Beyond Nonstructural Quantitative Analysis - Blown-Ups, Spinning Currents and Modern Science

BLOWN-UPS, SPINNING CURRENTS AND MODERN SCIENCE

Yong WU YiUN

World Scientific

BEYOND HOHSTRUCTURHL QUHNTITHTIVE flHHLYSIS BLOWN-UPS, SPINNING CURRENTS AND MODERN SCIENCE

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BLOWN-UPS, SPINNING CURRENTS AND MODERN SCIENCE

Yong WU National Soil Bureau of Fuling District, China

Yi LIN International Institute for General Systems Studies, USA

V f e World Scientific lflfe

• Hong Kong New Jersey • London •Singapore Sir

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

BEYOND NONSTRUCTURAL QUANTITATIVE ANALYSIS Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4839-3

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Foreword

At the beginning of the 1980s, an oversea scholar, who was a natural scientist, came to China and wanted to discuss about the Book of Change and Lao Tzu. However, to his surprise, very few Chinese natural scientists, which may be because he only met a limited few, ever studied these Chinese classics. Through great difficulty and some luck, he found me. Of course, as a Chinese scholar of my age, specializing in a branch of the natural sciences, I never had a chance to receive a systematic training of any ancient Chinese classics. At about the same time, during that special historical moment, the Book of Change was seen as a religious book by the government, and consequently became a banned book. It was only when I was a labor worker in a factory during the so-called Cultural Revolution, a fellow factory worker brought a really deteriorating copy of the Book of Change to me to discuss the meanings of some words, which could hardly be found in ordinary dictionaries, that I had an informal opportunity to read the classic. And, very soon I became totally immersed in the contents. As a characteristic of strong figurative thinkers, I naturally realized the fact that the Book of Change and Lao Tzu describe the physical world through processes of figurative recognition and the fact that they analyze materials' and events' evolutions through figurative structural analyses. Specifically, Lao Tzu emphasized on how to study and how to understand the world through fluids by introducing the concepts of "rong" and "constant rong" so that the concept of Tao could be comprehended. What has influenced me the most is Lao Tzu's teachings: "All things are impregnated by two altering tendencies, the tendency toward completion and the tendency toward initiation, which acting together, complete each other" and "far away

VI

Foreword

means reversing". It can also be said that Lao Tzu's teachings have played a crucial role in the changes of my thinking logic and have greatly supported the formation of a figurative thinking logic of the early years of my professional career. Evidently, in order not to disappoint the visitor from far away, I did all I could to explain the origin and methods, based on my own superficial understanding of "abstracting quantities (numbers) into shapes (figures)" (figurative structures) of ancient China, developed about 3,000 years ago. Then, I emphasized on the difference between this ancient method and modern theories and methodological systems of the western civilization established in the past three hundred years. Surely, as expected, we touched on my various first-hand experiences, consequent thinkings and reasonings, relevant epistemological problems on figurative and logical analyses applied in scientific explorations, and some hot topics existing in the modern theories and methodological systems. For example, our discussion touched on the crisis of mechanics, the "hands of God" of the first push, the materials' structures of the second stir, nonlinear evolutions, reversal changes of evolutions — blown-ups, whether or not "God plays dice", the debate between quantitative determinacy and indeterminacy, the definiteness of materials' structures, chaos theory, equal-quantitative effects and inequal-quantitative effects, economic uncertainties, and other problems alike. The topics, dealt with in our discussions, interested so many people that they recommended that I write down these ideas on paper. Unfortunately, due to various administrative responsibilities of the time, the fact that some influential Chinese scholars thought that my opinions were ridiculous, and the reason that the Book of Change could not be formally mentioned in any form of scholarly works, I did not dare to write anything down. In early 1990s, after all administrative responsibilities were removed, I started to speak up about my true beliefs gradually. Yong Wu, one of my former students during 1981-1985, spent most of his college life and years later on the study of theoretical physics and became quite an expert in the area of quantum mechanics. Even though he did not pursue an academic career in physics or meteorology, he has never stopped his own scientific exploration, and has published numerous high quality scholarly works. Even though some colleagues have tried to convince him to pursue an academic career, he has decided to enjoy a self-fulfilling career not for any personal gains. Mr. Wu is a person who thinks quickly and independently so that he has never been lured into studies of currently hot topics. Even

Foreword

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though he has gained much respect and appraisals from his colleagues for his achievements and organizational abilities, he has never been satisfied and still continues his self-fulfilling scientific pursuit. As a faithful follower of Albert Einstein's theories, when he heard about my recent work, he used his personal leave to come to Chengdu, traveling thousands of miles to discuss epistemological problems with me. Especially, when having learned that I did not have the necessary funds to carry out what I was working on, he unexpectedly left a considerable amount of cash for me to complete the project successfully. Needless to say, it was difficult to find such a young and strong supporter. At the same time, his ability to quickly understand and master the works, which took me over 30 years to fathom, has made me see the hope in the coming generations. Later, he joined my efforts in organizing the special volume, entitled "Mystery of Nonlinearity and Lorenz's chaos", which was edited by Yi Lin and published in "Kybernetes: The International Journal of Systems and Cybernetics" in 1998. Mr. Wu's scientific career path is different from those of many other scholars. His success once again proves the fact that the institutional mode of teaching and learning is not the only or effective way leading to intellectual growth and scientific discoveries. Besides personal efforts and hard work, what is more important is the thinking based on practical applications. In terms of the development of human civilizations, creativity is the key of all relevant factors. However, true creativity comes from practice and bold self-negations. Mr. Yi Lin is an American Chinese, a mathematician by training, and a famous systems scientist and applied mathematician. In his studies of mathematical and systems engineering problems, he realized the importance and depth of Lao Tzu's teachings and thinking. In the studies of topology, he understood the significance and effects of "abstracting numbers into shapes". As soon as he saw the first abstract of my work, he decided to devote his efforts into the study of blown-up systems. In his charactersitic swift manner, he has successfully pushed the blown-up theory and its methodology forward to the international level. Obviously,without Mr. Yi Lin, especially without his deftness, his super organizational ability, and his great oral and speaking language skills, it would be impossible to convince scholars from England and Germany, the birthplaces of calculus, to accept our opposing logics. It can also be said that it was due to Mr. Yi Lin that scholars of the western world started to notice what the Chinese have been doing and have accepted the phrase — blown-ups, created

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by Chinese. As for the formation of my own conceptual foundations, it is truly a story that makes it difficult for me to decide whether to cry or to laugh. It is because the era, in which I grew up, was an era without individual selves. Naturally, each person was a walking pile of quantitative cells without any organizational structures, any personal thoughts and purposes. All these walking piles of quantitative cells had been so fine tuned that they could only meet the calls and needs of the above. This era may very well be written in history as a silent era with lack of creativity. Due to an accidental opportunity, I was picked by my art teacher in junior high school to help him with his duty of teaching the societal need of his time. So, unconsciously, in my young mind, a figurative thinking logic was formed. And, unintentionally, I helped some fellow students solve one of their so-called tricky problems in spatial geometry. Consequently, their mathematics teacher (mistakenly) thought that I had the gift of learning mathematics. (As a matter of fact, my unintentional help came as a consequence of my figurative thinking logic.) To help overcome a national shortage of experts in science and technology, I was forced to study natural sciences. Since my figurative thinking logic had long ago been formed and rooted in my head, the now-classical quantitative logic thinking had always been difficult for me to grasp and to understand. So, over time, I have been trained to "translate" concepts and consequences of the logic thinking into those equivalent forms in my figurative thinking. As a consequence, my "beliefs" and "opinions", in meeting the need for such necessary translation, were thrown into a "painful chasm". In the language of religions, my soul was thrown into the Nether World. This end later became my motivation for nearly 30 years of thinking and practice in order to save my own "soul" from the Nether World. However, in the end, my figurative thinking logic, which was formed in my head first, has not changed a bit. Instead, it has been greatly developed in my pursuit of natural sciences. In the area of arts, I had been taught to "merge objects into the nature and create your own distinguished styles". And also, "all, who are similar to my style, will have to die", as said by Qi Bai Zhi, and "only those, who are different and have been looked down by others, will be teachers of others", as said by Chen Dayu. So, in my head, these kinds of explorations of figurative structures have formed a fundamental difference from those nonstructural quantitative studies of the natural scientific particle systems, developed in the past 300 plus years. The assumptions of particle's evenness

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IX

and calculus's continuity have made variable mathematics a regularized computational scheme. So, such schemes will naturally disagree with the realistic unevenness, discontinuities and irregularities. This end leads to what Godel's incompleteness theorem is about: "The whole of mathematics cannot be established on any orderly axiomatic system. Each mathematical system (theory), no matter how complicated it might be, cannot avoid all paradoxes". As a consequence of my reasonings, I gradually formed my trust in the figurative thinking logic: "Figures not only process the scientificality, but are also more complete than nonstructural quantitative analysis systems". This is where my later concept of "abstracting numbers back to shapes" and its methodology came from. For example, the research on nonlinearity in the past nearly 240 years has mostly focused on the analysis of the forms of motion, while my research has just been in the opposite direction with focus on the analysis of the structure of acting forces — the "God" of modern science. As a consequence of my figurative thinking logic, by emphasizing on the structural unevenness of "time and space" of the acting and the reacting objects, my colleagues and I have revealed the fact that what's described by nonlinear evolution models are eddy motions instead of wave motions. This end agrees well with the fact of spinning materials in the universe. Especially, the structural unevenness of materials has to cause irregularities of eddy motions so that different directional eddies and various eddies, sub-eddies, sub-sub-eddies of different levels are naturally formed. In terms of transformations of energies, the existence of these different eddies helps to complete all heat-kinetic energy transformations. Also, because of these eddies, the concepts of complexity in the form of motions, quantitative complexity, multiplicity, and structural uncertainty have been introduced. Thus, many currently, widely studied problems, such as the nonlinear evolution form, quantitative complexity, multiplicity, etc., are only about formal concepts without touching on the essence of the underlying problems. It is because when seen from the angle of the vectority of materials' structures, the world is simple, which is not the same as the quantitative abstraction, in terms of the summary of structural functions and effects. Since the vectority consists of only two directional rotations: clockwise and counterclockwise, our simplicity of the world coincides with the concept of Yin and Yang, as introduced in the Book of Change over 3,000 years ago, and also agrees with Lao Tzu's "rong" of the world when seen through fluids. Especially, in practical applications, evolution problems of motion can be analyzed through

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Foreword

differences shown in functions and effects of the relevant vectorities. This method has been shown to be more effective than the quantitative analysis under equal-quantitative effects and constitutes a geometric structural method of analysis, which is different from the methods based on either equations or statistics. When materials' figurative structures are employed as the theory and methodology to trace the root of problems of interest, one can naturally realize the fact that the core of the modern scientific system, developed in the past 300 plus years, is the "abstraction of shapes into numbers". Since the establishment of the particle theory and the quantitative analysis method, in terms of their forms, is extremely clear, they have been seen as magnificent achievements in mechanical analysis. However, their essence is the typical artifact of "quantification" so that the research of the past 300 plus years, in terms of quantitative comparability, has played an important role in the development of modern science and technology. That is also the origin from which "quantitative comparability" has been equated with scientificality. However, evenness and continuity are products of the thinking logic of nonstructures, on which the relevant epistemology is incomplete, the relevant methodology is also conditional. Such methodology can only be applied to inequal-quantitative effects and becomes invalid where equal-quantitative effects are concerned. What needs to be pointed out is that due to successes of the thinking logic of particles in studies like celestial bodies and movements of solids (except break-offs of solids) under inequal-quantitative effects, some scholars have been misled to believe that such thinking logic can be applied to any situation satisfying no specific sets of conditions. They always wish to generalize the past successes into new territories, including situations under the effects of equal-quantities. However, just as what Engel said, when he summarized the mathematical research of the nineteenth century: The absoluteness in solids ... met great difficulties in fluids, ... Even though the 20th century is almost over, no fundamental changes have occurred to correct the situation. And, all generalizations of the particle theory are still continued in the form of assumed infinitesimal increments. For example, the Rossby's long wave theory in the meteorological sciences is still seen as a classical and important theory. As a matter of fact, with the figurative thinking logic, it can be very easily and quickly seen that it is impossible for unidirectional waves to exist on a spherical surface. And, even hydrological surveyors have the common knowledge that the

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property of uni-directionality implies flows. Therefore, it is very clear that the Rossby's long wave theory does not truly hold true in reality. However, not only is this long wave theory seen as a classical and important theory in modern meteorology, but also baroclinics are treated as wave motions. Evidently, since each baroclinic is an eddy source, its resultant movements can only be eddy motions. Treating these movements as wave motions can only be an artificial extrapolation of the "form of the formal" particle doctrine without any practical significance. This result is also the very reason why meteorological scientists do not generally have the ability to predict weather changes and where the following irony in the community of meteorological sciences come from: Those specialists, good at truly forecasting weather changes, can hardly be seen as meteorological scientists. Since the science of fluids can no longer progress meaningfully by continuously employing the nonstructural particle mechanic system, as its methodology, weather evolutions are problems under equal-quantitative effects. As a second example, let us look at the concept of gravitational waves, which was introduced by Albert Einstein in 1916 based on his linear gravitational wave equation, established on his curved "time and space". However, after over 80 years of attempts of several generations of scientific minds, no one has successfully found or measured the existence of the so-called gravitons. In fact, the curved "time and space" do have a structure and Einstein did not end the era of the particle doctrine. Secondly, curved "time and space" are not linear, or at least Einstein did not know the fundamental difference between linearity and nonlinearity. What is more important is that the speed of light should not have been treated as a constant. Therefore, even assuming that we still employ the "gravitational wave" equation, due to changes in the speed of light and unevenness in energy tensors, an eddy source is formed. So, eddy motions would surely be the consequences of the structural unevenness of materials. So, nonlinearity is a problem about figurative structures instead of that of linear, nonstructural quantities. Obviously, problems of structures can no longer be described faithfully by quantities only. It is a real pity to see that Einstein, as such a great scholar that he had determined the beginning of a brand new era, and had proved the curvature of light-rays, and proposed the unevenness of "time and space" as the origin of all gravitations, repeated the path of (nonstructural) quantities without structural time and space. So, the current crisis of the mechanics is not from within the mechanics. Instead, it originates from the quantitative analysis system, established in

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the past 300 plus years. Newton completed the quantification of the classical mechanics through his particle doctrine. Albert Einstein established the quantification of "mutual reactions" of fields through transportations of energies. However, what is more important is that mutual reactions of materials involve not only problems of quantities, but also problems of structural transformations. The completion of their book Beyond Nonstructural Quantitative Analysis implies that Mr. Yong Wu and Mr. Yi Lin can not only enter the space of the "quantitative void", but also return to the humanly "broken time and space". By facing some problems of natural evolutions, they proposed their points of view different from tradition. Their purpose of doing so is to study theories faithfully describing nature instead of fitting the tools employed. Especially, they faced the phenomenon widely existing in some theoretical studies: Instead of explaining an inexplicable natural event, a nonexisting inexplicable is explained inexplicably. This widespread phenomenon naturally leads people to question the objectivity of the foundation of the relevant classical theories, such as continuous particles and media, developed on the thinking logic of nonstructural particles and the assumption of continuity. When Newton, as an epoch-making scientist, was analyzing solids based on quantitative analysis, Lao Tzu, as a nonscientific scholar, had long ago, more than 2,500 years ago to be more specific, had studied fluids figuratively and clearly said that "the one, who is good at thinking, does not need to plan". When Newton's mechanics only works under the first push of the "God", the Book of Change, written over 3,000 years ago, had clearly spelled out how to apply figurative structural analysis to foretell changes and had pointed out to the existence of such a law of structural conservation. Even when Shakespeare knew that "quantitative comparisons" were only a relative method, numerical comparability became the only standard for scientificality. In other words, if no two identical objects can be found, how can one tell any scientificality? When effective experiences are treated as nonscientific, the "scientific" first push has never worked. What a paradox exists in modern science! What should be admitted is that the quantitative analysis, developed in the past 300 plus years, has indeed played an important role in modern civilization. However, it has also left behind a huge amount of unsettled problems and paradoxes. Of course, it is reasonable to expect the appearance of unrealistic conclusions and beliefs in the process of explorations and the appearance of paradoxes and seemingly unsolvable problems. However,

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when the realisticity and the objectivity of the relevant epistemology are at stake, it is no longer as simple a problem as whether the results are correct or incorrect. As soon as Beyond Nonstructural Quantitative Analysis is published, it will naturally lead to debates involving whether or not the quantitative analysis is the only methodology which can be and should be employed to comprehend the world (or whether or not the science is not limited to the quantitative analysis). That might be the authors' purpose of writing such a book, since when a new publication leads to no debates, it must belong to well-accustomed subject matter without any future significance. As a teacher, I like to see young people thinking. As unevenness of materials, thinking is also a form of accumulation of energy. Its consequence can only be blown-ups of those well-accustomed beliefs and prejudices. As a person who has gone through thorny bushes, what is important for me to do is not to become drunk with the memories of the past. It is because the future will never be identical to the past, since the essence of evolutions is "rolling forward" instead of backward. The development of science has always left marks behind, labeling different eras. It is this reality that spells truthfully the meaning of scientific explorations.

Shoucheng OuYang Chengdu University of Information Technology Chengdu 610041, PR China This article was written when I was visiting the International Institute for General Systems Studies, Inc., thanks to Professor Yi Lin, the president and director of the Institute (http://www.iigss.net), Grove City, Pennsylvania, USA, during July-September, 2000

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Contents

Foreword

v

Chapter 1 Introduction 1.1 Scientific Discoveries 1.2 Nonlinear Science 1.3 Some Ancient Thoughts of the East and the West 1.4 Determinacy and Randomness 1.5 Equal Quantitative Effects 1.6 Organization of This Book 1.7 References

1 1 4 6 8 11 13 15

C h a p t e r 2 N o n l i n e a r i t y : T h e C o n c l u s i o n of C a l c u l u s 2.1 A Brief History of Calculus 2.2 T h e Method of Differential Analysis 2.2.1 Functions and Their Properties 2.2.1.1 Representations of Functions 2.2.1.2 General Properties of Function 2.2.2 Limits of Functions 2.2.3 Continuous Functions 2.2.4 T h e Concept and Properties of Differentials 2.3 T h e Well-Posedness and Singularity of Differential Equations . 2.4 Discontinuity: T h e Mathematical Characteristic of Nonlinear Evolutions 2.5 Question the Traditional Treatments of Nonlinearity 2.5.1 Linearization

17 17 21 21 22 23 24 26 28 31

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38 47 47

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2.5.2 2.5.3

2.6

Stabilization Comparison between Spectral Method and Numerical Schemes 2.5.4 Limitations of Lyapunov Exponents References

Chapter 3 Blown-Up Theory: The Beginning of the Era of Discontinuity 3.1 Looking at Whole Evolutions 3.2 Mathematical Physics Meanings of Blown-Ups 3.3 Nonlinear Transitional Changes: A Mathematical Character of Blown-Ups 3.3.1 Blown-Ups of Quadratic Nonlinear Models 3.3.2 Blown-Ups of Cubic Polynomial Models 3.3.3 Blown-Ups of nth Degree Polynomial Models 3.3.4 Blown-Ups of Higher Order Nonlinear Evolution Systems 3.3.5 Blown-Ups of Nonlinear Time-Space Evolution Equations 3.4 Mapping Properties of Blown-Ups and Related Observ-Control Problems 3.5 Spinning Current: A Physics Characteristic of Blown-Ups . . . 3.5.1 Bjerkness Circulation Theorem 3.5.2 Eddy Motions of the General Dynamic System 3.5.3 The Concept of "Wave Motions" and That of Eddy Motions 3.6 Equal Quantitative Effects 3.7 References Chapter 4 Puzzles of the Fluid Science 4.1 Fluids: A Historical Perspective 4.2 Convergency (Divergency) of Moving Fluids 4.2.1 Blown-Ups and Characteristics of Reversal Transformations of Converging Fluids 4.2.2 Blown-Ups and Transformational Characteristics of Spinning Currents 4.3 Dynamic Characteristics of Navier-Stokes Equation 4.4 Some Problems about Atmospheric Long Waves 4.5 Problem of Objectivity of Hadly Circulations 4.6 Problems Related to the Modeling and Solution of KdV and Burgers Equations

47 48 49 51

53 53 54 57 58 62 64 66 69 70 75 76 81 83 87 90 91 91 95 95 97 100 102 105 107

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4.6.1

4.7

4.8

KdV Equation and Related Problems .• 107 4.6.1.1 Problems in the Modelling of the KdV Equationl07 4.6.1.2 Some Problems in the Solutions of the KdV Equation Ill 4.6.2 A General Discussion on the Solution of Burgers Equationll3 Is Meteorological Science a Part of Geophysics or Atmospheric Science? 114 4.7.1 The Problem on the Atmospheric Temperature Structure 115 4.7.2 The Problem of the Planetary West Wind Circulation . 116 References 117

Chapter 5 Questions about Nonlinear Macro-Evolution Theory 119 5.1 Introduction 120 5.2 Does "Chaos" Objectively Exist? 121 5.2.1 About the Meaning of the Concept of Chaos 122 5.2.2 Determinacy and Randomness 123 5.2.3 About the Non-Realisticity of Lorenz's "Chaos" . . . . 127 5.2.4 Some Problems with Regard to Classical "Chaos" Models 130 5.2.4.1 General Analytic Characteristics of the Population Model 130 5.2.4.2 Problems Related to the Discreticization of the Population Model 132 5.3 Some Problems about the One-Dimensional Iteration Formula . 134 5.3.1 Iteration Properties of the One-dimensional Iteration Formula 134 5.3.2 Some Problems Existing in Discussions of "chaos" . . . 134 5.4 Problems Related to Thermodynamics 136 5.4.1 Entropy Changes in "Linear and Nonlinear Irreversible Processes" 139 5.4.2 About the Entropy Equilibrium Equation of NonEquilibrium Thermo-Local Fields 146 5.5 Methodological Problems of Synergetics 148 5.5.1 Classification of Fast and Slow Variables 150 5.5.2 The Problem of Expansion of Fast Variables 152 5.5.3 Explosive Evolutionary Growth 152 5.6 Some Problems Regarding Catastrophe Theory 153 5.6.1 The Model of Folding Catastrophe 153

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5.6.2 Blown-Up Properties of the Folding Model References

154 155

Chapter 6 Problems Existing in Theories of Microscopic Evolutions 157 6.1 Some Relevant Historical Recall 157 6.2 Rules of Quantum Motion and Characteristics of Singularity . 159 6.2.1 Born's Probabilistic Explanation 160 6.2.2 The Singular Characteristics of Quantum Movements . 162 6.3 Inaccurate Measurements and Equal Quantitative Effects . . . 164 6.4 Discussion on "Quantum Chaos" 167 6.5 Blown-Up Theory of Laser Emission 169 6.5.1 A Simple Laser Evolution Model 169 6.5.2 Blown-Ups of the Simple Laser Evolution Model . . . . 171 6.5.3 Laser Field Equation 174 6.6 References 175 Chapter 7 Some Problems Existing in the Field Theory 7.1 The Concept of Fields and Its Historical Importance 7.2 The Blown-Up Problem of the Evolution of Higgs Field . . . . 7.3 Some Problems about the Evolution of SU (2) Gauge Field . . 7.4 Do "Gravitational Waves" Objectively Exist? 7.5 Some Problems about "Universal Gravitation" 7.6 References

177 177 179 181 183 188 190

Chapter 8 Difficulties Facing the Dynamics of Nonlinear Chemical Reactions 193 8.1 Chemical Reactions and Their Rates 193 8.2 The Blown-Up Problem on Gaseous Chemical Reactions . . . . 195 8.3 The Blown-Up Problem on Liquid Chemical Reactions 196 8.4 The Blown-Up Problem on Schlog Reaction Model 198 8.4.1 F has a real root and a pair of complex conjugate roots 199 8.4.2 F has real roots and of multiplicity 2 and 1, respectively 200 8.5 The Blown-Up Problem on Chain-Reaction Models 201 8.5.1 Chain Reactions without Any Mutual Reactions . . . . 203 8.5.2 Chain Reactions with Mutual Reactions 205 8.6 References 208

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Chapter 9 Nonlinearity and Problems on Theories of Ecological Evolutions 211 9.1 The Population Evolution Equation 211 9.2 Evolution Problems of Logistic Models 212 9.3 Blown-Up Characteristics of Improved Logistic Models 215 9.4 References 220 Chapter 10 Nonlinearity and the Blown-Up Theory of Economic Evolution Systems 221 10.1 The Start of Economics and Inherent Difficulties 222 10.2 Evolution Problem in Merchandise Prices 224 10.3 The Evolution Problem on Competitions between Economic Sectors and Individual Enterprises 228 10.4 References 231 Appendix A Records of Shoucheng OuYang's Thoughts and Dialectical Logics (1959 - 1996) 233 A.l The Chapter on Thoughts 233 A.2 The Chapter on Search and Refinement 247 A.3 The Chapter of Attachment 251 A.4 Non-Ending Conclusions 255 A.5 Afterword 259 Appendix B A n Interview with Shoucheng OuYang

261

Appendix C Four Main Flaws and Ten Major Doubtful Cases of the Past 300 Years 269 C.l The Four Main Flaws of the Past 300 Years 269 C.2 The Ten Major Doubtful Cases of the Past 300 Years 270 Appendix D Structural Rolling Currents and Disastrous Weather Forecasting 273 D.l The Beginning of the Whole Story 273 D.2 The Discovery of the Ultra-Low Temperature 274

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D.3 Weather Situational and Factor Predictions Based on Structural Currents D.3.1 Weather Situational Predictions D.3.2 Factor Predictions Based on V-30 Rolling Currents . . D.3.3 Forecasting the Precipitation D.4 Some Final Words D.5 References

277 277 283 286 287 287

Appendix E Afterword E.l Existence of Numerical Quantities E.2 Unification of Units E.3 Materialism of "Time and Space" E.4 How Does Our Book Come About?

289 290 295 297 299

Bibliography

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Index

311

Chapter 1

Introduction

Even though nothing specific can be seen clearly, there exist figurative structures in the fuzziness. - Lao Tzu When we enjoy the "sunshine" and the "mountain dew" of the modern science, as infants sucking their mothers' milk, when we are comforted by the plentiful supply of materials, offered by the modern science and technology, and when we are faced with the current worldwide "knowledge and informational explosions", as a consequence of the computer technology, how many of us would seriously and carefully think about the limitations of and the problems existing in the science and technology we have inherited from the generations before us?

