 Range of λ ................... 46 3.2.2.1.2 Fixed Points and Stability Condition . 46 3.2.2.1.3 Steady State Solutions Versus System Parameters 46 3.2.2.2 Mostly Positive Logistic Map f (x;λ,b) = −λx(1 − bx) . 48 3.2.2.2.1 Range of λ ................... 48 3.2.2.2.2 Fixed Points and Stability Condition . 48 3.2.2.2.3 Steady State Solutions Versus System Parameters 48 3.2.2.3 Maximum Lyapunov Exponent . 49 3.2.3 Zooming xn+1 = ±λxn(a ± xn)...................
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