1.1

Scientific Discoveries

From the primeval to the modern civilizations, the mankind has gone through a history of development for over several millions of years. However, a relatively well-recorded history only goes back to as far as about 3,000 years BC. From the beginnings of the ancient Babylonia, who lived along the reaches of Tigris and Euphrates Rivers (around current Iraq), the ancient Egyptians, who lived along Nile River, Mayans, who lived in America, and the ancient Chinese, who lived along Yangtze and Yellow Rivers of the east, to the rise and fall of the ancient Greek Civilization, from the revival of the human civilizations, characterized by the European Renais1

2

Introduction

sance, to the modern society with highly developed science and technology, the mankind has grown out of its knowledge infancy and entered its period of youth with highly advanced science and technology. From the past of passively accepting bestows of the n a t u r e to t h e present of employing scientific knowledge to initiate changes of the n a t u r a l environment in which the mankind lives, all the h u m a n history has shown the fact t h a t progresses of the modern science have brought forward prosperities to the materialistic and spiritual civilizations of the mankind. Therefore, the development history of science is also t h a t of h u m a n civilizations. However, there never exist any absolute t r u t h and any absolute authority. Each scientific progress has always been the h u m a n consequence of persistently pursuing after the ultimate t r u t h . Each progress of pursuing after the ultimate t r u t h is also a process of continuously getting rid of the stale and taking in the fresh and a process of making new discoveries and new creativities. In the scientific history, each time when the authority was repudiated, the science was reborn again. Each time when the " t r u t h " was questioned, a scientific prosperity appeared. Ancient Greek astronomer C. Ptolemy (around 85 - 165 A. D.) proposed the "geocentric theory" so t h a t he could plausibly explain lunar and solar eclipses a n d planetary movements observable with the then-technology. This end could definitely be seen as a scientific progress of the historical moment. Soon after the population accepted t h e "geocentric theory", the theory was assigned with the holy "spirit" by churches and governed the h u m a n race as the doctrine of authority for the next 1,400 years. During the E u r o p e a n Renaissance, Polish astronomer N. Copernicus (1473 - 1543) gave u p the "geocentric theory" and introduced t h e "heliocentric theory" which basically agreed with the objective reality. Even though Copernicus's "heliocentric theory" suffered from extreme objections from t h e churches, the scientific t r u t h eventually won the victory against religious forces and all prejudices of the tradition; and the t r u e science took the place of the "authority" . Therefore, when people praise authorities, they are also praising ignorance. In t e r m s of the history of the modern science, one should not forget about t h e eve of t h e forthcoming scientific revolution at the end of the nineteenth century. At t h a t time, Newtonian (I. Newton, 1642 - 1727) mechanics (or classical mechanics) had been developed for over two centuries. And, a complete theoretical system a n d a mechanical view of the n a t u r e had been firmly established and permeated into almost all corners of the

Scientific

Discoveries

3

natural sciences. Such a mechanical view had deeply affected the development of epistemology, methodology and the entire philosophical view of the nature. In such a scientific atmosphere, the majority of the scientific community, including such big names as C. Huygens (1629 - 1695) and L. Kelvin (1824 - 1907), believed that on the basis of Newtonian mechanics, the natural science had developed to its final stage. The only remaining works were some minor computations of the last a few decimal places. However, such victorious atmosphere was not absolute. It was similar to the quiet time period before the arrival of a major thunderstorm. On this quiet eve of a major scientific revolution, there, as always in the history, appeared some pioneers of thoughts who broke the dead silence before others did. For example, E. Mach (1838 - 1916) and J. H. Poincare (1854 - 1912) were among the pioneers. They took the position of criticizing the classical physics and believed that it was a prejudice if all physical phenomena had to be explained by employing the mechanical point of view. Especially, Mach criticized Newton's absolute time and absolute space. At the same time, laboratory works also revealed the inability of the then-most current physics in terms of explaining the experiment of heat radiations and Michelson-Morley experiment. Therefore, these two experiments were playfully called two "black clouds" floating in the clear sky of the classical physics. Accompanying the chimes of the tower clock for the arrival of the 20th century, the scientific discipline, physics, was added with a brand new page. In the year of 1900, M. K. E. L. Planck, a German physicist (1858 - 1947), proposed his quantum theory of black body radiation, which was different of the physics of his time, and lied down the foundation for the birth of the future quantum mechanics. In the years of 1905, Albert Einstein (1879 1955) published his narrow relativity theory and completely criticized the concepts of absolute time and space of the classical physics. Therefore, it was the two "black clouds", a series of newly designed experiments at the end of the nineteenth century, and a group of scholars, such as Planck, Einstein, etc., who were not afraid of authorities and who were intelligent and brave enough to bring forward pioneering works, that the torrential rains of the 20th century revolution of physics had officially arrived. This scientific revolution pushed the entire spectrum of natural sciences into a brand new era. The purpose of looking back to the history is to better understand and predict the future. When the walking steps of the history have stepped

Introduction

4

over t h e threshold of the 21st century, and when the morning sun of this new century is slowly and gradually rising, what kind of a new century are we going t o face and to expect?

1.2

Nonlinear Science

In China, t h e east of the world and t h e birthplace of Lao Tzu, there has lived a scientific worker, named Shoucheng OuYang. W i t h his profound insights, independent creativity, and the courage a n d willpower of taking all h u m a n "wastes" to rest, he proposed against all "tradition" the blown-up theory of nonlinear evolution problems based on a reversed thinking logic, a factual evidence, and over 30 years of reasoning a n d practice. Based on the n a t u r a l phenomenon of evolutions - "All things experience growth and declination; and all people have to be born and die", combined with t h e core statements of the Aristotelian formal logic, he discovered a series of weaknesses existing in the "form of forms". At the same time, he conducted a kind of harmonic analysis on the formal evolutions studied in dynamic systems and properties of "dynamics". First, he pointed out some improper aspects of the proposal of a "nonlinear science". Then, he reasoned and was led to the conclusion t h a t the most researches on nonlinearity, published in the past 300 plus years since the time of Newton, did not go beyond the Aristotelian thinking logic of t h e "form of forms". Therefore, he concluded t h a t in t e r m s of the formalism, nonlinear evolution models are the singularity problems of m a t h e m a t i c a l blown-ups of uneven formal evolutions; however, in terms of physical objectivity, nonlinear evolution models have described mutual reactions of uneven structures of materials, which is no longer a formal quantitative problem under t h e slaving first push; instead, it is a problem on m u t u a l slaving materials' structures. Evidently, uneven structures are eddy sources, which inevitably lead to eddy motions instead of waves. T h u s , the mystery of nonlinearity, which has been bothering the mankind for over 300 years, is resolved at once b o t h physically and mathematically. Such understanding reveals the essence of nonlinear evolutions - destruction of the initial-value automorphic structures and appearance of discontinuous singularities, a n d provides a theoretical analysis tool for studying objective transitional and reversal changes of materials and events. W h a t ' s more important is t h a t the concept of blown-ups not only p u t s a brake on t h a t of continuity, but also points out t h e various limitations of non-structural

Nonlinear

Science

5

quantitative analysis science, developed in the past 300 years on the basis of particles since the time of Newton. It can also be said t h a t at the beginning of a new era, the study of nonlinearity has led the science t o the direction and territory of materials' structural, figurative evolutions, which possess more objectivity and scientificality. Currently, the "nonlinear science", as known in n a t u r a l sciences, has been developing like raging fires with various theoretical branches popping up as quickly a n d as massively as b a m b o o shoots after spring rains. For example, I. Prigogine established a theory of dissipative structures. R. T h o m introduced t h e theory of catastrophe. H. Haken studied the theory of synergetics. E. N. Lorenz started the chaos theory. At the same time, nonlinearity, complexity, multiplicity, and "chaos" have been confused or mixed together. W h e n nonlinearity is connected with t h a t of randomness, the concepts of determinacy and stochastics are confused or tangled together so t h a t the study of dynamic systems has been t r a p p e d in difficult situations. Such awkward situation has affected many other relevant scientific disciplines a n d formed a roadblock in the development of n a t u r a l sciences. By looking back at the history of n a t u r a l sciences, it is not difficult to find t h a t the so-called "nonlinear problem" appeared at almost the same time as when calculus was born. During the time of Newton over 300 years ago, and the time of Euler (1707 - 1783), there h a d existed the so-called "three-body problem" and the "problem of fluid motions". It has only been due to historical reasons and conceptual limitations t h a t the method of solving analytically equations cannot be employed to study properties not determinable by the initial values. Consequently, nonlinearity has been seen as a specially difficult problem with extremely unexpected features until today. As time goes on, a n d as the development of the theories, methodology and thinking logic of linear problems become more complete and more m a t u r e , the desire of being able to resolve nonlinear problems increases drastically. Traditionally, a quite wide-ranging belief has been t h a t nonlinear problems are not much more t h a n generalizations and extensions of linear problems. Therefore, the well-developed theoretical methods and thinking logic for linear problems can be extended to the case of nonlinear problems with no or little modifications and proofs. As a m a t t e r of fact, it is a serious mistake. T h a n k s to t h e rapid development of the computer technology t h a t vari-

6

Introduction

ous numerical methods can be employed to solve nonlinear evolution problems. Relatively typical examples at this junction include V. Bjerknes, who proposed at the beginning of the 20th century the idea of treating problems of weather forecasting as those of solving initial value problems of m a t h e matical physics equations. L. F . Richardson later built up his courage and conducted a set of numerical experiments (1916 - 1918). Since his experiments met with the problem of computational instability of nonlinearity, they were seen as "failures". In 1952, after over 30 years of silence, A. Dauglas, a mathematician, proved t h a t nonlinear evolutionary initial value problems are probmes of singularity. So, the whole existence theorem cannot be obtained from any local existence theorems. However, his work did not caght much attention. In 1963, ten years after Dauglas's work, based on a model of spectral truncations, established for atmospheric convections, Lorenz discovered in his numerical analysis "nonperiodic flows" - chaos. In 1978, when he was computing the one-dimensional nonlinear population model using t h e iteration formula, Feigenbaum also discovered Lorenz's "nonperiodic flows". Therefore, the development of the chaos theory has started to become a pronoun for the study of nonlinear evolution problems a n d become fashionable in t h e studies of n a t u r a l sciences a n d philosophy for over two decades now. Even though the chaos theory has met difficulties in practical applications in the past a few years and s t a r t e d to cool down slowly, from the high of being the "3rd scientific revolution", bridging the gap between "determinacy and randomness", as said in t h e eighties, which led to the conclusion of " us being living in a probabilistic world", to the current low of internal randomness a n d t h a t the chaotic indeterminacy is not the same as stochastic uncertainties, etc., its spirit still floats over the magnificent scientific palace. T h e situation not only makes people not realize the essence of nonlinear evolution problems, but also makes people be limited by some historical concepts having passed down t h r o u g h the past 300 plus years.

1.3

S o m e A n c i e n t T h o u g h t s of t h e E a s t a n d t h e W e s t

In the ancient scientific history of t h e western civilization, there existed two opposite points of view on materials' structures. One point of view believed t h a t materials were made u p of uncountable a n d invisible (through h u m a n eyes) particles, named " a t o m s " , meaning in Greek "inseparable".

Some Ancient

Thoughts of the East and the West

7

T h e representative of this school of thought is the ancient Greek philosopher Democritus (about 460 - 370 B. C ) . T h e other point of view was t h a t "all materials are continuous". T h e representative of this school is the ancient philosopher Aristotle (384 - 322 B.C.). This school of thought had seen " e a r t h , water, gas, and fire" as the fundamental components of all materials. Since the superficial appearances of solids m a d e it easy for people to accept t h e Aristotelian concept of continuity, the theory of " a t o m s " was not treated with any validity until the early p a r t of the nineteenth century when J. Dalton (1766 - 1844) established relevant evidence. Not only are materials discontinuous, but also is time discontinuous, as pointed out by Shoucheng OuYang. It is because time comes from rotation movements of planets, and the rotation movements experience "vibrations". This end establishes the necessary foundation for the concept of "broken time and broken space" and clearly points to the materialisticity of " time and space". In principle, Leibniz (1663) and Newton's calculus was originated from the achievements of thoughts of Aristotle. After calculus was well established, Newton was t h e first person to build his laws of mechanics on the computational basis of calculus and accomplished the first successes in applications in celestial movements under inequal-quantitative effects. T h r o u g h over two centuries of development, t h e classical mechanics has gradually formed a whole set of analysis m e t h o d s based on continuity so t h a t even after q u a n t u m mechanics and relativity theory were established nearly a century ago, the thinking logic and methods, developed on continuity, have still be employed solely until today. More specially, after the modern physics has criticized almost all basic concepts of the classical physics, the concept of "continuous fields" of t h e classical physics are still widely employed. This fact indicates the belief t h a t t h e thinking logic system of the past 300 plus years is still about "continuous quantitative time and space", which implies the non-materialisticity of " t i m e and space". T h e difference between the ancient eastern a n d t h e western civilizations was mainly caused by differences in the environmental conditions and living circumstances. Since the western civilization was originated from castle-like environments, it naturally led to the m e t h o d of understanding the world through solids and easy accesses to the epistemological thinking of continuous materials. As a consequence, t h e western civilization put more emphasis on the aspect of continuous changes of materials. So, there existed a solid mental and cultural foundation for calculus and quantitative analysis to appear in the west. In comparison, t h e eastern civilization was

8

Introduction

originated from big river cultures with agriculture and water conservation as the foundation of their national prosperities. So, naturally, Chinese people had been more observatory about reversal and transitional changes of weather and rivers. Since the form of fluid motions is irregular and difficult to compute exactly, it naturally leads to the consequence that Chinese people has always had a strong ability to understand the world through the distinguishability of figurative structures of materials or events. That is the reason why the Book of Change and Lao Tzu appeared in China. The most obvious characteristic of the Book of Change is its way of knowing the world through materials' images, which constituted the origin of the figurative writing language of China, and of analyzing materials' changes through figurative structures. Herein, what's more emphasized are materials' irregularities, discontinuities, transitional and reversal changes. Even with our most current point of view, one can see that all eddy motions, as described with nonlinearities, are irregular. And, regularized mathematical methods become powerless in front of the challenge of solving discontinuously quantified deterministic problems of nonlinear evolution models. Therefore, from the discussion above, it can be seen that OuYang's work has essentially formed a challenge to the system of continuity and to the "God" implicitly existing in the works accomplished from the time of Aristotle to the time of Newton. Especially, what needs to be pointed out is that the conclusions that nonlinearity is a betrayal of the continuity and that nonlinearity, in terms of physics, represents eddy effects, have started to catch more attention of the scientific community. An interesting historical episode, which we like to mention here, is that when Professor OuYang and his colleagues did a huge amount of numerical calculations with their eyes on possible practical applications, they discovered that the "chaos" theory does not really agree with the objective reality. Since such a conclusion has made some supporters of the "chaos" doctrine very upset, we will have to consider the debate between determinacy and randomness in a little more details.

1.4

Determinacy and Randomness

This debate was originated from Laplace's initial value determinism (P. S. Laplace, 1749 - 1827). Such determinism has been badly criticized in the

Determinacy

and

Randomness

9

past one hundred plus years by the scholars specialized in the stochastic systems and been named as the point of view of "fatalism". Over time, this debate has spread into many scientific disciplines and been an important topic in the research of philosophy. In the past nearly 30 years, due to the appearance of the "chaos" doctrine, this debate has become a negation of Engel's statement that "accidents are born out of the inevitable". Evidently, Laplace's initial value determinism (of equations) was from the successful applications of calculus and classical mechanics in the situations under the effects of "inequal quantities". These success stories have formally hidden the non-objectivity of the assumptions of continuity and even particles so that calculus and mechanics have become the "cornerstones" of all studies of motions and related matters. It was only because in applications, irregularities and non-expendabilities of the initial value systems were met in studies of objective evolutions, Laplace's initial value fatalism was faced with criticisms from pursuers of indeterminacy. However, all the criticisms, as much as what we have read, have not truly touched on the essence of the system of continuity and the classical mechanics. Also, since in practical implementations of the stochastic system, the "quantitative comparability" is still the only standard, and since statistical methods cannot truly explain the causal relation of the matter of interest, and more reliable conclusions can only be drawn on stable time series, statistical methods are also a regularized mathematical method. In practical applications, compared to the equation systems of particle mechanics, statistical methods have a slightly larger range of applicable information. However, the truly irregular information still has to be thrown out. Since statistical methods cannot tell what the true past was, it is natural for people to feel the uncertainty about their statistically significant predictions of the possible future. Therefore, the debate between determinacy and randomness has evolved into an epistemological problem, which has been unsettled up to now. In the blown-up theory, what is pointed out the first, in terms of the mathematical form, is that due to the escape in an uneven form from continuity, the evolution of any nonlinear evolution model is no longer a problem of simply expanding the initial values. What is interesting here is the fact that through nonlinear evolutions, the concept of blown-ups can represent Lao Tzu's teaching that "all things are impregnated by two altering tendencies, the tendency toward completion and the tendency toward initiation, which acting together, complete each other", and agrees with non-initial-

10

Introduction

value automorphic evolutions as what the Book of Change describes: "At extremes, changes are inevitable. As soon as a change occurs, things will stabilize and stability implies long lasting". What is practically meaningful is that the blown-up theory employs calculus as its tool to end the era of Laplace's initial-value determinism. Since nonlinearities describe eddy motions and the totality of all eddy motions are constant, there must exist different eddy vectorities, and consequently irregularities. That is, the phenomenon of "orderlessness" is "inevitable". When looking at fluid motions from the angle of eddies, one can see that the corresponding quantitative irregularities, "orderlessness", "multiplicities and complexities", etc., are all the multiplicity of rotating materials. Thus, there exist underlying reasons for the appearance of quantitative and formal irregularities, multiplicities and complexities. The underlying reasons are the unevenness of " time and space" of the evolutionary mateirals' structures. So, all objective irregularities are consequences of some definite causes and must be deterministic, which is not in terms of equation-determinism. The corresponding concept of randomness is originated from quantitative irregularities without having led to any further understandings about the essence of materials' evolutions. The kind of determinacy as employed in the particle mechanics is only a special case with even or relatively even materials, which is a quantitative determinacy without any structure. This end is the very reason why Laplace's initial value determinism does not really talk about any commonality and does not truly agree with any part of the objective reality. Therefore, it has not been very convincing to criticize Laplace's " fatalism" from the angles of stochastics and probability. To this end, OuYang has clearly said: "Determinacy is a problem about materials' structures and is not simply a problem of formal mathematical quantifications". We think that such conclusion should be an important epistemological progress of the 20th century; and at the same time, it ends the hundred-year old debate between determinacy and indeterminacy. What's more important is that this conclusion points to a forthcoming revolution in scientific methodology: Dwelling exclusively on the traditional equation-type quantitative relations and statistical schemes will not reveal the essence and origins of materials' evolutions. Obviously, such opinion will surely make scholars with strong faith in the traditional believes astonished or "angry". However, if one can go beyond the concept of narrowly observing the world through quantities and relevant forms, and think deeply on the level of generalized observ-controls, this opinion seems

Equal Quantitative

Effects

11

to be reasonable and natural. With this end in mind, when we look back on the various red-faced discussions and debates of the past, we truly feel the uncertainty of whether to cry or to laugh. At this junction, we would like to mention that the first person, who discovered the distillation of OuYang's thinking logic and world view, is Professor Xuemou Wu, the father of pansystems theory. His comments are that OuYang is a figure with one foot standing in the Nether World and the other in the Heaven. The conceptual statement of "materials' structural determinacy" points out not only the problems, existing in Laplcae's initial value determinism, but also the fact that it is not convincing enough to criticize this determinism through the use of only the concept of randomness. At the same time, this conceptual phrase confirms the correctness of Engel's statement that "Accidents are born out of inevitable" and Einstein's belief that "The God does not play dice", and the incorrectedness of S. W. Hawking's conclusion that "The God plays dice". (Hawking has been seen by some as the Einstein of our time".)

1.5

Equal Quantitative Effects

Another important contribution of OuYng's blown-up system is the introduction of the so-called "equal-quantitative effects", which describes the conclusion of quantitative analysis under equal quantitative effects. Even though the concept of equal quantitative effects was initially proposed in the studies of fluid motions, it actually represents the fundamental and universal characteristic of materials' movements. What is important here is that this concept reveals the fact that nonlinearities are originated from the figurative structures of materials instead of non-structural quantities. When summarizing the situation of applications of the nineteenth century mathematics, Engel mentioned that "mathematics is absolute in the study of mechanics of solids, is approximate in the studies of gases, meets difficulties in the study of fluids, , and equal zero in the study of biology". (Quoted from Natural Dialectics). In the past one hundred plus years, scholars had tried whatever they could to change this situation. Especially after the 1970s, with the rapid development of computer technology, a heat wave of mathematical modeling and computer simulation has been spreading across the entire spectrum of science and technology to such a degree

12

Introduction

that to some people, it seems that all natural events and matters can be modeled and simulated by employing mathematics and that all problems can be resolved by using numerical computations. Such a wave of modeling and simulation has gone beyond natural sciences and spread into social sciences so that quantitative comparability has been seen as the only standard for scientificality. It is just because the world of learning has ignored the fact that all mathematics of variables is about mathematically regularized computational schemes, and the reality that the evolutions of objective matters have always been irregular quantitatively, for a relatively long period of time, scholars have ignored the studies of the relevant physics essence of irregularities. As a consequence, all high-speed computers, as the characteristics of the modern science and technology, have become "beautiful displays", since they cannot be or have not been any meaningful use in front of irregularities, appearing in objective evolutions. The thoughts, which have brought forward societal changes, are always valueless. Corresponding to the evolutions under equal-quantitative effects, OuYang proposed a method to compare structures. With this new approach as guidance, and by the analysis method of "converting numbers back into figure", he applied available meteorological information of ultralow temperatures, reversed ordering of quantities, uneven discontinuities, and structures of vectorities, in his real-life weather predictions, and has achieved breakthroughs on the problem of forecasting disastrous, reversal and transitional weather changes. This work not only greatly improves the accuracy rate of weather forecasting, but also practically spells out the fact that the chaos doctrine has nothing to do with the weather predictability. What's more important is that the method of structural analysis reveals the fact that evolutionary materials do not have any macro- and microcosmic difference. The concept of differentials and theories of continuous particles and media all belong to the realm of human illusions without much objective realisticity. What might be truly useful, left behind these human illusions, are the relevant computational schemes, which are only approximations under special conditions. Besides, the method of structural analysis has opened up a brand new direction for the theory and application of an evolution science. And, in terms of methodology, it might have located a playground for the modern topology to further develop and to be practically applied. Or, in other words, the overall incomputability of nonlinear evolutions has on one hand made the quantitative comparability of dynamics meet great difficulties,

Organization

of This Book

13

and on the other hand, provided a wide open field for an informational topology to rapidly develop. Therefore, the blown-up system has constituted two great leaps, one in the area of epistemology, and the other in that of methodology, showing a great future and potential.

1.6

Organization of This Book

In this book, based on the nonlinear evolutionary characteristics of the blown-up system, we will analyze many problems of nonlinear evolutions, appearing in various branches of natural sciences and social sciences. Our purpose of taking on such a huge job is to, hopefully, peal off and clean up the chaos of all the false illusions so that the situation of the current studies in dynamics and in applied mathematics, specifically in the areas of mathematical modeling and computational schemes, can be improved. If successful, we would hope to reach the goal that scientific explorations will ultimately describe the Tao of the nature instead of trying to fit the Tao into the tools employed. The organization of this book can be outlined as follows: Chapter two is devoted to the presentation of the so-called mystery of nonlinearity and how the modern study, entitled "nonlinear science", has been originated, after a brief introduction to an array of basic concepts of calculus. After showing that discontinuity is a fundamental characteristic of nonlinear evolutions, it is concluded that in order to revolve problems of nonlinearity, new methodology and thinking logic are needed. At the end, some traditional treatments of nonlinearity are analyzed. Chapter three is entirely occupied by the introduction of the blown-up theory, as the title suggests. It is argued that a fundamental mathematical characteristic of blown-ups is nonlinear transitional changes. Then, a geometric illustration is developed to explain why and how mathematical singularities could have and might have occurred, if one strictly stays within the domain of differential equations. By assigning physical meanings to relevant studies, it is shown that a physics characteristic of blown-ups is spinning currents and eddy motions. It is this place that the important concepts of the second stir and equal quantitative effects are introduced and the conceptual difference between wave and eddy motions are clarified. In the rest of the book, we look at various disciplines of the classical science in the light of blown-ups so that many interesting and intriguing

14

Introduction

problems and insights can be raised or obtained. For example, In Chapter 4, we look at some age-old problems, which have been puzzling the scientific community for a long time. In Chapter 5, we will present our positions and the questions on nonlinear macro-evolution theories. Similarly, theories of microscopic evolutions are scrutinized in Chapter 6. Chapter 7 is devoted to the field theory. Chapter 8 is focused on difficulties facing the research of nonlinear chemical reactions. Chapter 9 aims at the nonlinearity and problems appearing in the studies of ecological evolutions. Since nonlinearity implies eddy sources and spinning motions, in Chapter 10, we point out the fact that the blown-up theory may bring forward impact on the research of economic evolution systems. At the end of this book, we included three appendices with the first one devoted to the introduction on how the thinking logic and methodology of blown-ups have helped and guided the development of a brand new procedure to conduct real-life weather forecasts aiming at major natural disasters. Here, what we like to emphasize is that this method has greatly improved the forecasting accuracy rate. The second appendix contains a list on how Professor Shoucheng OuYang, the founder of the blown-up theory, has developed his ideas leading to the establishment of the blown-up theory, and where he will travel to, if he chooses to continue his scientific pursuit. The third appendix summarizes some personal opinions and believes of Professor OuYang, as his final words about the science and scientific research, since he has decided to retire from science so that he could have time to pick up his true love of life: drawing and creative arts. The arrangement of the contents of this book is originated from Professor OuYang's blown-up system, augmented with our generalizations and specifications. At this special moment, we would like to express our appreciations to Professor Shoucheng OuYang for reading through our manuscript and for writing a preface for us. Also, during the creation of this book, we have been so fortunate that many colleagues and friends have unselfishly provided their comments and suggestions. Our special thanks go to, especially, Roman DeNu, Taihe Fan, Kimberly Forrest, Qingping Hu, Sifeng Liu, Donald McNeil, Zhenqiu Ren, Hector Sabelli, Achim Sydow, Story Troy, Vladimir Tsurkov, Victor Ven, for their inputs and encouragements at various times and locations.

References

1.7

15

References

The presentation in this section is mainly based on works of H. Haken (1978), G. Jarmov (1981), M. Kline (1983), H. M. Liang (1996), J. Lin and S. C. OuYang (1996), Y. Lin (1988), Y. Lin (1989), Y. Lin (1990), Y. Lin, Y. Ma and R. Port (1990), S. C. OuYang (1994), I. Prigogine (1967), R. Thorn (1975), L. Y. Xu (1983), Y. Z. Zhu (1985). For more details, please consult with these references.

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Chapter 2

Nonlinearity: The Conclusion of Calculus

In this chapter, we will look at a brief history of calculus, its achievements, fundamental concepts and results of the differential and integral analysis. As soon as the concepts of well-psedness and singularity of differential equations are looked at, one starts to see the limitations of calculus and all theories developed on calculus. Then, by considering a list of examples, it is argued that discontinuity is a mathematical characteristic of nonlinear evolutions. With this understanding in mind, we will look at an array of traditionl treatments of nonlinearity, including linearization, stabilization, spectral expansion and numerical schemes. At various locations, our opinions and positions are clearly laid out. 2.1

A Brief History of Calculus

Three hundred years ago, when Newton and Leibniz individually established calculus, other than influences of the thinking logic of continuity of the ancient Greek, there were other historical reasons, too. During the time period of the European Renaissance, mathematics had also gained much progress accordingly. French mathematician R. Descartes (1596 1650) invented the Cartesian coordinate system so that geometry and algebra were mingled together. This accomplishment was another leap in the development of mathematics since the time when Euclidean geometry was written (Euclid, around 330 - 275 B. C ) . Soon, the concept of functions was introduced, studied and found a wide range of applications. All these developments had lied down the foundation for the establishments of mathematical analysis and particle mechanics in order to "abstract shapes into 17

18

Nonlinearity: The Conclusion of Calculus

numbers". In the area of physics, the research focus switched from the studies of statics to those of dynamics and achieved many breakthroughs. First of all, Galileo (1564 - 1642) uncovered the law of free falling objects. Based on the huge amount of observational data, J. Kepler (1571 - 1630) summarized a set of empirical laws describing planets' movements. Scholars became extremely interested in the studies of varying relationships between "quantities" or the studies of dynamics and kinematics. Besides, many mathematical problems, such as finding the extrema of given functions, finding the conditions under which a launched cannonball can reach its (potentially) maximum distance, finding the distance a special planet would travel within a known time period, reflections of light rays off a surface, especially when it is known that the displacement of an object is a function of time, find the speed and acceleration of the object at any given time moment, etc., were placed in front of the scientific community. Even though at the time, many scholars had contributed to the studies of these and related problems, the most successful and most systemic results had been credited to Newton and Leibniz. That is, these two men and their followers had individually established the regularized methodological system of calculus so that the slaving system of the first push was successfully completed through the particle-non-structural quantitative analysis and the belief that "numbers are the Tao of all things". Here, by first push, it is meant to be a push which is needed for Newtonian mechanics to work. It is because in Newtonian physics, such as Newton's second law of motion states that "acceleration equals mass times force", a miracle force is always assumed to exist. After calculus was "officially" established, Newton was the first person, who applied calculus in the laws of mechanics and studies of celestial movements, which he discovered, under "inequal-quantitative effects", and achieved a great amount of successes. At the same time, he and his followers met the difficulty of being unable to solve the "three-body" problem under "inequal-quantitative effects". This difficulty was one of the earliest nonlinear evolution problems first met after calculus was officially born. During the second half of the seventeenth century, since there existed a need to solve dynamic physics problems, the study of differential equations became a focus. The class of physics problems, which could be studied with a degree of success by employing differential equations, was the theory of elasticity and the theory on celestial movements. The theory of elasticity

A Brief History of Calculus

19

is developed on Hooke's law (R. Hooke, 1635 - 1703), which is stated as follows: The restoring force of a spring is directly proportional to the displacement of the spring from its equilibrium position and is directed towards the equilibrium position, representing the recovery force of a stretched or pressed spring directly proportional to its length of stretching or pressing. In terms of differential equations, Hook's law can be written as follows:

where x stands for the displacement of the spring and R the spring constant. As for the theory of celestial movements, it is mainly developed on the basis of Newton's law of universal gravitation. The (two-body dynamic) problem about the relative motion of the sun (M) and a planet (m) can be written as the following evolution equation under inequal-quantitative effects: d2!^ _

~W

=

Mm_>

~rlr

r

For this mathematical model to hold true, both the sun and the planet have to be seen as dots without any shape, any structure, and any size with their entire masses located at the corresponding dots, respectively. This is the so-called assumption of particles. The reason for introducing such an assumption is that when the diameters of the sun and the planet are compared with the distance between these celestial bodies, the diameters are so small that they can be ignored so that the relevant evolution equation can be simplified. However, in the light of the blown-up theory (see next chapter for more details), the movements of the sun and the planet of interest are consequences of inequal-quantitative effects. So, in terms of materials' evolutions, these two points of view have constituted a huge epistemological difference. Swiss mathematician J. Bernoulli (1687 - 1759) was one of the pioneers in the study of analytic solutions of differential equations and one of the earliest mathematicians who had systematically studied nonlinear differential equations. The few earliest classes of nonlinear differential equations, such as Bernoulli equation and Riccati (1676 - 1754) equation, can all be written and treated as linear equations after some special variable transformations are applied. Even as of now, these equations still belong to the small group of nonlinear differential equations whose analytic solutions can

20

Nonlinearity: The Conclusion of Calculus

be obtained precisely. The concept of partial differential equations, involving several independent variables, was naturally introduced when some physical phenomena were studied. These physical phenomona include the vibration of strings, the problem of wave motions, and the problem of fluid motions. In general, the vibration of a string and wave motions are described using linear differential equations with continuous evolutions. However, the problem of fluid motions is relatively more difficult. At around the mid-eighteenth century, the study on fluid motions had been mainly on several special cases of the movements of non-viscous fluids. During the beginning years of the nineteenth century, scholars started to look into the problem of viscous fluid motions. All achievements along this line have been about as far as the establishment of Navier-Stokes system of equations about fluid motions. Thus, in terms of the theory of fluid motions, we do not really know much more than what was known during the era of Euler. Since the time when calculus was officially established, more than 300 plus years have passed by. There is not doubt to say that during these 300 plus years, calculus has played an important role in the development of the theories of natural sciences. However, at the same time, calculus has also met "difficulties" in the studies of nonlinear problems. When the essential characteristics of nonlinear evolutions are not truly understood, it is easy for people to attempt to comprehend these evolutions on the basis of the traditional concept of continuity, on which calculus is based, leading to epistemological mistakes. Based on what has been done in the area of nonlinear science, now is the time for us to completely organize the methodological system of the thinking logic of continuity. Scholars have finally realized the fact that calculus is indeed a powerful mathematical theory and method; however, it is not the ultimate methodology, which can be applied to resolve all problems under the Heaven. It is because the method of calculus is a computational system developed on continuity and regularization, while the objective world behaves continuously and regularly only under very special conditions. And, discontinuous and irregular changes and evolutions exist in the objective world widely and universally. This chapter is devoted to a brief summary of the methodological system of calculus. At the same time, we will point out limitations of and difficulties met in the study of nonlinear evolution problems by all calculus-based methods. At various locations of the presentation here, we will present our points of view on relevant traditional concepts and methods.

The Method of Differential

2.2

Analysis

21

T h e M e t h o d of Differential Analysis

T h e differential analysis is also called the analytic manipulation, which is the foundation of all higher level mathematical theories. This section is only a s u m m a r y of relevant concepts and operational laws so t h a t we do not intend to pursue any theoretical rigor. All the readers, who want to study the relevant rigorous theories, can make u p such a deficit by consulting with other widely available college mathematics books. Since the m e t h o d of differential analysis was initially introduced to study m a t h e m a t i c a l physics problems, it can only be described through the use of a series of m a t h e m a t i c a l concepts, such as those of functions, continuity, limits, etc. So, let us start from the concept of functions.

2.2.1

Functions

and

Their

Properties

T h e concept of functions is about relationships between variables. It is one of the most i m p o r t a n t concepts being studied in college and graduate school, and an object on which differential analysis exists and is about. In other words, without the concept of functions, there would be no need to talk about the differential analysis. In the studies of n a t u r a l phenomena and engineering problems, people deal with various different quantities, such as weight, time, length, area, volume, t e m p e r a t u r e , pressure, speed, acceleration, etc. These quantities can be classified, based on the specific studies of the phenomena or processes, into two classes: constants and variables. In general, all the quantities, which stay t h e same in all phenomena or processes, are called constants. Those, which may take different values in different phenomena or processes, are called variables. Also, in a great number of situations of n a t u r a l phenomena or engineering technology, two or more variables exist and vary at t h e same time. In most of these cases, the variables do not change individually. Instead, their changes follow certain underlying laws and are related. For example, the atmospheric pressure changes with the altitude. So, the variables pressure and altitude vary at the same time and are related. This fact provides the n a t u r a l basis on which the concept of functions was introduced and studied. D e f i n i t i o n 2.1 ( D e f i n i t i o n of F u n c t i o n s ) : W h e n a variable x assumes a value out of a set of the real number line, if based on a certain rule, another

22

Nonlinearity: The Conclusion of Calculus

variable y has to take a certain number value or a certain set of number values, then the variable y is called a function of the variable x. In this case, x is termed as to an independent variable and y a dependent variable or a function of x. In general, such a function relation is written as y = fix). If the independent variable x represents the time t, the function y = fit) is called an evolution or a movement. However, in the definition of evolutions and movements, since the time t is only understood as a non-materialistic quantity, the form of the evolution of the dependent variable y is not clearly and adequately represented. 2.2.1.1

Representations of Functions

There are three commonly used forms to represent functions. They are analytic, graphic and tabular representations. (1) Analytic Representation. If a function relation between two (or more) variables can be written as a formula, then such a formula representation is called an analytic representation of the function relationship. For example, n- - x 2 y = sina;, y = W — — , y = x + x + 1 V 1 +x or the following piecewise defined function v -— fix)

1 — x, 1

1 + x,

when x < 1 when x > 1

etc. One main advantage of employing analytic representation of a given function is that such a representation is convenient in the study and analysis of the differential method. (2) Graphical Representation. This is a representation of a given function as a graph in the Cartesian coordinate system, where the horizontal axis stands for the independent variable x and the vertical axis for the dependent variable y. The advantage of this representation is that it is intuitive and visible. For example, when an automatic recording equipment can be used to represent the function relationship of atmospheric pressure, and temperature in terms of time as two-dimensional curves. The weakness of such a representation is that it cannot be effectively applied to reflect a function relationship in several independent variables.

The Method of Differential

Analysis

23

(3) Tabular Representation. This m e t h o d employs tables to represent t h e function relationship between the independent and dependent variables. Its advantage is t h a t even when no analytic representation of a function relationship is known, the necessary relationship can still be relatively clearly shown. 2.2.1.2

General Properties

of

Function

We now look at some main characteristics of functions. (1) Even or odd functions: Let y — fix) be a function. If when the sign of x changes, the changes occurring to the function value y are only a sign change, t h a t is

fi-x) = -f{x) then this function y = fix) is called an odd function. If when the sign of x changes, the relevant function value y stays the same, t h a t is

/(-*) = f(x) then this function is called an even function. If the function y = fix) does not satisfy either of these two conditions, t h e n it is neither an odd nor an even function. For example, y — x3 is an odd function; y = x2 is an even function; and y = sinx + cosx is neither o d d nor even. (2) Monotonic function: Let y = fix) be a function defined on an interval I. If when x increases in I, the function value y also increases accordingly, then the function y = fix) is called a monotonically increasing function on I. Conversely, if the function y = fix) decreases as x increases in the interval I, then the function y = fix) is said to be a monotonically decreasing function on I. Each monotonically increasing or decreasing function on the entire real number line ( - c o , +00) is called a monotonic function. (3) Bounded functions: Assume t h a t y = fix) is a function defined on a set 9 . If there is a positive number M such t h a t for each x in 3 , the corresponding function value fix) satisfies t h e following inequality: |/(a;)|<M then t h e function y = fix) is said to be bounded on 3 . Otherwise, the function y = f(x) is said to be unbounded. (4) Single-valued or multi-valued functions: Let y = fix) be defined as above. If for each x in 9 , the domain of the function, there is exactly

24

Nonlinearity:

The Conclusion

of

Calculus

one value y satisfying y = f(x), then the function y = f(x) is called a single-valued function. If for some x in 9 , there is a set of y-values each of which satisfies y = f(x), then y = f(x) is called a multi-valued function. 2.2.2

Limits

of

Functions

The concept of limits is also one of the most fundamental concepts in higherlevel mathematics. And, since the entire differential manipulations are established on this concept. There are several different ways to state the definition of limits of functions. In the following, we will look at the most common forms for the concept. Definition 2.2 (Definition of Limits (I)): If for any chosen small positive number e, there always exists a positive number 5 such that for all x satisfying the inequality 0 < \x — XQ\ < 5, the corresponding function value f(x) satisfies the following inequality: \f(x)-A\<e then the constant A is called the limit of the function y proaches XQ, denoted limx^xof(x) = A.

f(x) as x ap-

The geometry of the concept of limits can be given as follows: As shown in Fig. 2.1, for any chosen positive number e, draw the horizontal lines y = A + e and y = A— e, which form a horizontal band region. Based on the definition, for such a chosen e, there exists a neighborhood (XQ — 5, XQ+6)

y = f(x)

Fig. 2.1

The geometry of the concept of limits

The Method of Differential

Analysis

25

of the point XQ such t h a t when the first coordinates of t h e points of the graph of y = f(x) fall within the neighborhood (xo — S,XQ + S) and x 7^ Xo, the second coordinates of these points satisfy the following inequality: \f(x)-A\<e Therefore, these points fall within the horizontal b a n d region obtained above. T h e previous definition is the one for the finite limits of functions when x —> XQ. T h e definition for the case when x —> 00, we have the following: D e f i n i t i o n 2.3 ( D e f i n i t i o n o f L i m i t s ( I I ) ) : If for any arbitrarily chosen small positive number e, there always exists a positive whole number N such t h a t for all x satisfying the inequality |x| > N, the corresponding function value satisfies t h e inequality: |/(x)

-A\<e

then the constant A is called the limit of the function y = f(x) denoted limx-^00f(x) = A.

as x —> 00,

Based on the definition of limits above, one can define the concepts of infinity and infinitesimal as follows: D e f i n i t i o n 2.4 ( D e f i n i t i o n o f Infinities): If for any predetermined, arbitrarily large positive number M, there always exists a positive number S (or N) such t h a t for all x satisfying the inequality 0 < |x — xo| < 5 (or |x| > N), the corresponding function value / ( x ) satisfies the following inequality: |/(a;)| > M then the function y = f(x)

is said to be infinite, when x —>• Xo (or x —> 00).

In mathematics, the infinities, either (-00) or (+00), are not considered the same as a regular number. T h e y represent a state of the function y = f(x) when x —> XQ (or x —> 00). In the forthcoming chapters, it can be seen t h a t infinities stand for the state of discontinuities. As for the definition of infinitesimals, we will omit all the details here. W i t h the concepts of limits well in place, the concept of continuity can be proposed formally in m a t h e m a t i c a l analysis and modern theoretical physics.

Nonlinearity:

26

The Conclusion

of

Calculus

y = f(x)

• x0 Fig. 2.2

2.2.3

Ax

x

xo+Ax

The geometry of the concept of continuous functions

Continuous

Functions

The concept of continuous functions can be defined in many different ways. In the following, we will choose just one of these approaches. Assume that the function y = f(x) of our concern is well defined in a neighborhood of a point xo. If when the increment of the independent variable x is approaching zero (from the value of xo), the corresponding increment of the function value y also approaches zero, then we say that the function y = f(x) is continuous at the point xoIf Ax is used to represent the increment in x and Ay for the corresponding increment in the function value y, then based on the definition above, one has that Um^^Ay = 0. The geometry of the concept of continuous functions is given in Fig. 2.2. The concept of continuity was initiated from the concept of continuous materials of the ancient Greek philosopher Aristotle. In the objective world, there indeed exist materials and events, which change relatively continuously. For example, temperature generally changes continuously. The phenomena of lives evolve continuously. Objects move from one place to another continuously. However, as what has been shown in the literature, all these continuous movements are only local phenomena within certain time ranges. If we look at the whole evolutions of events or natural processes, discontinuities definitely appear. And, discontinuities are the origin and motivation for objective matters and events to evolve.

The Method of Differential

Fig. 2.3

Analysis

An infinite (or unbounded) discontinuous point of functions

Prom the definition of continuous functions at a point XQ, it can be seen that the following three cases describe how discontinuities for the function y = f(x) could appear at a special point x = xo(1) The function is not defined at the point x — XQ. For example, the function y = — is not defined at x = 0, and xz lim —^

+oo

x->0 x*

So, the point x = 0 is called an infinite (or unbounded) discontinuous point of the function (Fig. 2.3). (2) Even though the function is well defined at the point x = x$, the limit limx^Xof(x) does not exist. For example, the following piecewise defined function

{

-x + 1, 0 < x < 1 1, x = l

- a ; + 3, 1 < x < 2 is well defined at x = 1. However, this function does not have a limit at the point x = 1. So, this function is not continuous at this point (Fig. 2.4). (3) Even though the function is well defined at the point x = xo and the limit limx^Xof(x) exists, limx->Xof(x) ^ / ( x 0 ) . For example, the function

V = fix)

_ J x,

x^ 1 x =1

28

Nonlinearity:

The Conclusion of Calculus

y •

1

2

3

Fig. 2.4 A discontinuous function

1/2

Fig. 2.5 A discontinuous piecewise defined function is well-defined at x = 1 and limx^r\f{x) = 1. However, limx_>if(x) ^ / ( x o ) . Therefore, the point x = 1 is a discontinuous point of the function y = f{x) (Fig. 2.5).

2.2.4

The Concept

and Properties

of

Differentials

T h e concept of derivative of functions is defined as follows: Assume t h a t t h e function y = / ( x ) is defined in a neighborhood of the point x = XQ. Given a n increment A x t o t h e point xo such t h a t t h e point xo + A x is still located in the same neighborhood, let the corresponding increment in t h e function value be Ay = / ( x o + A x ) — / ( X Q ) . If when A x —• 0, t h e limit of

The Method of Differential

Analysis

29

the ratio of these two increments Iim

£/

Ax->0 Ax

=

Um

/(XO + A X ) - / ( X Q )

Ai->0

Ax

exists, then this limit value is termed as to the derivative of the function y = f[x) at the point xo, and the function is termed to as differentiable at this point XQ. The derivative function is generally denoted as one of the A

Af

following symbols: / (x), y , -7- or — . Obviously, the derivative of the ax ax function y = f(x) represents the rate of change of the dependent variable with respect to the independent variable. For example, the speed V of a movement equals the derivative of the distance with respect to time, that ds is V — —, where S = S(t) is the distance function. at If the limit (2.1) does not exist, then we say that the function y = f(x) is not differentiable at the point x = XQ or the derivative of the function does not exist at the point x = xo.The continuity of a function y = f(x) at a point x = XQ is a necessary condition for the function to be differentiable at this point, but not a sufficient condition. That is, if the function y = f(x) is differentiable at the point x = XQ, then the function must be continuous at this point. Conversely, the continuity of this function at this point does not guarantee the differentiability of the function at the point. The geometric meaning of the concept of derivatives can be given as follows: Let N (xo,yo) be a point on the curve of the function y = f(x). Now, pick another point M(xo + Ax, yo + Ay) on the curve near the given point. Draw the vertical lines PN and RM, and the horizontal line passing through the point N, which intersects RM at point Q. Then, we have NQ = PR = Ax, PN = y0, RM = y0 + Ay, QM = Ay Obviously, the slope of the secant line NM

Aw is ——.When the point M

slides along the curve of y = f(x) and gets closer to N as its limit, the limit position NT of the secant line NM is called the tangent line of the curve at the point N. Now, the tangent line has the slope given by t a n a = lim —— = / (xo) Ax->0 A x

Therefore, the derivative of the function y = f(x) at a given point x = xo is the slope of the tangent line of the curve of the function y = f(x) at the point x = XQ (Fig. 2.6).

30

Nonlinearity:

Fig. 2.6

The Conclusion

of

Calculus

The derivative equals the slope of a tangent line

The concept of differentials is closely related to that of derivatives. If the function y = f(x) is difFerentiable at a point x, then the product / (x)Ax of the increment Ax in the independent variable x and the derivative / (a;) is called the differential of the function y = f(x) at the point x, denoted dy — y Ax = f (x)Ax = y dx

(2.2)

Evidently, if a function does not have its derivative at a special point, then the function does not have its differential at that point, either. Integration is the inverse operation of differentiation. For a given function y = f(x), if another function z = F(x) satisfies F (x) = f(x), then the function F(x)is called an antiderivative of the given function f(x), written as F(x) = J f(x)dx. The general antiderivative of f(x) is written as J f(x)dx + C, where C is a constant, called the integration constant. If a function y = f{x) is continuous on an interval I = [a, b], then the definite integral of the function exists on this interval, denoted [ f(x)dx

= J f(x)dx

= F(b) - F(a)

(2.3)

where the function F(x) is an antiderivative of the function y = f(x). The preceding paragraphs are only a summary of the fundamental concepts of the differential analysis, from which it can be seen that differential and integral operations of functions are conditional. First of all, the function relationship between variables must be definite and representable by employing analytic formulas. It would be extremely difficult to compute derivatives for functions which are only written in graphs or tables. Next,

The Well-Posedness

and Singularity

of Differential

Equations

31

the functions of concern must be continuous and smooth enough, such as being differentiable. Thirdly, the operations of differentiation and integration are introduced in terms of abstract "points" without any physical sizes. Therefore, these operations are specialized and regularized methods of manipulation, which provide an approximation for evolutions of the objective world. Especially, the difficulties, met in the studies of nonlinear problems, have revealed the fatal weaknesses of calculus - a combined theory of differentiation and integration.

2.3

The Well-Posedness and Singularity of Equations

Differential

At the time when a natural phenomenon or an engineering problem is studied, it is generally difficult to find the direct function relationship between the variables of concern. What has been done to overcome this difficulty is to establish necessary differential equations or systems of differential equations of the unknown function(s) based on some laws or principles developed in various ways. Then, by solving the equations or systems of equations, the unknown function(s) can be studied. These unknown functions, in general, provide the needed solution or understanding of the original problem of interest. However, due to techical (mathematical) reasons, most of these established differential equations or systems of defferential equations cannot be precisely solved. Because of this reason, the study on existence, uniqueness and stability of the solutions of differential equations become extremely very important. The problem of well-posedness of the solutions of differential equations is defined as follows: For a given differential equation or system of differential equations, if the solution satisfies existence, uniqueness and stability conditions, then the original differential equation (system) is said to be well-posed; otherwise, not well-posed. Commonly seen differential equations can be classified into two classes: ordinary differential equations and partial differential equations. The general form of an ordinary differential equation is shown as follows: u = f(t,u)

(2.4)

where u is an n x 1 matrix of unknown or state functions, f(t, it) an n x 1 matrix with functions in t and u as its entries, t the independent variable,

32

Nonlinearity:

The Conclusion of Calculus

which in the study of evolution problems, means the time, and u = — . at Eq. (2.4) is also called a non-autonomic system. If the function / does not contain t, then eq. (2.4) is called an autonomic system. The commonly seen partial differential equation can be written as follows: dtu = g(t,u,dxu)

(2.5)

where u is an n x 1 matrix of the state variables, g(t, u, dxu) a n n x l matrix with functions in t, u and dxuas its entires, dt and dx stand respectively the differentiation operations — and ——, t and x the independent variables, dt ox which in the study of evolution problems, means the time and an onedimensional space location. Euler system of fluid motion equations, Burgers and kdv equations are all examples of partial differential equations. In case of eq. (2.4) or (2.5), if the matrices / and g contain nonlinear terms of u or u and dxu, then they are called respectively nonlinear ordinary differential equations and nonlinear partial differential equations. For the existence of solutions of a nonlinear differential equation in the form of eq. (2.4), it can be given by employing local existence theorem as follows: If f(t, u) is well-defined and continuous on the open region D: a < t < b, (—oo < a < b < +oo), \u\ < +oo then through each point (to,uo) of the region D, there is at least one integral curve. And, each of these curves can expand in both directions until reaching the entire area of D and contains all closed sub-region containing the point (to,uo) of D. Since the solution of the general nonlinear equation is continuous only within finite time interval, no local existence theorem can be applied to guarantee the existence of the overall solution (that is, t e (—oo,+oo)) of the nonlinear equation. In the early 1950s, Dauglas proved that there does not exist any overall existence theorem for the initial value problem of quasi-linear mathematical models. In order for the solution of a nonlinear eq. (2.4) to exist and be unique, the unknown function u must satisfy Lipschitz (1832 - 1903) condition, which is defined as follows: There exists a constant L such that for each

The Well-Posedness and Singularity of Differential Equations

33

t € (a,b) and each pair u and u" of real numbers, t h e following inequality holds true: f(t,u')-f{t,u") 0 Cy/ab + b tanh v abt V — —7= 7=— Voi) + a coth v abt

(2.27)

Discontinuity:

The Mathematical

Characteristic

of Nonlinear Evolutions

43

when 06 < 0 C\/—ab + b tan J—abt V= —7= (2-28) r y/—ab + a cot y/—abt So, it can be seen that other than the situation when a = 0, Eq. (2.25) becomes a linear equation with a continuous solution, the solution of Eq. (2.25) experiences discontinuities. • Example 2.9

The point t = 0 is a singular point of the Riccati equation 9 . y + y2smt=

2sini j cos z t

, . (2.29)

l/ = » 7 ( 0 . £ = - c o s t

(2.30)

Through transformations

Eq. (2.29) becomes

which can be transformed into the following second order linear differential u equation through the transformation 77 = — : u u-^ru

2

=0

(2.32)

Now, it is not difficult to solve Eq. (2.32) and obtain Mi =

e «

u2 — e Vs

c

+« 3 /

So, the general solution of Eq. (2.29) is given by

„, _ „ (c\ — u _ 1 1 3g2 V2 -T){S) ~ u ~ -J + c+F

_ 1 1 3 cos 2 1 - c^sT + C-cosS t

(2.33)

Other than the singular point t = 0 of the original equation, this general solution also experiences discontinuities when the integration constant satisfies 0 < C < 1. •

44

Nonlinearity:

The Conclusion of Calculus

Even though previous examples are all about ordinary differential equations, similar results also exist for systems of nonlinear differential time and space evolution equations. For example, let us look at the following system of first order nonlinear differential equations * = - * ( * + »)

(2.34)

y = y(x + y)

By combining the first and the second equations, one can obtain yx + xy = 0. So, xy = C, where C is the integration constant. Now, Eq. (2.34) can be simplified as follows: i

+ f +C = °

"

(2.35)

X

Since the first equation of Eq. (2.35) is a special Riccati equation, whose general solution contains a discontinuity, the solution of Eq. (2.34) also experiences discontinuities. As an another example, let us look at the following quasi-linear advection equation ut + uux — 0

(2.36)

where «t = —- and ux = ——. Dauglas once used the method of charactered ox istic curves and obtained the solution of Eq. (2.36) as follows: ux = - ^ 1 + u0xt

(2.37)

«|t=o = u0(x)

(2.38)

satisfying the initial condition

where UQX =

. When t < t^ = ——, ux changes continuously. When 0l ax t = tb, ux —> +oo becomes a discontinuous point of ux. What's different here of the previous examples is that the first order derivative of the state variable experiences a discontinuity in the process of evolution. What needs to be pointed out here is that we must separate those nonlinear evolution equations, which can be simplified into linear evolution

Discontinuity:

The Mathematical

Characteristic

of Nonlinear Evolutions

45

equations, from the true nonlinear equations, by using the method of variable elimination. For example, the following system ' X = Y - Z < Y = X2-Y wz

(2.39)

2

=x + z

appears to be nonlinear. However, from X — Y + Z — 0, it follows that X — Y + Z = C\, where C\ is the integration constant. So, Eq. (2.39) can be rewritten as follows: ' X = C1 + C2et < Y + Y = X2 = (d + C2etf ^ Z - Z = X2 = (Ci + C2etf

(2.40)

where C2 is another integration constant. Now, the second and the third equations of Eq. (2.40) are linear differential equations, whose solutions do not contain any singularity. In the previous examples of nonlinear differential equations, the discontinuities have all appeared to be the situation of infinities. However, it is absolutely not the only situation facing nonlinear differential equations. For example, a solution of the nonlinear evolution equation u = — v 1 — u2 arcsin u

(2-41)

is

u = sin -

(2.42)

Evidently, u is discontinuous at the point t — 0. However, as t —>• 0,the values of s m | oscillate between - 1 and 1 forever. So, the point t = 0 is also called an oscillating singular point. Summarizing what has been said above, it follows that discontinuities appear in nonlinear differential evolution equations commonly. These discontinuities are not only limited to the understanding of solutions of nonlinear differential equations, but, in fact, also bring forward a series of conceptual challenges to calculus with great future significance. (1) Linear and nonlinear evolutions are two classes of fundamentally different evolution problems. It would be a methodological and epistemological mistake, if linearity is seen as an approximation of nonlinearity and

46

Nonlinearity:

The Conclusion of Calculus

if the thinking logic of linearity is employed to analyze and resolve nonlinear problems. (2) Calculus is a set of methods on regularized mathematical operations developed on the assumption of continuity. So, all methods and theories, developed on calculus, are also limited by this assumption. However, since the solutions of nonlinear differential equations contain discontinuous singularities, it reveals the fact that the theory of calculus itself suffers from internal difficulties. Consequently, it implies that the foundation of calculus is not very rigorous. (3) In general and especially in the situation of numerical approximations, the process of solving differential equations is carried out in the sense of Riemann integrations (G. F. B. Riemann, 1826 - 1866). One most important condition, under which Riemann integrals can be calculated, is the continuity of the integrand function. Since the solution or one of its derivatives of general nonlinear evolution equation contains discontinuity (-ies), the Riemann integration scheme would lose it validity. In other words, all methods of solving nonlinear evolution equations, such as numerical schemes and other approximation approaches like series expansions, etc., will fall into one of the following problems

(a) "explosive" increase or fall; (b) being trapped in error-value calculations; (c) the characteristics of discontinuity and singularity of the original problem are eliminated or lost so that the resultant conclusions really have nothing to do with the original problem. (4) Since the whole solution of the Cauchy problem, (A. L. Cauchy, 1789 - 1857), of nonlinear differential evolution equations cannot be obtained, Laplace's single-value determinism of initial values and evolution equations of the future is declared incorrect. This fact is different of the situation where Laplace's "fatalism" is negated by the probabilistic solutions of micro-dynamic quantum theory. Since nonlinear evolution equations represent deterministic singular movements (see next chapter for further details), they are problems of deterministic structural evolutions and are definitely not the deterministic random motions as what Lorenz's "chaos" has claimed.

Question the Traditional Treatments of Nonlinearity

2.5

47

Question the Traditional Treatments of Nonlinearity

Since the characteristics of nonlinear evolutions, different of those of linear evolutions, are an escape from continuity and the destruction of the structure automorphic to that of the initial values after the appearance of discontinuities, it naturally leads us to rethink about some traditional and widely employed methods, such as linearization, stabilization, numerical solutions, orthogonal expansions, etc., and their limitations, correctness and validities. In this section, we will propose our thoughts and opinions.

2.5.1

Linearization

From the discussions of the previous sections on nonlinearity and related evolutionary characteristics, it is not hard to see that linear and nonlinear evolutions are two different concepts and that linearity cannot be seen as an approximation of nonlinearity. The essence of linearizing a nonlinear problem is to eliminate all the discontinuous singularities of the underlying nonlinear evolution. As a mathematical method, if the underlying physical problem does not contain any singularity, linearization can surely be employed. However, if the physical problem itself contains singularity (-ies), such as reversal and transitional changes, then the method of linearization not only ignores all the existing singularities, but also distorts the evolutionary essence of the original problem.

2.5.2

Stabilization

The so-called stabilization is a method introduced to study long-term behaviors of evolutions. In terms of linear evolution equations, the relevant evolutions are periodic and continuous up-and-down changes. The longterm behaviors can be either stable or unstable. However, when one looks at nonlinear evolution equations, since discontinuities appear within finite periods of time, the structures, automorphic to those of the initial values, are destroyed. So, it is meaningless to talk about automorphic long-term behaviors of nonlinear evolution equations. We believe that in terms of nonlinear evolutions, the concept of stability and the idea of stabilization are not appropriate, since nonlinearity is essentially about evolutionary transitions, instabilities and sensitivities.

Nonlinearity:

48

2.5.3

Comparison Schemes

The Conclusion

of

between Spectral Method

Calculus

and

Numerical

Let us consider the following initial value problem of quasi-linear advection equation: (ut

+ uux = 0

(243)

[ u(0,x) = — sin a; Its solution is given by u = - sin(a; — ut)

(2.44)

or t, = x — ut = x + sin £i Eqs. (2.44) and (2.45) indicate that the speed and the characteristic curve of the traveling wave all changes with time. Now, applying Fourier (J. B. J. Fourier, 1766 - 1830) series expansion method on Eq. (2.43) gives oo

—u(t,x) = YJu„(t)sinna;

(2-46)

n=l

where

i r un{t) =

/ *

w(i, x)sinnxdx

(2-47)

J-a

Now, employing integration by parts provides u„(i) =

i

/

r

rnr J_v

cosnxdx

(2.48)

Substituting Eq. (2.45) into Eq. (2.48) provides 2 fn n(t) = — / cosn (£ — sin £i) cos t;d£

u

(2.49)

From Bessel integration, it follows that

i r Jn(z) = - / 7T Jo

cos ( n £ - z s i n £ ) d £

(2.50)

Question the Traditional Treatments of Nonlinearity

49

Now, from the recursive formula Jn(z) = ^[Jn+l(z)

+ Jn-l(z)}

(2.51)

one obtains

z r Jn(z) = — /

cosn (n£ — z sin£) cos£d£

(2.52)

By comparing Eqs. (2.52) and (2.49) provides , 2Jn(nt) Unit) = T-^ nt

. . (2.53)

So, the series solution of Eq. (2.43) is obtained as follows: oo

u= -2^2

(nt)'1 Jn(nt)smnx

(2.54)

71=1

As a m a t t e r of fact, from the solution Eq. (2.37) of Eq. (2.36), one has cos a; (2.55) 1 — icos x So, when t = */, =

, Eq. (2.43) experiences singularities. cosx If numerical integrations are applied directly to Eq. (2.43), not only will multiplicity appear, but also sharp-turning points. T h e results, calculated out of the series solution established on spectral expansions, contain neither multiplicity nor sharp-turning points, t h a t is, non singularity is obtained (Fig. 2.11). T h a t is because each integration without any smoothing effects can cause computational instability, i.e., singularity. If smoothing effects are introduced, other t h a n making the resultant solutions smooth, it tends to make all disturbances more harmonic. On the other hand, a consequence of employing the method of series expansion, or say, t h e key of Sturm-Lioville Theorem, is t h e elimination of all singularities. So, neither numerical solutions nor series solutions can correctly reflect t h e whole characteristics of nonlinear evolutions. 2.5.4

Limitations

of Lyapunov

Exponents

If X(t) is a trajectory in an m-dimensional phase space, then X G Rm. Let X(t) + 6X(t) stands for an arbitrary trajectory neighboring X(t). Then,

50

Nonlinearity: The Conclusion of Calculus t=l

+1

+1

+1

+1

Fig. 2.11 The result out of series expansions the Lyapunov exponent of the trajectory X{t) is defined as follows: LE=

lim In lim t-+oo t - t0 \\sx(t0)Ho

\5X{t)\ \5X(t0)\

(2.56)

If X(U), ti — 1, 2, ..., n, is a discrete time series, then the corresponding Lyapunov exponent is defined by 1

LE=

71-1

I

lim - V l n \X (t

(2.57)

n=0

where X (ti) is the value of discreticized first order derivative of X with respect to time. The physics meaning of Lyapunov exponent can be seen as the following approximation relationship, which is obtained from Eq. (2.56):

\5X(t)\^\5X(t0)\exp{(t-t0)LE}

(2.58)

Evidently, if LE < 0, the trajectory is locally stable. If LE > 0, then the trajectory is locally unstable. In general, LE has been employed as a

References

51

criterion for the existence of "chaos". It has been generally believed that if a dynamic system contains enough bounded trajectories of positive Lyapunov Exponents and the dimension of the phase space is greater than one, then the dynamic system must contain Lorenz's "chaos". However, from the definition of Lyapunov exponents, it can be seen that each LE value is dependent on taking limits twice and the differentiation of the trajectory with respect to time so that LE value is very closely related to the tangent space of trajectories. That is to say, using Lyapunov exponents to tell whether or not there exists "chaos" starts from the tangent space of trajectories. So, for this method to be valid, all trajectories of interest must be smooth. However, in terms of the whole evolutions of nonlinearity, such a condition cannot always be satisfied. Thus, the ideas of applying Lyapunov exponents to describe the overall characteristics of nonlinear evolutions and as a criterion to determine "chaos" loose their practical significance and validity. 2.6

References

The presentation in this section is mainly based on works of W. R. Ball (1960), R. Bellman (1953), C. B. Boyer (1968), F. Z. Chen (1979), S. S. Chen (1985), A. Dauglas (1952), J. C. Fan (1978), W. Hahn (1963), E. Kamkel (1980), M. Kline (1983), Y. Lin (2000), J. Lioville (1836), S. C. OuYang (1994), G. Sansone (1948), L. K. Teams and B. E. Berna (1986), C. Sturm (1836), Y. Wu (1992). For more details, please consult with these references.

This page is intentionally left blank

Chapter 3

Blown-Up Theory: The Beginning of the Era of Discontinuity

Based on discussions in the previous chapter, in this chapter, we will look at whole evolutions and point out t h a t singularities and discontinuities are t h e fundamental characteristics of nonlinear evolutions. So, t h e theory of blown-ups is introduced formally. Then, the mathematical physics meanings of blown-ups, and m a t h e m a t i c a l characters of blown-ups are studied. In terms of observation and control, a geometric explanation of the concept of blown-ups is derived in t e r m s of mapping properties. W h e n the physics characteristics of blown-ups are considered, it is argued t h a t spinning currents are t h e physical materials' movements underlying t h e concept of blown-ups. Built upon all these discussions, attention will be directed t o the concept of equal quantitative effects.

3.1

L o o k i n g at W h o l e E v o l u t i o n s

In all whole evolutions existing in the n a t u r e and h u m a n societies, there exist many different a b n o r m a l phenomena of discontinuity: convergences a n d divergences of fluids; vortex flows; spindrifts of water; fronts, jet streams, clouds and rains of weather evolutions; burnings and explosions occurring in chemical reactions; birth and death of life evolutions; appearance and disappearance of populations; creation and distinction of microcosmic particles; economic crises of a free society; rise and fall of stock market prices, etc.These are just some of t h e daily observations of discontinuous phenomena and transitional changes. As a m a t t e r of fact, what should be said is t h a t continuities, appearing in materials' evolutions, are simply special and relative cases with discontinuities more widely existing. Since the n a t u r a l 53

54

Blown- Up Theory: The Beginning

of the Era of

Discontinuity

law that all things go through stages of growth and declination, and all people have to be born and die, exists commonly in objective evolutions of materials, and since nonlinear evolution equations possess discontinuous, singular transitions as its fundamental characteristics, a mathematical physics foundation for the establishment of a "blown-up" theory has been firmly laid down. At the same time when such a theory is introduced to address discontinuities and singular transitions, the "blown-up theory" has also been introduced on the principle of establishing scientific theories to describe the Tao of the nature instead of just another theory. OuYang once said: "The concept of blown-ups is introduced on the realisticity of physical transitional changes with the borrowed form of the singular problem of nonlinear mathematical evolution models". Therefore, the concept of blown-ups reflects not only the singular transitional characteristics of the whole evolution of mathematical nonlinear equations, but also, more importantly, the changes of old structures being replaced by new ones, as seen in objective evolutions of the physical world. So, the ancient Chinese philosophy of "change - long lasting" is vividly seen in such a concept. The introduction of this concept has brought the study of discontinuous, transitional and reversal evolutions to the most dominant position in all scientific endeavors the very first time in the recorded history.

3.2

Mathematical Physics Meanings of Blown-Ups

Reversal and transitional changes of objective matters and events have always been the central and extremely difficult open problem of the prediction science. As a relatively more mature theory and methodology for linearity, which are extensions of the "rise-and-fall" changes of the structures automorphic to those of the initial fields, they are lack of the ability to predict forthcoming discontinuous transitional changes. In terms of nonlinear evolution models, they reflect destructions of the old structures automorphic to the initial values and discontinuous transitional changes of whole evolutions. Even though these models have revealed some problems existing in the analysis and methodology of the system of calculus, their form can still be employed to describe the realisticity of discontinuous transitional changes of objective matters. So, the concept of blown-ups is not purely mathematical. To represent this fact, OuYang

Mathematical

Physics Meanings of Blown- Ups

55

suggested to modify the phrase "blow-up" to "blown-up". Based on the background information which we like to represent by using the concept of blown-ups, let us define the concept as follows: If the Cauchy problem of nonlinear equations « = /(*,«) U|t=t 0 = U0

where u is an n x 1 matrix of the state variables, f(t, u) an n x 1 matrix dv, of functions in t and u, t the time variable, and u = —-, has a solution u dt' — u(t;to,uo)on the interval t e [£o,£fc)> *o < +oo, and when t —> tb, the following holds true: lim |u| = +oo

(3.2)

t—>tb

then u = u(t; to, uo) is called a blown-up solution; or the relevant movement experiences a blown-up. In this definition, we have assumed that the system Eq. (3.1) is a truthful description of the underlying physical problem. So, Eq. (3.2) implies that the mathematical model blows up at t = tb and at the same time the underlying physical system goes throough a transitional change. As for the nonlinear equations in the independent variables of both time and space, the concept of blown-ups is defined similarly. The difference is that blown-ups can occur to derivatives of some order of the state variable. If the time-space evolution equation is written as dtu = g(t,u,dxu)

(3.3)

where u is an n x 1 matrix of the state variables, g(t, u, dxu) a n n x l matrix of functions in t, u and dxu, dt and dx stand for the differential operations — and — , respectively. Assume the initial (or boundary) condition is u(t0,x)

= u0(x)

(3.4)

Then, when the solution u = u(t,x;to,uo) or ux = ux(t,x;to,uo) changes, for t e [to,tb), continuously, and when t —> tb, the following holds true: lim \u\ = +oo t~Ho

(3-5)

56

Blown- Up Theory: The Beginning

of the Era of

Discontinuity

or lim \ux\ = + 0 0 t-n0

(3.6)

t h e n either u or ux is called a blown-up solution, if the b o u n d a r y value problem, consisting of Eqs. (3.3) and (3.4), truly describes a physical syst e m and at the time moment t = tb, this physical system goes t h r o u g h a transitional change. In the rext of this chapter, we assume t h a t this assumption holds true. Based on the definition of blown-ups, some initial classifications can be done: transitional and non-transitional blown-ups, based on t h e evolutionary behaviors b o t h before and after the blown-ups. All transitional blown-ups are now classified into symmetric transitional blown-ups (Fig.3.1), asymmetric transitional blown-ups (Fig.3.2), and periodic t r a n sitional blown-ups (Fig.3.3). Non-transitional blown-ups are classified into normal non-transitional (Fig.3.4) and defective non-transitional blown-ups (Fig.3.5). Generally, for to < t < tb, the solution u is called the normal branch solution, for t > tb, u the solution of the singular branch. In t h e previous Figs.of classification for blown-ups, we only depicted the situation where t h e normal branch solution approaches + 0 0 . If the normal branch solution approaches -00 as t —•

Fig. 3.8

t

A blown-up solution for the case of ai < 0

each iteration step, no matter which smoothing scheme is employed. In this sense, it can be said that various integration schemes, designed to solve nonlinear evolution equations, cannot really avoid facing "explosive" growth or being trapped in "error-value spiral" computations due to the evolutionary singularities. That is, the paradoxes and misleading conclusions, drawn out of Lorenz's "chaos" doctrine have been originated from this place. It is a problem worthy deep and further consideration. (3) When A = a\ - Aa0 < 0, Eq. (3.8) becomes (u+-ai\

+-(4ao-a?)

(3.16)

Integrating this equation provides u = 2 y 4 a o - a? tan f - yj4a0

a\t + A0

:ai

(3.17)

When t->tb =

(- +717T- Aoj V4ao

Eq. (3.17) contains periodic transitional blown-ups. From the previous discussion, it can be seen that under different conditions, the solution of the same nonlinear model can either be continuous and smooth or experience either blown-ups or periodic transitional blown-ups. So, even for the simplest nonlinear evolution equations, the well-posedness of their evolutions in terms of differential mathematics is conditional.

62

3.3.2

Blown- Up Theory: The Beginning

of the Era of

Blown- Ups of Cubic Polynomial

Discontinuity

Models

If n = 3, Eq. (3.7) becomes u = a0 + aiu-ra2u2

+ u3 = F

(3.18)

We now discuss its evolutionary characteristics in several cases. 1. When F has a root u\ of multiplicity three, then the solution of Eq. (3.19) can be written as follows: u = ui ± {AQ - 2t)~*

(3.19)

Evidently, Eq. (3.19) is a defective blown-up solution. 2. If F has a real root u\ and two conjugate complex roots, that is F = (u — Mi) (y? + p\u + U2- Then Eq. (3.18) becomes u = (u - ui)2 (u - u2)

(3.23)

Nonlinear

Transitional

Changes: A Mathematical

Character of Blown- Ups

63

Once again, by the theorem on partial fractions, one has 1

A

(u - ui) 2 (u - u2)

B

u — u\

h

U-U2

(3.24)

u-u2

-2 , C = (u\ —\ 1x2)

where A = (ui — u?) , B = — (ui — U2) grating Eq. (3.23) provides

+ Cln

C

+ u-m

(u-ui)2

. Now, inte-

t + A0

U — U\

(3.25)

Evidently, when u > U\ or u < U2,Eq. (3.25) becomes

+ Cln

=t + A0

(3.26)

U — U\

u — u\

whose evolution experiences a blown-up. If the movement is within a bounded region, that is u\ > u > «2, then Eq. (3.25) becomes A

_,,

U — U\

u — U2

+ C In U —

U\

(3.27)

=t + A0

Substituting the expressions of A and C into this equation produces u2 + u i e ^ • e ( U l - U 2 K t + y l °) \ _|_ g » l - « .

(3.28)

2

e(u!-U2)

(t+A0)

When t e (0, 00), u is continuous and bounded. And, u does not experience any blown-up. 4. If F = Ohas three distinct real roots u\, U2, U3, and without losing generality, assume that u\ > u2 > W3. Now, Eq. (3.18) becomes ii = (u — Mi) (u — 1x2) (u — W3)

(3.29)

Now, the theorem of partial fractions and integration provide the following: In u - u\ + l n u - 1x31 U\ — U2

In hx - u2 \ = — (t + Ao) U\ — U2

C

(3.30)

64

Blown- Up Theory: The Beginning

where A =

C, B = Ml -

U2

of the Era of

Discontinuity

C. If u > ui or u < W3, then Eq. U\ — U2

(3.30) becomes l n

( , -

M l

) ^ ^ - U

(u - u 2 )

U1

""

3

)

lft

+ A))

(331)

2

which contains blown-ups. However, a careful analysis of Eq. (3.31) reveals that the evolution of u shows a "semi-broken" state, since when t —> — j4o,the corresponding ubecomes multi-valued: (u — « 1 ) u i-"2 . (u — u 3 ) = (u — ii2)"i-2

(3.32)

and u ->• +00

(3.33)

The former situation represents a bounded movement, while the later situation shows a blown-up. As a matter of fact, the phenomenon of "semibroken" does exist in evolutions of the nature. For example, spindrifts of water, produced by two or more flows colliding into each other, and materials' partial damages all belong to this situation. 3.3.3

Blown-Ups

of nth Degree Polynomial

Models

For the case of nth degree polynomials, even though the analytical solution cannot be found exactly, its properties of blown-ups can be studied through qualitative means. In the field of real numbers, based on the fundamental theorem of algebra, Eq. (3.7) can be written as u

=F = (u- Ul)Pl (u2 + b1u + c1)qi

• ... • (u - ur)Pr • • ...• (u2 + bmu + cm)qm

wherepj and qji i = 1,2, ...,rand j = 1, 2, ...,m, are positive whole and n = £ [ = 1 Pi + 2 ££=1 Qj, & = h) ~ 4c ? < °> (j = 1. 2 . - . m loss of generality, assume that u\ > u2 > ... > ur, then the properties of the solution of Eq. (3.34) are given by the following

{

'

'

numbers, Without blown-up theorem:

Theorem 3.1 The condition under which the solution of an initial value problem of Eq. (3.34) contains blown-ups is given by (1) When Ui, i = l,2,...,r, does not exist, that is F = 0 does not have any real root; and

Nonlinear

Transitional

Changes: A Mathematical

Character of Blown- Ups

65

(2) If F = 0 does have real roots Uj, i = 1,2,..., r (a) When n is an even number, if u> u\, then u contains blown-up(s); when u < ur,no solution exists. (b) When n is an odd number, no matter whether u > u\ or u < ur, there always exist blown-ups. Proof.

l°If F = 0 does not have any real solution, Eq. (3.34) becomes ii = {u2 + b\u + ci)

1

... (u2 + bmu + c m ) "*

(3.35)

From b2- — 4cj < 0,it follows that yj = u2 + bjU + preaches its absolute minimum ctj — \ (4c, — b2) > 0. So, Eq. (3.35) can be estimated as follows: > Po {u2 + b\u + ci) 2 . „ >A)(u+^6i)2>0

u

(3-36)

where /3Q = . So, u is a monotonically increasing function. Solving Eq. (3.36) provides that u>--6i+, "° (3.37) 2 1-uoA)* where un is the initial value. When t —> th = —^-, u —>• oo. So, u contains a blown-up. And, the statement (1) holds true. 2° If F = 0 does have real solutions Uj, i = 1,2,..., r, then we have (a) When n is even, since q = 2 X)'j=i 9i i s e v e n i P = ]Cj=i Pi *s a ^ s o even. If u > u\ or u < u r ,then the following holds true: u >0

(3.38)

So, u is a monotonically increasing function, which contradicts the fact that u < ur.So, when u < ur, there does not exist any solution u.In the following, we prove that when u > u\, the solution contains blown-ups. From ui > «2 > ••• > u r ,it follows that Eq. (3.34) can be estimated as follows: u>pi(uwhere (3 — aq? • off? •... • a^. u>

Ul

+

p

Ul)

(3.39)

Solving this inequality produces ;

*-V-[4> + (P-I)/M

(3.40)

66

Blown- Up Theory: The Beginning of the Era of Discontinuity

where Ao = a constant, determined by t h e given initial value. T h a t is, under certain conditions, Eq. (3.40) experiences blown-ups. b. W h e n n is odd, since when q is even, then p is odd. So, when u > u\, we have u >0

(3.41)

T h a t is u is a monotonically increasing function a n d has t h e following estimate u >P\{u-

U]f > 0

(3.42)

Solving this inequality a n d choosing t h e " + " branch provide t h e following u >m +

. P-V-[A> + (P-I)/M]

(3.43)

So, u contains blown-up(s). W h e n u < ur, one has ii < 0

(3.44)

So, u is a monotonically decreasing function a n d has t h e following estimate u 0 and either 4c2 - h\ > 0 or 4c2 — b\ < 0, integrating Eq.

(3.52) all provides c2 (±t + A>) -In

(3.54)

Evidently, this equation contains blown-ups. (4) If E = 0 has four distinct real solutions, and assume u\ > u^ > u^ > U4,then Eq. (3.50) becomes u = ± v (u — ui) (u — U2) (u — W3) (u — Ui)

(3.55)

So, the continuous bounded movements of Eq. (3.55) can be written by using an elliptic function. As for the unbounded movement, we will have to apply the method of estimation to prove that it contains blown-ups. As a matter of fact, when u > u\ and take the "+" on the right hand side of Eq. (3.55), one can obtain the following estimate: u>{u-ui)2

(3.56)

It is not difficult to prove that in this case, u contain blown-ups. Similarly, it can be shown that when u < U4, u also contains blown-ups. As an application, let us look at the nonlinear elasticity model X = -fX

+ X3

(3.57)

where X stands for displacement of position, / the linear elasticity coefficient. As what was done to Eq. (3.47), one obtains E2 = X2 = -^XA - l-fX2 + hQ

(3.58)

Nonlinear

Transitional

Changes: A Mathematical

Character of Blown- Ups

69

Now, Eq. (3.57) can be reduced to the following first order nonlinear evolution equation X = ±VE

(3.59)

From the previous discussions, it follows that under certain conditions, the evolution of X contains blown-ups. Therefore, the nonlinear elasticity change is fundamentally different of that of linear elasticity change. That is, within the range of elasticity, linear elasticity models evolve continuously and smoothly. And, the evolution of nonlinear elasticity models contain discontinuous singular blown-ups. 3.3.5

Blown-Ups tions

of Nonlinear

Time-Space

Evolution

Equa-

Nonlinear evolution equations involving changes in space have wide range of applications in physics. To a certain degree, these equations can directly and intuitively reflect the physical meanings of the blown-ups. The one-dimensional advection equation is one of the simplest nonlinear evolution equations involving changes in space. Its Cauchy problem can be written as follows: (ut

+ uux = 0

{ U\t=0 = U0

du du where u stands for the speed of a flow, Ut = — and ux = —— with t repot ox resenting the time and x the one dimensional spatial location. By applying the method of separating variables without expansion, as first introduced by OuYang, that is, let u(t,x) = A(t)v(x),

u0(x) = A(0)v(x)

(3.61)

Now, Eq. (3.60) can be reduced to - ^ = -vx = - A

(3.62)

A + XA2 = 0, vx = A

(3.63)

where A is a constant. Then

70

Blown- Up Theory: The Beginning

Integrating Eq. Ao provide

of the Era of

Discontinuity

(3.63) by separating variables and taking A(0)

A ° 1 + A0v xt

Multiplying both sides of this equation by ux and taking UQX =

=

(3.64) d{A0v) dx

AQVX produce UOx

1 + u0xt

(3.65)

This solution is the same as that obtained by applying characteristic curves as shown in Eq. (2.37) in Section 4 of Chapter 2. Based on the definition of the degree of divergence, ux is the firstdimensional degree of divergence. When UQX > 0, that is when the initial field is divergent, ux declines continuously with time t until the diverging motion disappears. If UQX < 0, that is when the initial field is convergent, then evolution of ux —»• oo experiences a discontinuous singularity with time £—*•£(, =

.Evidently, when t < tb-, ux < 0 evolves continues ously. When t > tb, ux > 0. So, the convergent movement of the initial field (uox < 0) can be transformed into a divergent movement (UQX > 0) through a blown-up. The characteristics of this kind of movement cannot be truly and faithfully described by linear analysis or statistical analysis. They reflect the fundamental characteristics of nonlinear evolutions. As a matter of fact, studies on bifurcation (broken) solutions of some nonlinear time-space evolution models, such as Shrodinger equations, and some classes of parabolic and hyperbolic equations, have been very active in the area of mathematics. All interested readers should consult with the relevant references. As for the commonly seen Burgers and kdv equations, the relevant problems of modeling, blown-ups and various evolutionary discontinuity characteristics in various nonlinear fields of physics will be addressed in the forthcoming chapters and sections. 3.4

Mapping Properties of Blown-Ups and Related ObservControl Problems

Through the discussions up to this section, we have gained some initial understanding on the fundamental characteristics of nonlinear evolutions.

Mapping Properties of Blown-Ups

and Related Observ-Control

Problems

71

J

Fig. 3.9 The Riemann ball shows the relation between the infinity of the plane and the three dimensional north pole

However, the symbol "oo", appearing in the definition of blown-ups, means "indeterminacy" mathematically. So, this end naturally leads to the problem of how to comprehend such a m a t h e m a t i c a l symbol. First of all, let us observe the mapping relation of the Riemann ball, well studied in complex functions (Fig.3.9). This so-called Riemann ball relatively and intuitively illustrates the mapping relation between the infinity on the plane and t h e n o r t h pole N of t h e ball. Such a mapping relation connects -oo and + 0 0 through a blown-up. Or in other words, when a dynamic point travels through the n o r t h pole N on the sphere, the corresponding image of the point on the plane shows u p as a reversal change from -00 to + 0 0 t h r o u g h a blown-up. So, treating the planar points ± 0 0 as indeterminacy can only be products of the thinking logic of a narrow observ-control, since, generally speaking, these points stand implicitly for direction changes of one dynamic point on the sphere at the polar point N, or in other words, the phenomenon of directionless, as shown by blown-ups of a lower dimensional space, represents exactly a direction change of the movement in a higher dimensional space. Therefore, t h e concept of blown-ups can relatively a n d specifically represent implicit transformations of spatial dynamics. T h a t is, through blownups, problems of indeterminacy of a narrow observ-control are transformed into determinant situations of a more general observ-control system. In this sense, the concept of blown-ups has reflected the practical meaning of the

72

Blown- Up Theory: The Beginning of the Era of

Discontinuity

old saying "it is difficult to appreciate the fine without knowing the coarse", and practically shrunk the distance of reaching meaningful epistemological conclusions. In the following we will focus on the analysis of blown-ups and dynamic transformations of quadratic evolution models. In the third section of this chapter, we have already studied the problem of blown-ups of quadratic models. Here, our discussion will focus on the essence of singularities by looking at spatial dynamic transformations. Let us first cite some results from Section 3 of this chapter on quadratic evolution models. For the evolution described by u = ao + a\u + u

(3.66)

one has three possibilities. (1) when a\ — 4ao = 0,

U

= -(M^"H

(3 67)

"

(2) when d\ — 4ao > 0, there exist two possibilities: 1) if \\a\ + u\ < \\Jd\

— 4ao,then

u= - 2 V a i - 4 a 0 t a n h f ^\J a\ - 4a 0 i + -A0 J - - a x 2) if \\a\ + u\ > \yja\ u

= 2Vai ~

4a

(3.68)

— 4ao, then

o c o t h ( ~ o V a i ~ 4 a o i ~ 2 A° ) " 2 S l

^ 3 ' 69 ^

(3) when a\ - 4a 0 < 0, then u = -y4a0

- a\ tan f - ^ / 4 a 0 - a\t + - A 0 ) - -ax

(3.70)

This series of results indicates that other than Eq. (3.69), representing a local smooth continuity, all other cases, as shown in Eqs. (3.67), (3.68) and (3.70), experience discontinuous blown-ups in their whole evolutions. In the following, we will explain in detailed steps on how these results on evolutions of quadratic evolutions correspond to the dynamic implicit transformations of the projection mapping between a planar circle and its projection on a straight-line.

Mapping Properties of Blown- Ups and Related Observ-Control

Problems

73

N

Uo Fig. 3.10

P'i

Implicit transformation between a circle and a tangent line

(1) Implicit transformation between a planar circle and a line tangent to the circle Fig.3.10depicts the dynamic relation between the point p,on the circle and the projection point p{ on the tangent line. That is, the set of all point Pi is one-to-one corresponding to the set of all point pi. Now, we let the point N on the circle corresponds to the singular point +oo on the tangent line. So, when the point pt travels directly from the singular point +oo to the other singular point -ooon the straight-line, it simply reflects the traveling of the point pi on the circle through the polar point N with a change in direction. Combining this explanation with the situation of two equal real roots of the quadratic form Eq. (3.67), the movement of the point Pi is limited by UQ (a real root or an equilibrium state). When the point Pi travels upward on the right hand side of the circle, the corresponding point PjOn the tangent line travels to +oo.When the point p^goes across the polar point N, the corresponding point pi leaps from +oo directly to -oo. That is, a direction change on the circle stands for a discontinuous singularity on the line, which is shown as a blown-up in Eq. (3.67). (2) Implicit transformation between a planar circle and a line secant to the circle If we focus on local bounded movements, such as the point pn in Fig.3.11 whose movement is limited by the distinct real roots ui and U2,then the point pn cannot go across the boundaries u\ and u^ and would not be able to travel over the polar point N. So, the corresponding projection point p i x on the secant line will not approach either +oo or -oo. That is, the point

74

Blown- Up Theory: The Beginning of the Era of

Discontinuity

N

\

\

P'il

Pi2

/

Ui

U2

P' 12

Pil

Fig. 3.11

Implicit transformation between a circle and a secant line

pa does not experience any singularities. This fact corresponds exactly to the smooth continuous solution Eq. (3.68). If we do not limit ourselves to bounded movements, such as the movement of the point pa in Fig.3.11, which can go across the polar point N, then the corresponding projection point pi2 can approach +ooand be transformed into -oo when the point Pi2 goes across the point N. This situation corresponds to the blown-up solution in Eq. (3.69). (3) Implicit transformation between a planar circle and a line disjoint from the circle This situation corresponds to the periodic blown-ups of Eq. (3.70) with a\ - 4a 0 < 0 and i/4a 0 - a\t + ^A0 = ( | + 2k) IT, k = 0, ± 1 , ±2,... Since the movement of the point p^ is not limited by any condition, it can go across the polar point N repeatedly, the corresponding projection point pi3 on the line detached from the circle experience periodic transformations from +ooto -ooor from -ooto +oo. (See Fig. 3.12) From our discussions above, it can be seen that the traditional view of singularities as meaningless indeterminacies has not only revealed the obstacles of the thinking logic of the narrow observ-control, but also the careless omissions of spatial or dynamic implicit transformations. What is more valuable is that in practical applications, the concept of blown-ups has also been applied to employ the strengths and avoid weaknesses of the traditional tools. For example, in real life practices, predictions of transitional changes are an extremely difficult problem, such as in the situations of major natural disasters and sudden nose-dive of the stock market.

Spinning

Current: A Physics Characteristic

of Blown- Ups

75

P.3

P',3

Fig. 3.12

Implicit transformation between a circle and a disjoint line

However, by predicting the blown-ups of the narrow observ-control systems (differential evolution equations), the forthcoming transitional changes can be forecasted, if the systems applied have captured the main characteristics of the underlying physical system of the interest. What is very interesting here is that blown-ups are described by singularities of low dimensional spaces or discontinuous reverses of high order derivatives. So, this fact provides no doubt a hope for predictions of any forthcoming transitional changes. For instance, it has been an employment of discontinuities of high order derivatives that OuYang and his colleagues have achieved successes in forecasting reversal and transitional weather changes in the past nearly 20 years. With the successes, he pointed out that "disorder" is an omen of a transitional change.

3.5

Spinning Current: A Physics Characteristic of BlownUps

Mathematically speaking, nonlinearity is shown to be singularities. In terms of physics, nonlinearity stands for eddy motions. This is the marrow of the blown-up theory. It reflects the universal law for the motion of all things in the universe, and constitutes a challenge to the traditional theory of wave motions and a significant blow to Newton's "particle" mechanics, developed in the past 300 plus years. Eddy motions are a problem of structural evolutions, which is a natural consequence of uneven evolutions

76

Blown- Up Theory: The Beginning

Fig. 3.13

of the Era of

Discontinuity

The definition of a closed circulation

of materials. Nonlinearity accidently describes the discontinuous singular evolutionary characteristics of eddy motions from the angle of a special narrow observ-control.

3.5.1

Bjerkness

Circulation

Theorem

At the end of the nineteenth century, Bjerknes (V. Bjerknes, 1898), discovered the eddy effects due to changes in the density of the media in the movements of the atmosphere and ocean. And, he established the wellknown circulation theorem, which was later named after him. Let us first introduce this theorem. By a circulation, it is meant to be a closed contour in a fluid. Mathematically, each circulation T is defined as the line integral about the contour of the component of the velocity vector locally tangent to the contour. In symbols, if V stands for the speed of a moving fluid, S an arbitrary closed curve, 5~r* the vector difference of two neighboring points of the curve S (Fig.3.13), then a circulation T is defined as follows:

r = <j> V W

(3.71)

Its rate of change with respect to time is given by

^=*j;V5l>

(3.72)

dt

K

dtjs

'

Spinning

Current: A Physics Characteristic

of Blown- Ups

77

The right hand side of this equation can be written as

Based on the assumption of continuity, one has

dt

(3.74)

\ dt I

Substituting Eq. (3.74) into the second term on the right hand side of Eq. (3.73) produces

//•^-/.'t?)-

0

(375)

So, Eq. (3.72) becomes

dv

i dV

which is also called an accelerating circulation. The Euler equation in fluid mechanics is given by _ _ = --S7p-2~d dt p

x V + ~ct

(3.77)

where p stands for the atmospheric pressure, p the density, i f the gravitational acceleration, Tf the earth's rotational angular speed. The first term on the right hand side of Eq. (3.77) is called a pressure gradient force, the second term the Coriolis force. Substituting Eq. (3.77) into Eq. (3.76) produces

^ = I f --Vpj 5y> - I 2 (it x VY Si* + I ~f • 51*

(3.78)

Assume that A is the area enclosed by the closed curve S (Fig.3.13). Then the second term on the right hand side of Eq. (3.78) is

I 2 (it x VY 5^ = 2 (t • I V • 61* = 2It • I ltVr • Sr

(3.79)

78

Blown- Up Theory: The Beginning of the Era of Discontinuity

where it is the unit vector perpendicular to the plane generated by V and 6~r*, Vr the component of V in the direction perpendicular to 81*. That is, \v\ sin OSr = VrSr So, we have

I 2 (jt x V) • 51* = 2 ft • I TtVr • Sr = 2fl^-

(3.80)

where a is the projection of the area A on the equator plane. If i f is taken to be the gradient of the potential function tp, that is i f = — V tb, Rx = -r— —> oo, that is, the first order derivative of ox the wave number with respect to space experiences a blown-up, then, from 27T

R = — , it follows that Li

«-, we have

Evidently, eq. (4.14) is a blown-up problem. If A^o > 0, that is, the initial state is a positive vorticity, then when t < tb, the movement of the fluid is a continuation of the initial positive vorticity. When t = tb, the fluid movement experiences a blown-up. When t > tb, since 1 — j - < 0, Atp < 0. So, due to the reversal change of the blown-up, the initial positive vorticity movement is transformed to one with negative vorticity. If A^o < 0, that is, the initial state is a negative vorticity, then it can also be transformed to a positive vortical movement through a blown-up. If the vortical advection qualitative analysis, developed on meteorological principles, is employed, we can assume that the spatial flow function *f!(x, y) is continuous and bounded. That is, | * y | = \U{x,y)\ < \c\\ , \^x\ = IV(oc, 2/)j < JC21, for any x,y G (—00, +00). {This assumption is not equivalent to that ipy — —u{t,x,y) and ipx = v(t,x,y) are continuous and

Convergency

(Divergency)

of Moving

Fluids

99

bounded}. Therefore, if {A^!)x and ^y have the same sign or the second term (A\&) in eq. (4.10) and tyx have opposite signs, we have

(A*)x + A A * M

=

o, (A*) + A y

A

$ =0

(4.15)

|c 2 |

Conversely, if ( A $ ) x and $ y have the opposite signs or the second term (A\P) and ^>x have the same sign, then we have ( A * ) x - A A * = 0, ( A * ) y - A A * = 0 (4.16) |ci| » \c2\ From the given boundary condition eq. (4.7), and by solving eqs. (4.15) and (4.16) for A, one obtains A^T^lnA^L

(4.17)

where % = 1, 2. The "-" and "+" signs are respectively corresponding to eqs. (4.15) and (4.16). By analyzing eq. (4.10) carefully, one obtains: (1) If Vv (or ci) > 0, ( A * ) x > 0; ¥ „ (or cx) < 0, ( A * ) x < 0; * * (or c2) > 0, ( A * ) y < 0; * B (or c2) < 0, ( A ¥ ) y > 0,then mt,x,y)

=

* 0) through a blownup. Conversely, Aipo > 0 (a positive vorticity) can also be transformed to a negative vorticity (At/) < 0) through a blown-up. And, the time moment when the transitional blown-up occurs is located right ahead of the positive vorticity in the direction of evolution and right behind the negative vorticity. This fact accidently agrees with the fact that on weather maps, clouds and rains occur in the transitional zones ahead of trough(s) and behind ridge(s). (See Figure 4.2 (a)). That is, rainfalls just happen to appear on the "broken" zones in the atmospheric evolutions. From eq.

100

Puzzles of the Fluid

(a)

Science

(b)

Fig. 4.2 (a) The transitional zones, ahead of trough(s) and behind ridge(s), of clouds and rains, (b) The zones, behind trough(s) and ahead of ridge(s), of clear and sunny weathers.

(4.19), it follows that Atp weakens continuously with time, indicating the fact that clear and sunny weathers appear at the continuous zones located behind troughs and ahead of ridges. (Figure 4.2 (b)). Summarizing what's been analyzed above, it can be said that the results of our mathematical physics analysis agree well with those, produced by employing vortical advection qualitative analysis, developed in meteorology, which are about the local strengthening of the vorticity (in the regions between the front of troughs and behind ridges) or the local weakening of the vorticity (in the regions between the back of troughs and the front of ridges). What's shown more vividly here is the characteristics of the discontinuous reversal transformations in the areas before troughs and on the back of ridges, and of the continuous evolutions in the areas behind troughs and in front of ridges. Since the blown-ups of the spinning movements were initially introduced by OuYang, we will name them as OuYang blown-ups.

4.3

Dynamic Characteristics of Navier-Stokes Equation

In Euler language, the general Navier-Stokes equation is given as follows:

Vt + V -VV = --Vp-2lt

x V + ^+F^

(4.20)

P where V is the velocity of the current of interest, vt = ——, V the gradient at operator, p the atmospheric density, p the pressure, TT the angular velocity of the earth's rotation, i f the gravitation, and F the viscosity force, eq. (4.20) was a kinematic equation of fluids originated from the model

Dynamic

Characteristics

of Navier-Stokes

Equation

101

with separate "objects and forces" of Newtonian mechanics. In the study of atmospheric dynamics, the density p and the angular velocity TT have been often seen as constant. (In fact, the actual value of TT is never fixed. It leads to the occurrence of such natural disasters as earthquakes, torrential rains, etc., due to major changes in mutational moments) And, the force ~F is often seen as a linear viscosity force. In this way, each acting force on the right hand side of eq. (4.20) is linear, while the term on the left hand side is nonlinear. Due to the complete difference in evolutionary properties and characteristics between linearity and nonlinearity, the modified eq. (4.20), as mentioned above, contains inconsistency in terms of properties of physics described. In his book, Weather Evolutions and Structural Forecasting, OuYang has pointed out the inappropriate places in the concept of "pressure gradient forces". Here, we will give NavierStokes equation a complete analysis and provide the following opinions. 1. In the atmospheric science, ( — V p ) is called a pressure gradient force, is originated from the assumption that the density p is a constant so that ( — V p ) describes the effect of an elastic pressure in the linear situaP tion, which, in physics, is shown as a wave motion or a spraying current. OuYang pointed out that if p is not a constant, even though its change could be very small, (

Vp) will become a nonlinear term and its corresponding

movement will be an eddy motion. Therefore, ( — V p ) should be called the gradient stirring force of the density pressure. So, it can be concluded that treating p as a constant is not a general problem of approximation. As a matter of fact, such an assumption leads to the epistemological problem of altering evolutionary properties. 2. In general, (—2$T x V) is call a Coriolis force. Its effect is a spinning movement. If this term is expanded at the latitude ip, one has -2ft

x T^ = (fv + Jw\ ~t - fuf

+ Jut

(4.21)

where / = 20,simp, f = 211 cost/? and V = u i + v j + w k . In the current studies of the atmospheric science, / and / are generally seen as constants. A consequence of such an approximation is that the eddy effect of the Coriolis force becomes that of fluids' resistance (or pushing), which has directly changed the original significance of physics. Therefore, it is absolutely a mistake to treat / and / as constants.

102

Puzzles of the Fluid

Science

3. The force F on the right hand side of eq. (4.20) has been often written as V • (fi • V K ), where p, is the viscosity coefficient. If fi is seen as a constant, then becomes a linear force whose effect is to dissipate the underlying energy of the movement. Thus, the force F has been called the dissipative term in studies of the atmospheric science. If /x is not a constant, then F will take a form of nonlinearity. That is, F = V • (/iV V J. In this case, its meaning of physics is no longer to dissipate the underlying energy. Instead, it is about the realization of the heat-kinetic energy transformations in the form of sub-eddies, which is surely different of the afore-mentioned single duty: dissipate the energy underlying the relevant movement of fluids. A realization of heat-kinetic energy transformations is also the major effect and function of the turbulences existing in the nature. 4. Finally, if if is not taken as the gradient of the potential function, that is if ^ ~VS, instead, in terms of a unit mass of the object being acted upon, one can have if = — V S , then when pis not a constant, if P is also a nonlinear term, which is also an eddy source. So, it would only be correct to use the assumption that p = a constant as a local approximation. Combining what has been said above, we have the following conclusions: All terms on the right hand side of Navier-Stokes equation (4.20) are nonlinear. So, they must represent non-uniform eddy motions so that discontinuous "birth-death"-like evolutions are created, and are definitely different of any continuous "rise-and-fall" changes under local circumstances. That is not simply a difference in forms and will lead to reforms in terms of concepts and methodologies. When connecting to the current state of studies in the atmospheric science, such a reform seems to be more important and desperate than anything else.

4.4

Some Problems about Atmospheric Long Waves

The atmospheric long waves, also known as Rossby (C. G. Rossby, 1940) waves, are the central theory of the atmospheric science. Let us in the following look at the basic situation of this theory. The following equation Ct + < x + vCv + f3v = 0

(4.22)

Some Problems about Atmospheric Long Waves

103

is known as the vorticity equation of a horizontal, nondiverging atmosphere under a positive pressure, where C = vx — uy stands for the vertical vorticity, P = ~al'f = 2fisin(y9, C* = wr, Cx = -5-, and (y = — . Assume that the at at ox oy average wind speed in the latitudinal direction is u and that no average wind speed in the longitudinal direction exists. Let u and v be respectively the latitudinal and longitudinal components of the vibrating wind speed. That is,

u = u + u,v

—v

(4-23)

Substituting this into eq. (4.22) and deleting the quadratic term of the vibration produce Ct + Kx + Pv = 0

(4.24)

By assuming £ = Ae^ and £ — kx + ny — ct, one obtains the so-called Rossby's long wave formula c= «-TT^-o

(4.25)

which is also known as the "wave motion" solely existing in the current fluid dynamics of geophysics. Based on his many years of practical experience and on the viewpoint of his blown-up theory and the evolutionary characteristics of nonlinearity, OuYang pointed out some fundamental problems existing in the atmospheric long wave theory. In the following, we will introduce his positions to our readers. 1. About the assumption of positive pressures. In reality, it is very difficult to see a positive pressure in the atmosphere. At least, this assumption does not possess the generality. An atmosphere under a positive pressure is not equivalent to an even atmosphere. In general, p is not a constant. So, eq. (4.22) does not hold true. Or, at least, it should be rewritten as Ct + U(x + VC,V + 0V

dx \p))

dy

\dy \pjJ dx

(4.26)

which is more feasible mathematically, since the left hand side describes the evolution of £, the vorticity, and the right hand side the solenoid term of the circulation. Evidently, eddy sources lead to eddy motions. So, both math-

104

Puzzles of the Fluid

Science

ematically and physically, eq. (4.26) sounds correct. So, the assumption that p is a constant has betrayed the objective reality. 2. According to Bjerknes' Circulation Theorem, the nonlinear term ( — V p ) stands for an eddy effect instead of an elastic pressure effect. So, as a mathematical problem, this term cannot be linearized. To this end, the corresponding Rossby's mathematical treatment has confused the fundamental difference between linearity and nonlinearity. 3. In eq. (4.22), (/3v) comes from the term (2IT x V) of Navier-Stokes equation. Evidently, this term is about the eddy motion caused by the earth's rotation. So, /3 is not a constant. It is because no mathematical treatment is allowed to alter the fundamental characteristics of physics and mathematics of the protocol (2TT x V), (/3v) should still be a term about df dip . rotations. As a matter of fact, (5v = -^--^— is indeed an eddy. ay ox Especially,

P = dy

-2f2cos %, C changes its direction. So, eq. (4.50) contains a blown-up.

Puzzles of the Fluid Science

112

(2) If A > 0, eq. (4.49) becomes C = a C+

4a

2a

(b2 - 4oc)

(4.51)

Integrating this equation produces In

b \

c + 2a

y/b2 — 4ac 2a

In

b\ 2 ^

C +

y/W-Aac +

VP

2a

4ac

£ + £(>

(4.52)

And, when

< — s/b2 — 4ac, one has 2a

— \ / b 2 — 4ac, one has of the movement, that is when 2a 2a

c = IxA^ccoth (_ 1 v ^ r ^ _ | & ) _ ^

. 54) (4

So, eq. (4.54) indicates that without limitations on locality, the evolution of ( contains blown-ups. Besides, what needs to be pointed out is that a "solitary wave" exists in the partial differentiation of eq. (4.28) with respect to £. This "solitary wave" is different of that contained in the unknown function of the KdV equation. (3) If A < 0, then eq. (4.49) contains periodic blown-up solution C = l v

/

4 ^ ^ t a n ( l y

4

^ ^ e

+

^ o ) - A

(4.55)

As for the method of solving the KdV equation by applying Backlund transformation, it is because the transformation Px

U

1

(4.56)

is a quadratic equation in P, its distribution in the x-space is a discontinuous piecewise defined function. So, only under local conditions, a solitary

Problems Related to the Modeling and Solution of KdV and Burgers Equations

113

wave solution can exist. Since the method of solving the KdV equation using the reduction of order must result in an equation of order greater than two, the general characteristics of movement of the KdV equation are not only solitary wave solution(s), but more generally are also blown-ups. Summarizing what has been said above, it can be seen that there is a need to reconsider the modelling and solution of the KdV equation by using the series expansion or any other expansion method for approximation. The so-called smooth solution is only a special case under local conditions without any general significance. Since the central idea of any expansion method is to eliminate the singularity (-ies) of the protocol, even when the KdV equation is correct, it still cannot be seen as a simplification of the Navier-Stokes equation. In other words, the evolution of the KdV equation cannot be determined as the evolution of fluid movements. 4.6.2

A General Equation

Discussion

on the Solution

of

Burgers

Burgers equation is known for its shock solution. As a matter of fact, the shock solution of Burgers equation is a continuous smooth solution under local conditions. It does not represent the general characteristics of the evolution of the equation. In the following, we will focus on the discussion of this claim. The so-called Burgers equation is written as follows: ut + uux - vuxx = 0

(4-57)

where v is a constant. It is a weak nonlinear evolution equation containing "dissipations". Let u = u(£) and £ = x — ct, where c is a constant. Now, eq. (4.57) becomes u, = — (u2 - 2cu + 2i/£o) (4-58) ' 2v x ' where £o is the integration constant, eq. (4.58) is a typical quadratic equation, whose solutions are respectively given as follows: (1) If A = c2 - 2i/f0 = 0, one has u = c-

V

, 2iM 0 + £ (2) If A > 0, one has respectively n

A

(4.59)

114

Puzzles of the Fluid Science a. If \u — c\ < \Jc? — 2 ^ o j one has u = -yjc2

- 2i/fotanh (^Vc2

- 2i/f0£ + ^ o ) + c

(4.60)

b. If \u — c\ > \Jc2 — 2i/£o, o n e has u = V c 2 - 2 ^ 0 c o t h ( - i - Vc 2 - 2U&Z +^Ao)+c

(4.61)

(3) If A < 0, one has u = y/2v^0 - c2 tan f — ^ 2 ^ 0 - c2£ + -A0 J + c

(4.62)

where Ao is the integration constant. Evidently, due to the appearance of blown-ups in eqs. (4.59), (4.61) and (4.62), the whole characteristics of nonlinear evolutions are clearly shown. Only eq. (4.60) is a smooth solution within a bounded region and describes the evolution of the shocks in the classical studies. However, from the point of view of blown-ups, shocks are only special cases of local problems. In other words, even weak nonlinear equations under the effect of "dissipation" also contain discontinuities causing Riemann integrations to be invalid. That is also why in numerical integrations, "decrease" cannot cancel "increase" so that relevant parameters have to be adjusted frequently. This end also explains the fact that smooth shocks are not the commonly seen phenomenon in fluid evolutions. Instead, the commonly seen phenomenon is broken "spindrifts".

4.7

Is Meteorological Science a Part of Geophysics or Atmospheric Science?

It does not seem to be questionable whether meteorological science is a part of geophysics or a part of the atmospheric science. It does not seem to have any background to raise such a question. However, if one carefully consider the atmosphere in which the mankind exists, he/she will discover some phenomena unexplainable in terms of either geophysics or the atmospheric science. OuYang has had a very comprehensive understanding of this problem and casted the two important problems: "Atmospheric temperature" and

Is Meteorological Science a Part of Geophysics or Atmospheric

Science?

115

"West wind circulation". These two problems have touched on issues which cannot be satisfactorily explained by employing either geophysics or the atmospheric science. Even though these two problems have described two phenomena which have been customarily seen, they have revealed a serious problem existing in the development of the current meteorological science. In the following, let us look into the relevant details.

4.7.1

The Problem ture

on the Atmospheric

Temperature

Struc-

To most people, the earth, on which we live, seems to be under the sunny sky, covered with gentle breezes, and shades of green trees, entertained by various singing birds, and perfumed with many kinds of flowers' fragrance. However, the temperature of the magma flows under the earth's surface can reach as high as 1000°C. In the atmosphere, below the altitude of 85 kilometers, there exist two tropospheres, one stratosphere, and a heat sphere, where the temperature is way above 1000°C. So, our human beings live in a space between two layers of extremely high temperatures, which is similar to a roasted chicken hanging in the middle of extreme heat. So, a natural question is: How can a cool temperature space exist between extreme heat so that the phenomenon of life could appear? The traditional theory believes that the atmosphere absorbs the solar radiation reflected off the surface of the earth. If it is so, how can the temperature on the top of the lower troposphere drop to as low as (-60° ~ —70°C)? How can the temperature in the stratosphere increase to 0°C, while the temperature at the top of the upper troposphere (85 kilometers above the sea level) once again drops to as low as (—83°C ~ — 113°C)? So, a reasonable conjecture is that the atmosphere must have the function of some kind of "refrigerator", i.e., the photochemical reactions. On the top of the lower troposphere, one has 4H20

uitraiJ

^et

li9ht

40H~ + 4tf+

where 4 —> 8 volumes of lowered temperature by absorbing heat, in laboratories, can reach the level of decreasing the temperature by as much as 25° ->• 3 0 ° C .

Since O H - expands, as influenced by the effect of floatation, these elements float up to the stratosphere, where when mixed with clouds, they

116

Puzzles of the Fluid

Science

participate in slow reduction reactions As-^TT— reduction reaction

AOH

- > •

nTT

/. , «

.

2H20 + 02 + 4e

then 4 —>• 3 volumes give off heat causing the temperature in the stratosphere to increase. That is, water molecules in the lower troposphere become partially ionized, which lowers the temperature in the troposphere. At the same time, the partially ionized water molecules, due to expansions, rise into the stratosphere, where they are reduced back to water molecules while increasing the surrounding temperature. The real-life situation is that the phenomenon of increasing temperature occurs after thunders. So, the physics of molecules cannot truly explain the problem of the atmospheric "refrigerator". As for the temperature drop in the upper troposphere, it might be caused by a process of complete ionization. 4.7.2

The Problem

of the Planetary

West Wind

Circulation

Based on the line speeds of the earth's rotation at different latitudes, say, 40° N(S): 0° N(S):

354.72 m/s or 1277 km/h 463.61 m/s or 1669 km/h

at the altitude near the region of 50 kilometers above the earth, the corresponding wind speeds at different latitudes with respect to the earth are given by 40°N(S): 0° N(S):

80 m/s ~ 100 m/s or 288 km/h ~ 360 km/h - 2 0 m/s or -72 km/h (east wind)

So, at different latitude, the wind speed with respect to the outer space, such as the sun, is given by 40°N(S): 0° N(S):

434.72 m/s ~ 454.72 m/s or 1565 km/h ~ 1636 km/h 443.6 m/s or 1597 km/h (east wind)

That is, from the outer space, the visible earthly strong west wind is about 1565 ~ 1597 km/h. Now, one faces the following problems: (1) Where is the west wind on the earth from? Since the earth rotates from the west to the east, the expected wind should be from the east, which

References

117

is at least as strong as 1400 meters / second. So, the earthly west wind is not a consequence of the earth's rotation. (2) As of this writing, there does not exist any evidence showing the effect of the temperature difference between the south and the north on the earth's surface on the temperature at the altitude of 85 kilometers. At the same time, the temperature of the lower troposphere over the equator is less than (—70°C), and around (—55°C) at the high latitudinal areas. So, with the cold south and warm north, the consequent wind should be from the east. (3) When the earth finishes its rotation 243 times, Venus completes its rotation once in the opposite direction. That is, compared to the earth, Venus is nearly still and spinless. And, since the longitudinal south -north winds dominate Venus, its temperature difference between the south and the north of Venus is also smaller than that of the earth. However, according to the information collected by "Venus-8" (1972) planetary detector, which landed on Venus during its daylight, it was indicated that the "atmosphere" of Venus contains also a west wind circulation. At the same time, the cloud charts of Venus, produced on the earth over the years, also indicate a west wind circulation on Venus. Even Neptune, with a average surface temperature less than (—200°C), is also covered by a west wind circulation. Therefore, it is truly an important problem to ask where and how planetary west wind circulations are created? These two problems, as posed above, imply that the study of meteorology is not simply limited to those of geophysics and atmospheric science. Indeed, the study of meteorology touches on problems of physics and chemistry about the objective and materialistic evolutions of the universe. The meteorological science is closely related to the studies of the cosmos. Based on the discussion here, OuYang pointed out that each earthquake is in fact a heaven-quake, each torrential rain is in fact a heavenly rain, meteorology is about the studies of astrophysics and astro-chemistry instead of problems of the regularized mathematics.

4.8

References

The presentation in this section is mainly based on works of C. S. Gardner, et al.(1967), J. E. Guni (1981), B. L. Guo (1995), G. R. Guo (1990), G. Jarmov (1981), M. Kline (1983), B. Q. Liang (1980), Y. Lin (2000a), Y.

118

Puzzles of the Fluid Science

Lin, S. C. OuYang, Z. G. Liu and Z. Wang (2000), Y. Lin and C. H. Zhang (2000), S. D. Liu, et al.(1982), M. Miura (1968), A. S. Morin, et al.{1959), S. C. OuYang (1992), S. C. OuYang (1994), S. C. OuYang (1998), S. C. OuYang, Y. Lin and Y. Wu (2000), G. B. Whitham (1974), Q. Z. Zeng (1979). For more details, please consult with these references.

Chapter 5

Questions about Nonlinear Macro-Evolution Theory

The focus of this chapter is on open problems existing in and challenges facing various established theories we classify as parts of macro-evolution theory. These theories include "chaos" theory, one-dimensional iteration, thermodynamics, synergetics, and catastrophe theory, in terms of "chaos" theory, we look at the following problems and challenges: (1) Does "chaos" objectively exist? Based on numerical experiments and the point of view of blown-ups, it is strongly suggested to study the practical and theoretical truthfulness of chaos. (2) The meaning of the concept of chaos. (3) The debate between determinacy and randomness. It is shown that the solution of Lorenz's equation is deterministic instead of random. (4) The non-realisticity of Lorenz'ss "chaos". In the light of physics and practical applications, the modeling process and mathematical and physics properties of Lorenz's model are looked at. (5) Some problems with regard to classical "chaos" model. In terms of one-dimensional iteration, we take a closer look at (1) iteration properties; (2) several problems existing in discussions of "chaos". In the name of thermodynamics, we check on (1) entropy changes in "linear and nonlinear irreversible processes"; (2) the entropy equilibrium equation of non-equilibrium thermo-local fields. In terms of synergetics, we look at problems related to the classification of fast and slow variables, expansions of fast variables, and explosive exponential growth. When we turn our attention to catastrophe theory, we focus on differences between blown-ups and catastrophe, and blown-up properties of catastrophe models.

119

120

5.1

Questions about Nonlinear Macro-Evolution

Theory

Introduction

There does not exist any clear boundary between the macro-world and the micro-world. In the eastern epistemology, it has not been recommended to separate the two worlds, since what's been believed is that "each grain contain the whole world". If such a tradition continues, then each macroscopic movement of the western civilization should be seen as a massive movement of a huge amount of grains. The reason why the concept of the two worlds has been introduced in the western civilization is essentially because in order to conveniently borrow the thoughts and results of the continuity system and the quantified methods of the particle assumption, each moving object needs to be seen as a theoretically even thing. The evolutionary behaviors of a single macroscopic object or a simple system with very few degrees of freedom, in general, can be described by applying the classical Newtonian mechanics. The special characteristic here is that the time can be reversed. On the other hand, macroscopic movements of complex systems, consisting of a huge amount of particles or of a large degrees of freedom, are hardly reversible. For example, concentration diffusion, heat conduction, electric transmission, and their combined effects can not be reversed. The classical mechanics has failed to describe these phenomena successfully. These phenomena respectively follow Fick's Law, Fouier Law, Ohm Law, etc. Starting in the 1940s, along with the so-called systems movement, studies on macroscopic movements of systems, consisting of a huge amount of particles, have made a certain degree of progress. Especially, after the 60s, these research have become quite active, and some influential "theories" and "doctrines" have appeared one after another. These theories include dissipative structures, synergetics, catastrophe theory, chaos theory, and some resultant branches, such as bifurcation theory, fractals, etc. These theories or doctrines have been collectively called in the modern world of learning as the "nonlinear science". Some of these theories had one time or another seen as the "magic cure" for all scientific "diseases", and been cited by great many recent publications in theory or in applications. For example, the application of the chaos theory has married this theory with the fluid science, producing the "chaotic fluids", with laser research, producing the "chaos in laser beams", with the studies of quantum theory, producing the "quantum chaos", etc. The whole cloud has been called by some as the "chaos" technology.

Does "Chaos" Objectively

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121

Each scientific exploration is a difficult a n d serious process. Due to the reason t h a t the essence of nonlinear evolution problems was not truly understood, t h e current "nonlinear science" has been established on the thinking logic of continuity. Even though the catastrophe theory is an exception, it still contains a fundamental difference from the discontinuity appearing in t h e evolutions over time or over time and space. Thus, all these theories, as t h e products of their respective historical times, have their individual merits, limitations and even misleading conclusions. In this chapter, we will focus on the discussion on these problems a n d cast our different a n d relevant opinions and positions in the hope t h a t further studies and discussions can be resulted.

5.2

Does "Chaos" Objectively Exist?

In the year of 1963, E. N. Lorenz produced his nonperiodic flows with some numerical integration for a limited range of the parameters of the equations, established by t h e m e t h o d of spectral t r u n c a t i o n out of Saltzman's kinematic equations developed for fluid convections. This kind of nonperiodic flow is extremely sensitive to the initial value, and has been called "chaos" or a "strange a t t r a c t o r " . (Figure 5.1). In the year of 1964, astronomer Henon a n d his colleagues discovered an unintegrable Hamilton system of degree of freedom two. W h e n the energy increases gradually, t h e system's trajectory in t h e phase spaces looks more and more random. In 1971, Ruelle and others applied the chaos theory to the study of dissipative systems and established a mechanism for a turbulence to occur. In 1978, M. J. Feigenbaum discovered the scaleness and the universal constant appearing in the bifurcation phenomenon of multiple periods. Conceptually, chaos was imagined as t h e "randomness, contained in deterministic equations" and as the "bridge over t h e gap between the determinism a n d probabilisticity". Currently, t h e chaos doctrine has been widely affecting many scientific disciplines, including, b u t not limited t o meteorology, optics, q u a t n u m mechanics, biology, civil engineering, chemistry, etc. It has been honored to be one of t h e theories of t h e modern nonlinear science with the greatest "achievements". No m a t t e r whether it is Lorenz's model or other classical models studied in the chaos theory, t h e phenomenon of chaos has been a product of t h e numerical integration. Now, a n a t u r a l question is: C a n numerical inte-

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Theory

SQUARE SQUARE

12

16

20

SQUARE

SQUARE

12

16

20

-20 -16 -12 -B

12 16 20

Fig. 5.1 A strange attractor grations be applied to nonlinear evolution models? Since one fundamental characteristic of nonlinear evolution models is an escape from the continuity, artificially forced numerical integrations will surely lead to falsehoods. So, there is a strong need to study the practical and theoretical truthfulness of chaos. 5.2.1

About the Meaning

of the Concept

of Chaos

In the Webster's New World Dictionary of the American Language, (1959, The World Publishing Company), the word "chaos" is defined as (1) the disorder of formless matter and infinite space, supposed to have existed before the ordered universe; (2) any great confusion or disorder; (3) an abyss; chasm. When the word "chaos" is translated into Chinese, several translations appear. One is "Hun-Dun", another is "Hun-Luan", and several others, with the first choice being the most accepted. In Chinese dictionary, "Hun-Dun" stands for the special state before the earth and the heaven were formed.

Does "Chaos" Objectively

Exist?

123

As for the question whether "Hun-Dun" or "Hun-Luan" is a more appropriate translation of the English word "chaos", OuYang believes t h a t "Hun-Luan" is more appropriate. His reasoning is given as follows: Based on Lao Tzu (Chapter 25), by Hun-Luan, it is meant to be t h a t "something mysteriously formed, born before heaven and earth. In the silence and the void, standing alone and unchanging, ever present a n d in motion. Perhaps it is the mother of ten t h o u s a n d things." Also, Lao T z u said in C h a p t e r 42 t h a t "The Tao begot one. One begot two. Two begot three. And, three begot the ten thousand things.". Evidently, if "Hun-Dun" is t h e "Have Not" as what Lao Tzu said, then "Hun-Dun" is a kind of physics concept of morphological s t a t e which is at least fewer t h a n (or smaller t h a n ) "one", satisfying "in the silence and the void, standing alone a n d unchanging, ever present and in motion." So, "Hun-Dun" should be an indistinguishable mixture, which is different of distinguishable "Hun-Luan". T h u s , "HunDun" and "Hun-Luan" should be seen as two different concepts. If Lao T z u ' s "Have Not" is understood as "Hun-Luan", then it is equivalent to say t h a t "the myriad of things are born out of a myriad of things". So, "Hun-Dun" stands for "still, large and similar, and indistinguishable" under comparisons, while "Hun-Luan" means "moving, small and separable and distinguishable". Based on OuYang's analysis, Lorenz's "chaos" is nonperiodic flows, which should be uneven a n d distinguishable. So, Lorenz's "chaos" should be translated to "Hun-Luan", instead of indistinguishable even "Hun-Dun". As for whether the phenomenon of "chaos" appearing in nonlinear evolution models is "Hun-Luan" or agrees with the objective reality, we will go into further details in the following sections.

5.2.2

Determinacy

and

Randomness

In the preceding chapters a n d sections, we have talked a b o u t the fact t h a t nonlinear evolution equations are the problem of determinant singularities, instead of Laplace's single value determinism and of problems of randomness. Currently, some scholars believe t h a t "chaos" has established a bridge between "determinacy and randomness". T h a t is, "Chaos" s t a n d s for nonperiodic flows determined by deterministic differential equations. At this junction, instead of the realisticity of the "chaos" appearing in Lorenz's model, we will temporarily focus on the discussion of some problems existing in his model development.

Questions about Nonlinear Macro-Evolution

124

Theory

The so-called Lorenz model is the following X = -aX + aY Y = rX -Y-XZ Z = XY-bZ

(5.1)

where X = ——, Y = —r-, Z = —-, a = — is the Prandtl number with dt dt dt K v being the viscosity coefficient, K the heat-conduction coefficient, b = 2 Tf

4 (l + a 2 ) , r =

3, Re the Rayleigh number, a the expansion 7r4 (1 + a 2 ) coefficient. Three equilibrium states are given by

ro

(X, Y, Z) = (0,0,0)

A

(X, Y, Z) = Ub(r-1),

B

(X,Y,Z)

y/b(r-l),r

- l)

= (-N/&(r-l),-v'6(r-l))r-

(5.2)

l)

For the sake of convenience for our discussion, let us rewrite eq. (5.1) as follows: H dt

rxi y

. z_

=

—a r Y

0 -X -b

a -1 0

X Y Z

(5.3)

or more condensely as X = v4(X)X

(5.4)

where X =

X Y Z

A(X) =

—a r Y

a -1 0

0 -X -b

(5.5)

The equilibrium states A and B can be related by the following matrix - 1 0 0 0 - 1 0 0 0 1

(5.6)

If J stands for the 3x3 identity matrix, then the set {/,e}forms a linear group together with the matrix multiplication.

Does "Chaos" Objectively Exist?

125

Moreover, the group {/, e} is not only a connection between the equilibrium states eq. (5.2), but also a group of linear transformation for Lorenz equation (5.1). In fact, assume that eq. (5.1) has two solutions: " Xi "

Y1

X

,Y =

" x2 Y2

(5.7)

z2

Zi

Substituting eq. (5.7) into eq. (5.4) provides Y + B~1{Y)B{Y)Y

B~1{Y)A(B{Y)Y)B(Y)Y

=

(5.8)

To keep eq. (5.4) unchanged, one must have B~1{Y)A{B{Y)Y)B(Y)

- B~1{Y)B{Y)

= A{Y)

(5.9)

As a linear transformation, the transformation matrix B should not have anything to do with Y. That is B = 0. So, eq. (5.9) becomes B~1A{BY)B

(5.10)

= A

or A(BY)B

= BA

(5.11)

Assume that the constant matrix B is given as follows:

B

oi

a2

a3

6i c\

b2 c2

b3 c3

, where \B\ ^ 0

(5.12)

Substituting this expression into eq. (5.1) and writing out the details produce -a

a

0

r 61X2 + b2Y2 + 63Z2

-1 0

0 1 X 2 + 0 2 ^ 2 + 03^2 -b

B = B-

-a r Y2

a 0 - 1 X2 0 -b (5.13)

Based on the definition of matrix multiplications and equal matrices, one

126

Questions about Nonlinear Macro-Evolution

Theory

has -aai + ab\ = -a\(7 + a2r + a3Y2 -aa2 + crb2 = a\o - a2 -aa3 + ab3 = 02X2 — 036 rai - 61 + (aiX2 + a2Y2 + a3Z2) c\ = -b\o + b2r + b3Y2 < ra2 - b2 + (aiX2 + a2Y2 + a3Z2) c2 = ha - b2 ra3 - b3 + {a\X2 + a2Y2 + a3Z2) c3 = b2X2 - b3b (biX2 + b2Y2 + b3Z2) ai - bc\ — -C\a + c2r + c3Y2 (biX2 + b2Y2 + b3Z2) a2 - bc2 = cxo - c2 , (biX2 + b2Y2 + b3Z2) a3 - bc3 = c2X2 - c3b

(5-14)

iFrom the condition that det(B) ^ 0, it follows that not all a\, a2, and a3 are zero. Similarly, neither 61, b2, and 63 nor c\, c2, and C3 are all zero. By solving some easy systems of linear equations, one has the following ( 0 l = 62 = ± 1 I c3 = l { a2 = a3 = 61 = b3 = a = c2 = 0

(5.15)

So, the transformation matrices, satisfying the required conditions, are / , and e

(5.16)

Since the set {/, e} constitutes a linear group with respect to the matrix multiplication, it is the linear transformation group of Lorenz's equation. And, the three fixed constants can be related through the group {/, e } , indicating the fact that the solution of Lorenz's equation is deterministic instead of random. This kind of determinacy is different of Laplace's single value determinism. One should not draw the conclusion that the relevant causal relationship of Lorenz's equation is destroyed, because of the negation of Laplace's single value determinism. The evolution of Lorenz's equation should be a discontinuous, singular, and deterministic movement. This end is also the fundamental difference between linear deterministic and nonlinear deterministic movements. On the other hand, since Lorenz's equations are nonlinear, all computational schemes, developed on the assumption of continuity, are no longer appropriate. This end is also a problem worthy close attention when numerical schemes are employed. Especially, at this junction, we will look at the problem whether or not Lorenz's "chaos", which he established by

Does "Chaos" Objectively Exist?

127

applying smoothing computational schemes, truly agrees with the objective reality. 5.2.3

About the Non-Realisticity

of Lorenz's

"Chaos"

In the year of 1963, Lorenz established his model on the basis of B. Saltzman's model, which is in fact a two-dimensional system of Boussinesq equations approximating a moving atmosphere: ut + uux

= -PQXPX

+ wuz

wt + uwx + wwz =

~PQ1PZ

+

vAu

- git + cAw

(5.17)

1tt + UTtx + WTtz = — W + nAlt

where u and w are respectively the velocity components in the x- and z-

,.

•

,ro gd2e

P

o2e

,

,

V an directions, it = —, po& constant, N — 7:77-^+ TT^J d K the viscosity po 6 oxz ozz and heat-conduction coefficients, and the subscripts t, x, and z stand for the respective partial derivatives. In eq. (5.17), assuming that po is a constant has in fact changed the gradient stirring force of the density pressure to a pressure gradient force, which, consequently, has replaced the existing eddy source into a pressure source, and a dissipative term into a linear form. That is, the properties of physics of the original model have been altered. In principle, this mathematical model has already lost its objective realisticity. Since our focus here is a discussion on Lorenz's model, we will leave the problems existing in Saltzman's model aside. In terms of Lorenz's model, it touches on at least the following problems: 1. Disagreement between the effect of elastic pressure and eddy motions. If introducing a flow function tp, one can take u = ipz and w = —ipx.By substituting them into eq. (5.17), he obtains

Aipt + J (ip, Aip) = -gnx + vA2if> T,

,

^

N

,

A

(5-18)

7Tt + J (tp, It) = — V>x + KATT

where J ( , ) is the Jacobi operator. By comparing eqs. (5.18) and (5.17), the meanings of physics on the both sides of the equations can be seen more clearly. That is, the left hand side of eq. (5.18) is purely an eddy motion, while the first term of the right hand side has constituted a gradient of the non-dimensional ratio of densities, which is clearly an elastic pressure

128

Questions about Nonlinear Macro-Evolution

Theory

effect. Now, the natural question is: Can an elastic pressure produce eddy motions? Or, in other words, in terms of physics, eq. (5.18) experiences the problem of disagreement between the form of motion of materials and the acting force. 2. The problem of spectral expansion. Lorenz applied a truncated spectral expansion to eq. (5.18). That is, he took 2 2 ,-K (n + k ) ip = y/2—y— '-X sin kX sin nZ

Kn2 + k2 (n2 + k2)3 (5'19) TT = y/2 TV—Y cos kX sin nZ - Z± T ^ - sin 2nZ gkzn gkzn where K and n are the wave numbers in the x- and z- directions, respectively, and X, Y, and Z the relevant coefficient components of the spectral expansion, which respectively stand for the movement intensity, stratification intensity and nonlinearity intensity of the stratification. In terms of mathematics, Lorenz's previous treatment suffers from the problem of meeting the relevant conditions under which spectral expansions exist. In Section 5 of Chapter 2, we have already mentioned about the problem of spectral expansions. That is, to expand a function / using the spectral expansion, the function / must satisfy two conditions (SturmLioville Theorem): (1) The function / must have continuous first order derivatives; and (2) The function / must have piecewise continuous second order derivatives. Since, in form, eq. (5.18) is a weak nonlinear system of equations and the nonlinear parts stand for eddy motions, based on the fundamental characteristics of nonlinear evolutions, escaping from the condition of continuity, the continuity of the functions if) and TT can only be guaranteed within a finite time period without mentioning about the guarantee of continuity for the first order derivatives of these functions i/j and TT. Besides, Lorenz split the single quantity of stratification intensity into two phase space variables (Y, Z). In terms of physical structures of variables, his splitting of the variable is not warranted. Summarizing the discussions above, Lorenz's model involves the following problems: (1) eq. (5.17) is a "model of the model", in Euler language, established for fluid motions in terms of differential equations. If the Euler equation is denoted as M B , the Saltzman's model eq. (5.17) as Ms,and the Lorenz's

Does "Chaos" Objectively

Exist?

129

model as ML , then the modeling process ME —>• ML has altered the m a t h ematical and physical properties of the protocol, even though the process has followed t h e general procedure of the traditional theoretical research. (2) T h e problem of integrability of Ms- Even though Ms is twodimensional with the gradient stirring force of the density pressure replaced by a linear pushing force of the pressure gradient and with the nonlinear heating process replaced by a linear heating process, t h e form of t h e model is analogous to t h a t of Burgers equation with balanced growth and declination under t h e effect of a linear dissipation. If Riemannian integrations are applied, and if numerical schemes or special computational t r e a t m e n t s , such as imposed smoothing or artificially adjusting the parameters, are not considered here, t h e n t h e computer will soon stop computing due to "computational instability" a n d "spill", as N or kdecreases. In t e r m s of the analysis of travelling waves, w h a t ' s been met is the problem of the denominator involved in the characteristic line integral becomes zero. T h a t is t o say, in order for the integration to continue, one needs to assume t h a t the propagation speed is a constant. This end agrees with neither eq. (5.17) nor eq. (5.18). So, b o t h models Ms a n d ML face with t h e problem of integrability. Besides, if one claims t h a t it is not very obvious in Ms, it is then definitely clear for people to see the disagreement in the modeling process of ML between the form of motion of fluids - eddy motions, and the "external forcing" t e r m - elastic pressure effect. This disagreement has constituted a clear infeasibility of physics. Next, can spectral expansions be applied to eddy motions? T h e whole evolutions of nonlinear equations are t h e m a t h e matical problem of singularities and do not satisfy, in general, the conditions necessary for one to carry out spectral expansions. So, our discussion indicates t h a t ML is a model of the model Ms, which is a model of t h e model ME- Since in t h e modeling process of M L , some fundamental properties have been omitted, ML has nothing to do with t h e objective physics problem and the protocol. Therefore, even when Lorenz's model is correct in whatever imaginable sense, it in no way reflects any property of the protocol. As for t h e "butterfly effect" of sensitivity to the initial values, appearing in numerical computations of Lorenz's model, it is truly a numerical consequence of "large quantities with infinitesimal increments", t h a t is the difference of two quasi-equal quantities, which can only be a meaningless error-value computation. So, we t e r m the p a t t e r n , called "double spiral

130

Questions about Nonlinear Macro-Evolution

Theory

structure", appearing in the numerical computations of Lorenz's model, as to "error spiral". For a detailed account on the results of the "error spiral" with varying time steps and parameters, one can consult with either OuYang's book Weather Evolutions and Structural Forecasting or Lin's edited volume Mystery of Nonlinearity and Lorenz's Chaos. 5.2.4 5.2.4.1

Some Problems Models

with

Regard

to Classical

"Chaos"

General Analytic Characteristics of the Population Model

The population model, or the population evolution model, is a quadratic differential evolution problem. In 1978, M. J. Feigenbaum was the first person who discovered the same "double spiral pattern" in his numerical iterations of the discreticized population models as what Lorenz did in the computation of his model. Consequently, the population model has been widely cited as one classical "chaos" model. However, the major difference between the population model and Lorenz's model is that the analytic solution of the population model can be physically obtained. The population model is established on the assumption that the rate of change in a population size equals the product of the population size and the difference of the birth rate and the death rate. In symbols, the population model is written as follows: X = (r - 8) X

(5.20)

where X = X(t) stands for the population size at the time moment t, X = —— the rate of change in the population size at the time t, and r and at 5 the birth rate and the death rate, respectively. In general, the birth rate is written as r = r0-aX

(5.21)

where a represents a constant related to the birth rate and the current population size and ro is another constant. Substituting eq. (5.21) into eq. (5.20) produces X = bX - aX2

(5.22)

where b = ro — S = is a constant, eq. (5.22) is the so-called logistic population model used in ecology to describe the evolution of a population, being

Does "Chaos" Objectively

Exist?

131

it a population of animals or a population of virus. Since this logistic model was introduced into the discussion of "chaos" in the 1970s, it has become very well known in systems science. This model belongs to the class of first order quadratic forms, which has been studied in detail in Section 3 of Chapter 2. With regard to the specific model eq. (5.22), there are several situations involved. (1) If b2 > 0, the variables in eq. (5.22) can be separated as follows: dX

(6 - aX) X

= dt

(5.23)

Integrating the both sides individually provides X

In

= b(t + t0)

X-S.

(5.24)

where to is the integration constant. When t < - , this equation becomes a I n - 5 - ^ = 6(4 + t o )

(5-25)

a

which can be organized as follows: x

=

l +

(5-26)

%+t0)

Evidently, this expression represents a continuous evolution in the range t € (0,+oo) without any singularity. And, when t —>• +oo, x —>• -—.However, 2a when x > — ,one obtains from eq. (5.24) the following

X

b eb(t+t0) = ~- eKt+t0) _ i

( 5 - 27 )

which experiences a singularity when £—>•£& = —to- So, as the characteristics of the whole evolution of X, a blown-up exists from t < % to t > %. (2) If b2 < 0, separate variables in eq. (5.22) and integrating both sides of the resultant equation produce 2

t a n

- i ^ z | ^

= t

+

t0

(5.28)

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Questions about Nonlinear Macro-Evolution

Theory

or, with X being solved out, one has X =

b tan 2 (* + *) + b 2a

(5.29)

This expression implies that eq. (5.22) contains periodic blown-ups. Just as the fact that the general mathematical models of the first order quadratic form don't experience "chaos", the population evolution model does not experience any "chaos", either. Other than the smooth solution under some local and limited conditions, the general characteristic of this model is blown-ups. In other words, similar to the general nonlinear evolution models, the population model also contains continuous, discontinuous changes and reversal transformations through discontinuities. Since the focus of this section is about the problem of "chaos", other problems like evolutionary characteristics and modeling feasibilities will be addressed in future sections. 5.2.4.2

Problems Related to the Discreticization of the Population Model

If eq. (5.22) is discreticized by using the corresponding difference equation, one has Xn+i ~ Xn = bXn - aXi At

(5.30)

Xn+1 = (1 + bAt) Xn - aAtXl

(5.31)

eq. (5.31) is still an evolution model. By employing numerical methods, results, similar to eqs. (5.26), (5.27), (5.28) and (5.29), can also be obtained. However, if one assumes a = 1 + bAt, P = aAt

(5.32)

Xn+1 = aXn - $Xl

(5.33)

then eq. (5.31) becomes

which is the typical "one-dimensional "iteration model that is currently studied in "chaos" with At = 1 so that the dimension of time is eliminated. Even though eq. (5.33) seems to have the same form as that of eq. (5.31), in terms of mathematical rigor, their intensions are different. It is

Does "Chaos" Objectively

Exist?

133

because a and (3 cannot be seen as parameters of eq. (5.33) because At is taken as a constant. The increment At of time is an independent variable of both eqs. (5.31) and (5.33). Or, in other words, changes in Xn+\ of eq. (5.33) are determined by changes in time, which are reflected in the changes of a and (3. In terms of the whole evolutions of systems, the time variable cannot be limited. Therefore, the corresponding values of a and (3 cannot be confined, either. With this understanding in mind, it can be seen that when eq. (5.31) is rewritten as eq. (5.33), both Xn+\ and Xn evolve with a and (3. The Xn—value can only be the evolutionary value at a previous time moment before Xn+i, which might be called an initial value of Xn+\. Now, with changes in a and (3, which should be functions of changes in At, the value Xn+\ is produced. So, eq. (5.33) still stands for an evolution problem instead of an iteration problem. At this junction, there is a difference between evolution and iteration. Now, if a and (3 are assumed to be parameters, then both the evolutionary properties of eq. (5.31) and the properties of the independent variables are changes. Suppose that one ignores such a change, then, due to the fact that a and (3 are treated as constrained parameters, the resultant mathematical form of eq. (5.33) is no longer an iteration formula, either. It is because when a and (3 are seen as constrained parameters in eq. (5.33), in form, Xn has become the independent variable with Xn+\ being the dependent variable. As for the treatment of the subscript n a s t„, as done in studies of "chaos", one faces the problem of having the time variable t appearing at the same time in the parameters a and (3 and in the subscript. ^From eq. (5.33), it follows that X should be the dependent variable and that except XQ being the given initial value, both Xn and Xn+i are dependent variables at different time moments. In other words, even with given XQ, the initial condition, the unknown function of the equation will not change its properties because of the given initial condition. So, except the initial condition, all X—values at other time moments are still an unknown function. In this sense, eq. (5.33) is equivalent to

which constitutes an algebraic indeterminate equation in two unknown functions. Therefore, in terms of mathematical properties, eqs. (5.33) and (5.31) are fundamentally different. So, eq. (5.33), where the functions a

134

Questions about Nonlinear Macro-Evolution

Theory

and /3 of the time t are seen as constrained parameters, should not be seen as the relevant one-dimensional iteration formula of the original model.

5.3

Some Problems about the One-Dimensional Iteration Formula

Up to now, we have not talked about the problem of numerical iterations of iteration formulas. At this junction, we will base our discussion on the standard one-dimensional iteration formula Xn+i = \-nXl

(5.35)

This formula, currently, is an example frequently cited in the theoretical discussions of "chaos". Without dealing with the problem of its feasibility, we will focus our attention on whether or not such a model truly contains "chaos". 5.3.1

Iteration Formula

Properties

of the One-dimensional

Iteration

The mathematical structure of the iteration formula eq. (5.35) can be seen in at least two different ways: a. If |X n | > 1, the value of |X n +i| will increase without a bound. That is exactly the general property or the fundamental property of nonlinear evolution models. b. If | X n | < 1, then as |/U.X^| —» 1, |X n +i| decreases. Due to reasons like truncation or rounding errors, the relevant machine computation does not equal zero exactly. Instead, the computation is trapped in error value calculations. In other words, in this case, in numerical computations of eq. (5.35), one experiences the problem of uncertainty with "large quantities of small increments", which is similar to the situation when Lorenz's "chaos" falls into an "error spiral". Evidently, error-value computations are useless and meaningless and cannot be seen as a property of nonlinear models. 5.3.2

Some Problems

Existing

in Discussions

of

"chaos"

a. According to the population model, the parameter u should be a function of time. That is, \i = /x(Ai). So, /i should not be treated as a constant

Some Problems about the One-Dimensional Iteration Formula

135

V.....I i—

(

rzz.

\ r\..._ _

^ n

Fig. 5.2 Computational results of equ. Xn+i = 1 — n(At)X%. parameter. Also, limiting /z in the range (0, 2) is not a concept of evolution. Xn should be dependent of the evolutionary time value. If eq. (5.35) is seen as an evolution problem on alterations of generations so that it is transformed into an equation with varying parameters, then eq. (5.35) should be written as Xn+1 = 1 - ^t)X2n

(5.36)

Evidently, the factor n(At) in eq. (5.36) should not be limited. b. If eq. (5.35) is considered with a limited fi, then mathematically speaking, eq. (5.35) is an quadratic algebraic equation. Its multiple bifurcation is a must instead of random even when all values involved in computations are error-values. As for the graph (Figure 5.2) generated by computers, it should be the computational results of eq. (5.36) with varying //. So, it is not the result of eq. (5.35). According to eq. (5.35), for each given ^i-value, there can only be a graph with one fork. As for a varying /i-value, the fact that the number of forks increases is a consequence of different /i-values, and should not be understood as random even though the whole pattern could not be seen clearly, since each fork is definitely deterministic. Since in these computations, Xn has been seen as a dependent variable instead of the initial value, there exists a problem of confusing initial values with dependent variables. Even as an iteration formula, eq. (5.35) is no longer a relationship between Xn+i and Xn. Instead, it represents a graph or pattern with varying parameter. c. What's important is that in studies of "chaos", or in the studies of classical "chaos" problems, the "chaotic" phenomena have been obtained by restricting Xn £ (—1,1). For instance, Feigenbaum's results were devel-

136

Questions about Nonlinear Macro-Evolution

Theory

oped with this restriction. For eq. (5.35), it can be seen immediately that when Xn G (—1,1), the evolution can be easily trapped in the error-value computations of "large quantities with infinitesimal increments", losing all practical significance. This problem exists not only in nonlinear numerical computations, but also in linear cases. Practically more important properties of eq. (5.35) are the explosive rises and falls when |X n | > 1. However, such practically important properties of nonlinearity have been ignored by the "chaos" doctrine.

5.4

Problems Related t o Thermodynamics

The theoretical foundation of the classical thermodynamics is based on several axioms, also called laws, two most important of which are the first and the second laws of the thermodynamics. The first law is about the conservation of energy and the second law about the principle of increasing entropy of isolated systems. Since the first law is written in a clear and complete manner, no one ever questions the truthfulness of this law. However, the second law has had a completely opposite experience. Since the concept of entropy can be understood in many different ways, the principle of increasing entropy has undergone numerous discussions and debates. As a consequence, major progresses have been made in the past one hundred plus years. At the same time, some unsettled problems still remain unsettled. In the year of 1850, R. Clausius reconsidered S. Carnot's work on the basis of the first law of the thermodynamics, and discovered the fact that the heat energy does not automatically flow from a cold object to a warm object. In 1951, L. Kelvin also discovered independently that it is impossible to successfully absorb heat out of an object and to completely transform the heat energy to another form of energy. Even though Clausius' and Carnot's discoveries look different on the surface, they actually stated the same fact from two different angles. This same underlying fact was later formulated as the second law of the thermodynamics, in the year of 1854, Clausius introduced the concept of entropy and stated the second law of the thermodynamics with the following formula: dS > 0

(5.37)

where S stands for an entropy. The meaning of eq. (5.37) is that the

Problems Related to

Thermodynamics

137

entropy of an isolated system does not decrease. In a reversible system, the entropy stays the same, while in an irreversible system, the entropy of the system increases. This conclusion is also called the principle of increasing entropy, which reveals the fact that each irreversible process in a system develops automatically toward the direction of increased entropy. This second law of the thermodynamics states the first time in the history the dirction in which natural events evolve. However, more specifically, what is the entropy of a given system? The entropy of a system is supposed to be a state function about the system. So, it is a formal concept instead of a concept about the mechanism of materials' movements. In thermodynamics, it is defined as the theoretical measure of energy, as of steam, which cannot be transformed into mechanical work; in statistical physics, it is a theoretical measure for the microscopic states of the system of interest. L. Boltzmann (1844 - 1906) studied the concept of entropy from the angle of molecular movements, and derived an analytic relationship between the entropy S and the number W of the microscopic states, which is the now-well-known Boltzmann formula S = kmW

(5.38)

where k is the Boltzmann constant. In information theory, the concept of entropy represents a measure of the uncertainty of a random event. C. E. Shannon established the concept of entropy in the information theory and consequently, laid down the theoretical foundation of the modern information theory. The entropy H of an information source is defined as follows: n

H = -CY^Pi^Pi

(5-39)

i

where pt is the probability for the ith kind of signal to appear from the information source, (— In pi) stands for the amount of information carried by the ith kind of signal, and C a constant. So, at different circumstances, the concept of entropy stands for different entity. For example, it can be employed to stand for the degree of chaos of the state, the degree of orderless, indeterminacy, the lackness of information, unevenness, or the degree of richness. In the past one hundred plus years, practical applications of the concept of entropy have gone way beyond thermodynamics and statistical mechanics, and have touched

138

Questions about Nonlinear Macro-Evolution

Theory

upon great many scientific disciplines, such as mathematics, astrophysics, biomedicine, information theory, social sciences, etc. Albert Einstein once said: "In terms of the entire science, the theory of entropy is the number one law". From such a brief historical review, one can clearly see the importance of the theory of entropies. However, the theory of entropies, even though it has been important in the development of the modern science, is not completely spotless. Clausius generalized the principle of increasing entropy to such a huge system as the entire universe so that he established the "theory of dead heat". In his eyes, the entire universe, as an isolated system, will evolve in the direction of increasing entropy. So, eventually, the universe's entropy will have to approach infinity. The heat changes from high temperature state to a low temperature state. So, there will definitely be such a day that the temperature of the heat in the universe will reach its point of equilibrium without any further fluctuations. Thus, the whole universe will enter into a kind of inert still state - the dead heat state. At the time when the "theory of dead heat" was initially announced, the society went through a large scale panic. Later on, scholars tried to explain from various angles why and how the theoretical state of "dead heat" would not occur. For example, some reasoned that the universe is not an isolated system so that the second law of the thermodynamics cannot be applied to non-isolated systems. The belief that new materials are constantly created, as stated in the stationary state doctrine of the universe, can prevent the universe from falling into the "dead heat". Also, some scholars pointed out that since there exist gravitational fields (mutual reacting fields) in the universe, the so-called "dead heat" would never occur. As for how to explain the phenomenon of an orderly nature, there also seem to be great difficulties. From the point of view of thermodynamics, there are two classes of orderly phenomena in the nature: One class contains such orderly phenomena that without any exchanges of materials and energies with the external environment, they can be maintained in isolated systems under equilibrium conditions. The orderly structure at the molecular level, as found in crystals, is one example of such orderly phenomena. This class of orderly structures can be explained by employing the general theory of the thermodynamics. The other class of orderly phenomena contains such structures that they exist only under non-equilibrium conditions through exchanges of materials and energies with the external environment. Biological beings, in this case, belong to this class of orderly structures. As

Problems Related to

Thermodynamics

139

of this writing, the classical thermodynamics has not be successful in terms of providing satisfactory explanations for this class of orderly structures. Starting from the research of Thomson (1854) on the irreversible thermo-electric phenomenon, the study of the orderly structures of the second class has gone through the highs and lows of the past 100 plus years. Since the time when L. Onsager discovered the reciprocity relation and when Prigogine uncovered the minimum entropy principle, relatively major progresses have been made in the area of linear non-equilibrium thermodynamics. However, great difficulties have been met in the area of nonlinear non-equilibrium thermodynamics. Since in the area of the orderly structures contained in objective evolutions of materials, nonlinearity plays a significant role, results in linear non-equilibrium thermodynamics, in fact, have not truly touched on the problem of understanding orderly structures. In the following, we will look at this important problem from the point of view of blown-ups, and propose our positions and understandings. 5.4.1

Entropy versible

Changes in Processes"

"Linear

and Nonlinear

Irre-

Based on Newton's second law of mechanics,

H = m-^

(5.40)

where m is the mass, ~F the force, ~r* the position and t the time, it follows that when t is replaced by (—i),the form of eq. (5.40) stays the same. That is, laws of mechanics are symmetric with respect to time, no matter whether the time evolves forward or backward. In other words, the past of the system is equivalent to the future. This kind of evolution (movement) is called reversible. As for macroscopic movements of complicated, isolated systems, the past is not equivalent to the future. This kind of movement is irreversible. This class of movement has a fundamental difference from the class of reversible movements describable by the classical mechanics. In the following, let us look at a few examples of irreversible movements. 1. Heat conduction When different parts of a system has a different temperature, the heat will flow from the parts of higher temperature to the parts of lower temperature. Eventually, the temperature difference within the system will

140

Questions about Nonlinear Macro-Evolution

Theory

disappear. This is the so-called process of heat conduction. Through experiments, it has been measured that the amount of heat passing through a unit area over a unit time period is directly proportional to the temperature gradient. That is, if J g stands for the vector of a heat flow, x the heat conduction coefficient, then one has ~fg = -xVT

(5.41)

This relationship is called Fourier Law. 2. Viscosity When macroscopic movements exist in a fluid, and when some subcurrents of the fluid flow at different speeds, the parts with lower speeds will impose a viscous force on the parts with higher speeds. So, eventually, all parts of the fluid will flow evenly. This is the so-called phenomenon of viscosity. If a fluid moves along the a;-axis and there exists a velocity gradient in the z-direction, then in terms of the phenomenon of viscosity, the viscous stress Pxz, which is defined as the momentum transported through a unit area in a unit time period, is directly proportional to the velocity gradient. In symbols, one has P « = , ^

(5.42,

where r/ is the viscosity coefficient, eq. (5.42) is called Newton's Law of Viscosity. 3. Diffusion When different parts of a mixture have different densities, the materials will flow from the areas with high densities to those areas with low densities. That is the so-called process of diffusion. Experiments imply that the mass, flowing through a unit area in a unit time period, is directly proportional to the density gradient. In symbols, one has i t = -/iVjO

(5.43)

where Jp is the density vector of materials' flow, p, the diffusion coefficient, and p the density. This formula is called Fick's Law. 4. Process of Electric Conduction When different parts of a conductor have different electric potentials, a process of electric conduction will occur. The electric current flows from the parts of high electric potentials to the places with low electric potentials. Eventually, the entire conductor will reach a state where all parts have the

Problems Related to

Thermodynamics

141

same electric potential. Through experiments, it has been observed that the intensity vector Je of the electric current, which is defined as the amount of electric current flowing through a unit area during a unit time period, is directly proportional to the electric field intensity E or the gradient of the electric potential . In symbols, one has X = aE = -aVif

(5.44)

where a is the rate of electric conduction. Summarizing the previous four examples, all these irreversible processes can be unified by the following formula: ~f = Lit

(5.45)

where J stands for the quantity, such as energy, momentum, mass, electricity, etc., transported through a unit area during a unit time period, called the transportation quantity, X represents the cause underlying the transportation process, such as differences in temperature, velocity, density, or electric potential, which is generally named thermodynamic force or kinetic force, eq. (5.45) indicates that the transportation quantity is directly proportional to the thermodynamic force. Here, L is the constant which is fixed as soon as the transportation process is determined. In practical situations, a physical transportation process is not as simple as the ones in the previous examples, instead, each practical transportation process is a combined effect of several irreversible processes. Examples here include Soret effect, pyroelectric effect, thermomagnetic effect, etc. That is, eq. (5.45) can be generalized to such a way that the transportation quantity Ji is linearly related to the thermodynamic force Xi as follows: Jfc = 5 > i f c * i

(5-46)

k

where the coefficient L^ satisfies Onsager's reciprocity relation Lik = Lki

(5.47)

eq. (5.46) is a linear expression for irreversible thermodynamic processes. AQ

The rate of change — of the entropy is related to the thermodynamic at force Xi and the thermodynamic flow (transportation quantity) Ji as fol-

142

Questions about Nonlinear Macro-Evolution

Theory

lows:

f

= f > 4

(5-48)

i=l

Now, based on the following equation of continuity of a certain physical quantity a:i,such as density, temperature, etc., —Xi + V • ~fi = 0

(5.49)

Ti = LifcXfc = -/xVxi

(5.50)

and

one can obtain an evolution equation which Xi satisfies d , —Xi = ^ V Xi

(5.51)

where /^ is a constant. When the time evolves inversely, that is, t = —t, eq. (5.51) becomes —Xi = -/A7 2 Zi

(5.52)

So, each physical transportation process, satisfying Onsager's reciprocity relation, is irreversible. And, from eqs. (5.46) - (5.48), it follows that

i

k

That is, in isolated systems, each irreversible process makes the system's entropy go up. If an one-dimensional situation is considered, eq. (5.52) becomes d 82 Xi X dt = ~^ >

,

N (5 54)

'

Assume that the initial boundary condition is f Xi(0,t) = 0,Xi(t,t) \ Xi(x,0) = 0 0<xxiV>z)P*(t>x>Viz) (rd + r ) 2

( 7 47)

where rd = r\ + r2, r* is the radius of the object pi (t, x, y, z), i = 1,2, V — v\V2 is a constant. Now, eq. (7.47) indicates that the mutual reaction of p\ and p2 is a nonlinear stirring force. Evidently, structural unevenness is the fundamental attribute of materials so that materials' rotation is inevitable. So, the Newtonian quantitative "gravitation", obtained out of the quantitative "time and space" of the "universal gravitation", cannot reveal much of the fundamental attributes of materials. More specifically, the "universal gravitation" can illustrate neither what the "universality" is nor what "gravitation" is and how materials move. Because of the introduction of the sum r^, the unreasonableness of Newton's formula eq. (7.45), when t —> 0, is avoided. The age-old belief that the universal gravitation is a

Some Problems about "Universal

Gravitation"

189

straight fulling force, in symbols, ? = C ^ T >

(7.48)

is really a misleading consequence of the assumption of particles or infinitesimal differential units without considering the characteristics of materials' uneven structures. Besides, based on the principle of mathematical modeling, as proposed by OuYang, "nonlinear evolution models can only correspond to nonlinear stirring forces. And, linear evolution models should only correspond to linear pushes", on the curved space of the non-Euclidean geometry Albert Einstein should have produced nonlinear models. However, eq. (7.32) is a mixture of both linear and nonlinear models, constituting a problem of unreasonable modeling. Also, the speed of light should be a "curved" variable in the realistic "time and space , Kl pi/ — —^—-i/xf becomes an eddy source, causing eddy motions, since c is not a fixed number (it is because in the realistic "time and space", light rays travel along curves instead of straight lines). However, eqs. (7.43) and (7.44) have become linear equations which disagree with the underlying movement in a curved space. Evidently, the formal logic conclusion of "gravitational waves", derived out of a linear equation, is not the same as its realistic objectivity. So, what's shown above indicates that Albert Einstein was still under the influence of the thinking logic of continuity of the first push, although he had made his revolutionary contribution when he pointed out the fact that "gravitation" comes from the unevenness of "time and space". The proposal of the concept of "gravitational waves" is a step backward along the historical line of the quantitative logic thinking. Similar to Newton, Einstein did not escape the misleading constraint of the quantitative non-structural analysis, either. In our discussion here, the attracting component of stirring forces is exactly the place where "gravitations" come from and also illustrates the meaning of "universality". It is our belief that OuYang's statement - "the universal gravitation is in fact a component of a stirring force" - will make the "science of God" a true science of the mankind, and that his statement will be seen as one of the most important epistemological progresses made in the twentieth century. At this junction, we like to point out the fact that the mechanical system, established on the quantitative "time and space", has played impor-

190

Some Problems Existing in the Field

Theory

tant roles in the development of the methodology of the quantitative analysis. However, the pity here is that this methodology still cannot constitute an epistemology. To this end, Zhenqiu Ren, Yi Lin and Shoucheng OuYang proposed the law of conservation of informational infrastructure, indicating that fact that at the end of the twentieth century, the leading thinking logic of the world of learning has truly entered the era of a materialistic "time and space". It is expected that the beginning of such a new era may bring forward major reforms in the world of learning at the level of basic concepts. What's practically meaningful is that because of unevenness, existing in the movement of celestial bodies, especially those singular unevenness, rotation speeds of celestial bodies will surely be affected consequently. Now, a natural question is that the mutation moment, which changes the rotation speeds of celestial bodies, is huge and is far greater than the "gravitation" of the attraction component of the stirring forces. So, a series of changes will have to occur to the existing balance of the celestial bodies. This end will no doubt provide an objective condition for future celestial evolutions and transitional changes so that various abnormal phenomena would appear or occur, leading to "catastrophes" of different levels. Evidently, the establishment of the concept of uneven rotations has provided a brand new framework on which further exploration of the evolution science can be carried out. Such a concept touches on the fundamental understanding about the nature, and helps the transition from the methodological system of the thinking logic on continuity, developed in the past 300 plus years since the time of Newton, to a methodology of the thinking logic on discontinuity. This kind of transition can be expected to bring forward an epistemological revolution.

7.6

References

The presentation of this chapter is mainly based on the works of A. Beck (1995), G. Charmov (1981), Z. L. Chen (1998), A. Einstein (1993), A. Einstein and N. Rosen (1937), A. Einstein and M. Grossmann (1913), D. N. Fan (1979), M. Fierz and W. Pauli (1939), Y. G. Hu (1984), G. Kane (1993), Z. D. Li (1983), F. R. Pukalenis (1979), Z. Q. Ren, Y. Lin and

References

191

S. C. OuYang (1998), N. Rosen (1937), J. Weber (1961), S. Weinberg (1967), C. N. Yang and R. Mills (1954). For more details, please consult with these references.

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Chapter 8

Difficulties Facing the Dynamics of Nonlinear Chemical Reactions

As what the title suggests, in this chapter, we will look at the dynamics of chemical reactions. After the concept of rates of chemical reactions is introduced, we will look at the blown-up problem existing in the studies of gaseous, liquid and chain chemical reactions, and in the study of Schlog reaction model. Our analysis indicates that the blown-up theory can specifically describe all the chemical reactions with discontinuous processes, such as burning and explosions, which can not be well described by the current methods.

8.1

Chemical Reactions and Their Rates

In recent years, the dynamics of chemical reactions has played a controlling role in the studies of chemical processes. In general, all reaction rates, satisfying linear differential equations, change up-and-down continuously. And, if a reaction rate satisfies a nonlinear evolution, then the relevant chemical reaction should also contain the phenomena of discontinuous, singular, reversed changes. As a matter of fact, burning and explosion are two commonly seen discontinuous reaction phenomena. As of this writing, such a problem has not been carefully studied, leading to a gap existing between the theories and laboratory experiments. With the introduction of the blown-up theory, a new theoretical tool of analysis has been established to resolve some open problems in the area of nonlinear chemical dynamics. By a reaction rate, it is meant to be the speed at which a chemical reaction develops. In order to define the concept rigorously, let us consider 193

194

Difficulties Facing the Dynamics

of Nonlinear

Chemical

Reactions

the following elementary reaction:

where a, and (3j , i, j = 1, 2, ..., stand for respectively the coefficients for amounts of reaction chemicals Ai and-Bj products . Assume that at the moment when the chemical reaction starts, the particle numbers of the components Ai and Bj are respectively NAi and -Afe., and, at the time moment t, the elementary reaction has completed £ times. Since each reaction process consumes Qj.i4.i- particles and produces PjBj- particles, the particle numbers of Ai and Bj at the time moment t are respectively given as follows: / NAi = N%t - cai \ NBi = N°Bi + M

(8

-2)

where £ is called the degree of advancement of the reaction, which represents the number of time elementary chemical and physical reactions having been completed. Now, the concept of reaction rate can be defined as follows:

where V stands for the total volume of the reaction system. The reaction rates of Ai and Bj are respectively given by RAt = yNAi = ~cnR V '" 1 RBj = yNBj = (5jR

rs.4)

According to the law of mass actions, as proposed by Guldberg and Waage on the basis of the laboratory experiments, the equation for the reaction rate is d{Aj) d{Bj) n R--^r-^r=K[[(Ai)

„,

(8.5)

i

where n is a coefficient of the reaction rate with [density] ~^ n* • [time] ~ as its unit, rii an index, n = Ylni an< ^ the degree of the reaction.

The Blown-Up Problem on Gaseous Chemical

8.2

Reactions

195

The Blown-Up Problem on Gaseous Chemical Reactions

Let us consider the following type of gaseous chemical reaction: aA + bB + other chemicals -> products

(8.6)

According to the law of mass actions, the equation for the reaction rate should be d(A) dt

=

(8.7)

K(A)(B)

Let (vl) 0 and (B)0 stand for the initial densities of j4and B, respectively, and x and y the amounts of declination in the densities of A and B at the time moment t, respectively. That is, x = (A)0 -(A),y=

(B)0 - (B)

(8.8)

Based on the relationship between chemical measurements, d(A) _ dt

d{B) dt

(8.9)

has d(A) dt dx b

dx dt'

dy

=a

d(B) dt

_dy_ dt

(8.10)

ba = ay

M Tt>

So, eq. (8.7) becomes r o

(XX

"1

— = cr |z + px + q\

(8.11)

where Kb

>0

p=-[a(B)o

+

b(A)0}>0

(8.12)

q=l(A)0(B)0>0 Let Ai — p2 — Aq. Then, based on the circumstances under consideration, the nonlinear reaction equation eq. (8.11) will experience different evolutionary phenomenon. More specifically,

196

Difficulties

Facing the Dynamics

of Nonlinear

Chemical

Reactions

1. When Ai > 0, (1) if la; 4- \p\ < \ Ai,one has 1 i Ci P ^ v A 1 * - 1 1 ±^/A7— = i-ip

(8.13)

(2) If la; + \p\ > | A i , one has 1 / — c o e _ < T V ^ ' 4-1 1 a; = - V / A T ^ ~- ~P

(8.14)

where CI

=

P - VST p + v •== Ai

,

p + \/AT

a n d C2

p — vAi

2. When Ai < 0, the solution of eq. (8.11) is x= - ^ - A i t a n f -ay/-Ait

+ x0 J -

-p

(8.15)

where XQ is the integration constant. eqs. (8.13) - (8.15) describe the overall evolutionary behavior of the nonlinear reaction eq. (8.11). Evidently, eq. (8.13) is a solution for continuous changes. And, eqs. (8.14) and (8.15) are blown-up solutions, containing singularities of time. eq. (8.15) describes the phenomenon of periodic blown-ups, existing in the gaseous chemical reaction. The so-called "cool flames", appearing in chemical reactions, seem to have something to do with periodic blown-ups. 8.3

The Blown-Up Problem on Liquid Chemical Reactions

Let us consider the following simple, reversible, second order liquid reaction process: A + B^C

+D

(8.16)

(A) (B) + K2 (C) (D)

(8.17)

R2

The relevant reaction rate equation is d(A) dt

Kl

Let X be the degree of advancement of this chemical reaction, (A)0 and (B) 0 the initial densities of A and B, respectively. Assume that the initial

The Blown- Up Problem on Liquid Chemical

Reactions

197

densities of C and D are zero. Then, (A) = (A)0 — X and (B) = (B)0 • X. And, one has X = ^ =

K l

[(A)0 - X] [(B) 0 - X] -

2

K2X

(8.18)

Simplifying this equation produces X = 7 (X2 + 0X + a)

(8.19)

where 7:

Kl — K2

K\

(a + b)

Kl -

K2

(8.20)

K\ab Kl -

K2

Take o = (yl) 0 and b = (B)0. Without loss of generality, assume that 7 > 0 (similarly, the case 7 < 0 can be considered). Let A2 = 01 — 4a. Now, based on the value A2 takes, the solution of the nonlinear evolution equation eq. (8.19) will be different. More specifically, one has 1. If A 2 > 0, then (1) when x + \(3 < \\fEi, the solution of eq. (8.19) represents a smooth and continuous evolution. In this case, the chemical reaction is continuous. (2) When x + \(3 < \yfK2~, the solution of eq. (8.18) is

X=

1 r— b i e - T V S J t + l 1 2^A»6lC--rV2S*_i-2/3

(8.21)

/3 + A 2 > 1. So, eq. (8.21) implies that a blown-up occurs to /?-A2 the chemical reaction when t > %. Here, the time % of the blown-up is given by where b\

tb

1 •Anb 7VA2

(8.22)

2. If A 2 < 0, then the solution of eq. (8.19) is

= \ V/ZAltan Q 7v /3A^ + ^ j - 1/

(8.23)

198

Difficulties Facing the Dynamics

of Nonlinear

Chemical

Reactions

where x0 is the integration constant, eq. (8.23) experiences a blown-up at t = tj, with £;, defined as follows: - 7 ^ / - A 2 i b + x0 = - + rnr, (n is an integer)

8.4

3.24)

The Blown-Up Problem on Schlog Reaction Model

In 1972, Schlog introduced the following chemical reaction system, consisting of three participating components A, B and X:

(8.25)

Assume that during the reaction process, the component X does not interact with the external environment, while the components A and B can exchange with the external environment so that the densities of A and B are maintained constant. Under such circumstances, the state of the reaction system is described fully by the single variable X with the following equation X = -K3X3

+ K2AX2

- KIX

+ KQB

(8.26)

If the following parameters and scalar variables are introduced _

j K2A

^

r

>0, S

K3

Hi

K2 m z

< K3 m 3

'

1

m

1

(8.27)

T = Ksm2t then eq. (8.26) can be simplified as follows:

^

dt

-C 3 -6Z + 8' -5

(8.28)

which is a cubic form of differential equations. The general blown-up properties of such differential equations have been studied in details in Section

The Blown-Up Problem on Schlog Reaction

Model

199

3 of Chapter 3. In this section, we will focus on eq. (8.28) and present a specific discussion. Let F = £ 3 + 5i - S' + 5 and the initial value condition be 6=0 = Co

(8.29)

Then, the whole evolutionary characteristics of eq. (8.28) can be studied as follows. 8.4.1

F has a real root and a pair roots

of complex

conjugate

In this case, one has F = (Z-ti)(e+pit;

+ qi)

(8.30)

where p\ and qi are constants satisfying p\ — 4^1 < 0. Then, one has

£i+Pifi+9i

£-6

In

y/e+Pit

^ ~ ^ P l T " arctan + qi

,

=

*+

dICld.Il

V 9i t + A0

\P\

*Pl

.

v 9i -

\v\ (8.31)

where AQ is the integration constant, which is determined by the initial value as follows: A0=

-

1 H+piii+qi N — \p\m

arctan 1/91 -

io - 6

In

\P\

\/£o +Pi£o+tb

| P\lVx 7M W /

7T\

(8.36)

So, eq. (8.28) contains periodic blow-ups.

8.4.2

F has real roots and of multiplicity tively

2 and 1,

respec-

Since the discussions for the cases of £1 > £2 and £1 < £2 are similar, in the following, we will only consider the case of £1 > £2- That is, eq. (8.28) can be rewritten as (8.37) By integration, one obtains

l-li

cln

t + A0

(8.38)

The Bloxm-Up Problem on Chain-Reaction Models

201

where « ~i?a c =

(8.39) ~

where Ao is the integration constant determined by the initial value as follows: A

A0 = -

A

- c

G>-6 6>-6

(8.40)

i,From eq. (8.38), it follows that when t > tb = —A), a blown-up occurs. In other words, in order to have A0 < 0 and tb > 0, eq. (8.40) implies that

£o < 6 < 6 is a necessary condition for eq. (8.28) to experience a blown-up. For the case when F has three distinct real roots, we have considered the relevant details in Section 3 of Chapter 3. So, all the related discussions are omitted here. 8.5

The Blown-Up Problem on Chain-Reaction Models

Chain reactions are a class of important and relatively special chemical reactions. Their laws of reaction rates are different of those of general chemical reactions. The feature of chain reactions is that there exist active particles, called carriers of chains, in the reaction system. So, as soon as the reaction is started, if no external control is imposed, the reaction will evolve automatically in a fashion of a chain with one step connected to another until the reaction eventually ends. For example, styrene polyreactions, which are reactions of hydrogen and chlorine gases, etc., are all chain reactions. And, the nuclear explosions of atomic bombs and hydrogen bombs are also examples of chain reactions. Based on the development of chains, the totality of chain reactions is classified as straight chain reactions and branching chain reactions. The characteristic of straight chain reactions is that each active particle of the chain carriers produces only one new active particle after participating in the reaction. As for branching chain reactions, each active particle of the chain carriers produces, after participating in the reaction, two or more new

202

Difficulties Facing the Dynamics of Nonlinear Chemical Reactions

Fig. 8.1 Straight chain reactions

Fig. 8.2 Branching chain reactions active particles so that the pool of all chain carriers increases drastically. See Figs. (8.1) and (8.2) for more details. For the sake of convenience, let us assume that there is only one kind of chain carrier X. Its reactions of reproduction, propagation, branching, and eventually stopping the reaction can be respectively described as follows: >X + ---

x+

>x + ---

X + . - . - 2 X + ... x +

(8 41)

-

••••

with their respective reaction rates ro, 2lx, 1)x, and *Bx, where 21, 3D a n d 23 are directly proportional to t h e relevant densities of the chemicals excluding the chain carriers, t h e • • -terms on t h e left hand sides of eq. (8.41). T h e products of the symbols 21, 3D and 53 with t h e density x of t h e chain carriers stand for t h e relevant rates of reactions. Based on whether or not mutual reactions exist, t h e totality of chain reactions can be classified into two sub-groups: chain reactions without

The Blown-Up Problem on Chain-Reaction

Models

(a)

203

(b)

Fig. 8.3 (a) Chain reactions without mutual reactions, (b) Chain reactions with mutual reactions.

mutual reactions and chain reactions with mutual reactions, as shown by Fig. 8.3 (a) and (b). In the following, we will study the relevant rates of reactions for these two cases individually. 8.5.1

Chain Reactions

without

Any Mutual

Reactions

The rate of reaction of the chain carriers of non-reacting chain reactions can be described by a linear equation. For example, for a reaction with a single kind of chain carriers, the density xof the carriers satisfies the equation x = r0 - (03 - 2D) x

(8.42)

Assume that the initial value condition is x\t=o = x (0)

(8.43)

Such an initial value problem can be solved with the following solution: ro

-(93-X))t

03 -T)

+ x (0) e - ( < 8 - s ) t , when 03 > £

x = r0t + x (0), when 03 = D

(8.44)

(8.45)

and rn

03-D

3 (03-S))t

+ x ( 0 ) e ( s - : D ) * , when 03 < 3D

(8.46)

204

Difficulties

Facing the Dynamics

of Nonlinear

Chemical

Reactions

Based on the relation r = — between the rate of reaction r and the density IE of the chain carriers, where t stands for the average life span of the chain carrier x, and by introducing the non-dimensional time r = =, one obtains ro (3-5

+ r(O)e-{0-VT,

-(0-S)T

when/3 > T | + v (0) e - ^ - * > T , when (3 > 5

(8.50)

(8.51)

v = T + v (0), when (3 = 5 and e(P-S)r

(3-5 L

_{\

+ v

( 0 ) e (/3-*)T j

w h e n

p


p

Fig. 8.4 A non-fixed state rate curve for a single chain carrier chain-reaction without any mutual reactions involved.

8.5.2

Chain

Reactions

with

Mutual

Reactions

For t h e chain reactions with existing m u t u a l reactions between chains, t h e reaction rate of the chain carrier x is quadratic (nonlinear). In t e r m s of the situation of only one type of chain carriers, if the reaction rates of the j t h order ending and branching processes are respectively *BJX3 and DjX1, j = 1,2, and if t h e initial chain reaction rate is ro, then the density a; of the chain carriers satisfies t h e following reaction equation =

r0-( 0 and 772 >

206

Difficulties

Facing the Dynamics

of Nonlinear

Chemical

Reactions

0; and r?i = 0 and 772 > 0. Let us look at the details of each possibility as follows. 1. W h e n rji > 0 a n d 772 > 0, and A > 0, eq. (8.53) becomes .

2 . Vi

Vo

(8.54)

-m [ x H — x (a) If

m




xf+(t)

\ / A , then 2r?2

\/Acoth f -VKt 2r?2 \2

A0 ) 2 7

m_

3.56)

2r/2

where AQ is the integration constant to be determined by the initial density. Evidently, eq. (8.56) is a problem of blown-ups. W h e n t —> 00, b o t h eqs. (8.55) and (8.56) approach the fixed densities

and , 2r/2 2772 respectively. Especially i m p o r t a n t is t h a t the evolution of eq. (8.56) finishes t h r o u g h blown-ups. T h a t is why such reaction processes experience intensive explosions. 2. W h e n rji = 0 and 772 > 0, and A = 4770772 > 0, eq. (8.53) becomes x ~ -772

,

x

2

m

3.57)

V2 (a) W h e n \x\ < —-v/^o^i then the solution is

Xo+ = — V ^ o ^ t a n h ( -y/rj^t

+

-A0

(8.58)

The Blown-Up Problem on Chain-Reaction

Models

207

(b) When |x| > — y^o^i, 4 + = —VVom coth f V ^ o ^ i - 2^0 J

(8.59)

Evidently, other than different numerical values, eqs. (8.58), (8.59) and eqs. (8.55), (8.56) experience similar reaction processes. B. For the situation where branching process is more important than ending the reaction process, there are also two possibilities: T?I < 0 and 772 < 0; and 771 = 0 and 772 < 0. Let us now look at the details of each possibility as follows. 1. When 771 < 0 and 772 < 0, A = tf + 4770771 can be positive, negative, or zero. Corresponding to each one of these cases, the solutions are respectively as follows: x(l\ = —9i- - (A0 - 7/2*) -1 , when A = 0 2r?2 If A > 0 and x +

27?2

-(2)

2772

,( 2 )

2772

- — v ^ t a n h ( -VAt Ll)2 \Z

If A > 0 and x + HL

(8.60)

+ -A0 ) Z J

(8.61) 27?2

2772

- ^ V A c o t h ( - i ^ - ^

0

) - | L

(8.62)

and if A < 0, 1 / 1 (8.63) - — V ^ A t a n --y/~\t + A0 2772 2772 V 2 where A$ is the integration constant. Evidently, solutions eqs. (8.60), (8.62) and (8.63) all contain blown-ups, and eq. (8.61) stands for continuous changes. 2. When 771 = 0 and 772 < 0, eq. (8.53) becomes „(3)

-772

x

rjo

(8.64)

208

Difficulties

Facing the Dynamics

of Nonlinear

Chemical

Reactions

Its solution is

which experiences periodic blown-ups. T h a t is, when branching is more important t h a n ending t h e reaction process, the relevant chain reaction, other t h a n the case of eqs. (8.61 - 8.62) which describes a continuous evolution, experiences blown-up(s) with time. This end proves t h e fact t h a t in general, chain reactions are more speedy a n d more intensive t h a n other types of reactions. C. For the situation which is between the previous two situations, there are t h e following two cases: 771 < 0 and 772 > 0; a n d 771 > 0 and 772 < 0. T h e relevant details are: 1. If 771 < 0 and 772 > 0, then A = 77^+47/0772 > 0. So, the chain reaction has a solution similar to eq. (8.54) with different numerical values. 2. 771 > 0 and 772 < 0, then A = 77^ + 4770772 can be positive, negative, or zero. So, corresponding solutions of t h e chain reaction are similar to those when 771 < 0 and 772 < 0. Therefore, no m a t t e r which case is considered, the chain reaction always contains such intensive reactions as blown-ups. T h a t is a n important characteristic of chain reactions with m u t u a l reactions. In short, the blown-up theory of nonlinear models can specifically describe t h e fact t h a t chemical reactions can change their forms from one t o another through discontinuities, and can b e applied to check the feasibility of the model established earlier. T h a t is, if the form of transformation t h r o u g h blown-ups does not agree with the objective reality, then the model needs to be revised. Under t h e condition t h a t the model agrees with t h e reality, the concept of blown-ups reveals and provides the cause and a tool for the analysis of the mechanism of t h e reversal changes appearing in chemical reactions. Also, this concept provides a thinking logic and methodology for the analysis, c o m p u t a t i o n a n d design of controls for experiments of chemical reactions.

8.6

References

T h e presentation of this chapter is mainly based on the works of A. W. A d a m s (1982), L. Z. Jiang (1995), Y. Lin and S. C. OuYang (1998), N. Rala Flev (1954), J. Steinfeld, J. S. Franciso, a n d W . L. Hase (1998), R.

References

209

Wyatt and J. Z. Zhang (editors) (1996), X. Z. Zhou (1993). For more details, please consult with these references.

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Chapter 9

Nonlinearity and Problems on Theories of Ecological Evolutions

In this chapter, we look at ecological evolutions from the point of view of blown-up theory and nonlinearity.. Instead of the problem of "chaos" existing in the logistic model, we focus on the ecological contents of the model. When improved logistic models are considered, our focus is on the blown-up characteristics of these revised models. It is concluded that when the whole evolution of ecological models is the aim study, blown-ups are a commonly seen phenomenon..

9.1

The Population Evolution Equation

By an ecological system, it is meant to be a system consisting of some organisms and their environment. Each ecological system is an open system, since it has to exchange materials and energies with the external environment. When materials and energies are transported into the system from the external environment, the nutrition structure, made up of food chains, moves and changes within the system in an orderly fashion. Competitions within and between species of the ecological system, seeking for food, coexistence, etc., are all nonlinear mutual reactions. Generally speaking, the population evolution equation of each ecological system that contains several species with many mutual reactions is given as follows: d,Xi

at

Ni-Y^=iPijxj +FR({xj})

+

- diXi + Fc Fm({xj},{xf})

211

({XJ})

(9.1)

212

Nonlinearity

and Problems on Theories of Ecological

Evolutions

for i, j = 1, 2, ..., n, where ki stands for the growth coefficient of the population Xi, di the death coefficient, fy some coefficients to be determined, the nonlinear functions Fc and FR respectively describe competitions and adjustments not included in the logistic equation, and Fm represents the effect of populations on the rate of change, which, in general, has something to do with the density differences of the populations within and outside the system of concern. If only one species is considered, equ. (9.2) becomes the logistic equation dx — = kx(N-x)-dx

(9.2)

where fc and d stand for the constant birth and death rates of the population, respectively, and N the maximum population size supportable by the environment. For the case of two species, i = 1, 2, equ. (9.1) becomes dxi

—— = k\Xx (N — xi — X2) — d\x\ & v t* ^ , —— = K2X2 (iV - Xi - X2) - d2X2

(9

-3)

For the case of n species, the situation can be derived similarly. The research emphasis of the current studies on population evolutions has been on the growth and declination of the population sizes with the whole evolutionary characteristics and transitional changes of the population evolutions ignored. Especially, when nonlinear models of population evolutions are concerned, other than the problem of feasibility of the mathematical modeling, one should also put more emphasis on the physical essence revealed by the models. That is to say, since nonlinear evolution models contain "explosive moving forward" of the blown-up kind, the ability of objectively describing realistic population evolutions with this concept can be greatly deepened.

9.2

Evolution Problems of Logistic Models

In the study of ecology, the logistic model is one of typical models widely applied to study population evolutions. As for the point of view that the logistic model contains "chaos", we have given a detailed treatment in Section 3 of Chapter 3. So, in this chapter, we will omit all the related details.

Evolution Problems of Logistic Models

213

Instead, in the following, we will focus on the problem of blown-ups of the logistic model from the angle of ecology. In order to understand the physical meaning of the logistic model, there is no harm for us to first present the modeling process of this so-called logistic model. The elementary evolution equation for a population X is x = (r-S)x

(9.4)

dx where x = — represents the rate of change in terms of the population size x, r is the birth rate and S the death rate of the population. From equ. (9.4), it follows that if r > S, the population size increases. If r < S, the population declines. If r = 5, the population size stays the same. Assume that the birth rate is linearly dependent on the population size, that is r = ro — ax

(9-5)

where a is a constant which has something to do with the birth rate and the population size, and ro another constant. Substituting equ. (9.5) into equ. (9.4) produces x = fix — ax2

(9-6)

where j3 = ro — S is a constant. This is the well-known logistic model and is also called a population model. Evidently, equ. (9.6) is a quadratic nonlinear evolution model. Its mathematical properties can be analyzed directly. 1. When a,/3 ^ 0, the general solution of equ. (9.6) is (9.7)

a ( - l + ceP1)

where cis the integration constant. Assume that the initial condition is x|t=o = xQ

(9.8)

where XQ stands for the population size at the initial time moment. Subc stituting c = into equ. (9.7) pro produces OCXQ — p

/3x0 OLXQ + (/? — axo)

Now, our discussion is divided into three cases.

e-'3*

(9.9)

214

Nonlinearity

and Problems on Theories of Ecological

Evolutions

(1) When a > 0 and (3 — axo > 0, x, with time, increases and approaches a finite value. Since the overall pattern of a; is roughly an S-curve, this logistic model is more advantageous than the geometric and exponential models and has been widely employed in studies of the current ecology. When p < 0 and |/3 — axo\ > \axo\, the evolution of xshows a decreasing tendency. This is only one situation of population evolutions and cannot be seen as the whole evolution with t —>• oo. (2) When a < 0, one has

th=\\z(\-JL\ P \ ax0J

(9.10)

When t < tb, x increases continuously. When t = tb, x -> oo experiences a discontinuity, which is called the death of the continuous growth. In reality, it does not mean that the population size x indeed approaches infinity. This end is different of the concept of approaching infinity of the geometric or exponential model with time. Instead, it means an end of the continuous growth, and should be understood as the fact that the validity of the assumption of continuity is not infinitely applicable. When t > tb, x < 0, which is evidently and practically meaningless. As a mathematical property, it can be seen as an omen of the end for the continuous growth. Since the reversing characteristic, when a < 0, does not agree with the objective reality, it is called an "explosive moving forward" of the nonlinear model or a "blown-up" of the model. (3) Since the population size x cannot be less than zero in real life, and since mathematically one has |a;| > x, equ. (9.9) can be rewritten as follows:

px0 ax0 + {P - axo) e~0t

(9.11)

Therefore, for the case when a < 0, when t tb, the population size x evolves respectively as growth and discontinuity, termination of continuous growth, or declination after discontinuity. Based on the classification in Section 2 of Chapter 3, it belongs to the class of non-transitional blown-ups. 2. When a = 0 and P ^ 0, equ. (9.9) becomes x = xoe0t

(9.12)

That is, our logistic model is degenerated into an exponential model.

Blown- Up Characteristics

of Improved Logistic Models

215

3. When /3 = 0 and a ^ 0, equ. (9.9) becomes x0

1 + axot

(9.13)

Now, equ. (9.13) can be considered in three individual cases. (1) When a > 0, a; approaches zero continuously and decreasingly with time. (2) When a < 0, it is a typical form of blown-up evolution. Its blown-up occurs at t = tb =

— (9.14) axo That is, the appearance of a discontinuity has limited the continuous growth of x. (3) Considering the fact that x > 0, let us rewrite equ. (9.13) as follows: x0 1 + axot

(9.15)

That is, a; experiences nontraditional changes. Therefore, equ. (9-6) describes the end of the continuous growth of a population, supporting the fact of evolutions that "at extremes, things will have to develop in the opposite direction". It has also shown that if the evolution after a transitional blown-up change, does not agree with the objective reality, the blown-up analysis can be employed as a tool to check the feasibility of the mathematical model applied. That is, conclusions of mathematical models are not the same as the objective reality. One should be more open minded to adjust the model in use. 9.3

Blown-Up Characteristics of Improved Logistic Models

A characteristic of the logistic model is that the population growth rate decreases linearly with the increase of the population density. However, it has been clearly seen in real life situations that such a relation is nonlinear instead of linear. So, there is a need to modify the logistic model. One modification is to add a coefficient M on the basis of the original model, That is, x={Px-

ax2) (l - —\

(9.16)

216

Nonlinearity

and Problems on Theories of Ecological

Evolutions

x = -/3M + (f3 + aM)x-ax2

(9.17)

Its discriminant satisfies A = (/? + aM)2 - AapM = {/3 - aM)2 > 0

(9.18)

So, equ. (9.17) can be analyzed as follows. 1. If A = 0, that is, p = aM, equ. (9.17) becomes

pf

(9.19)

with the general solution given as follows:

where C\ is the integration constant, which can be determined from the initial condition as follows: Cy = — P -ax0

(9.21)

So, substituting equ. (9.21) into equ. (9.20) produces 1 ' x =— a

P-

ax0

1 - (P - ax0) t

(9.22)

Based on this expression, it is not hard to see that a blown-up occurs at t = tb=

(9.23) P — axo Further analysis reveals the fact that in order to obtain £& > 0, one must have P — axQ > 0. So, there exist two cases in the solution equ. (9.22). (1) If P — axo < 0, since M > 0 and p = aM, then a and P must have the same sign. When a > 0, x decreases as the time goes forward infinity. When a < 0, x increases with time. So, in this case, a; is a continuous evolutionary solution. See Fig. 9.1 (a) and (b). (2) If P — axo > 0 and when a > 0 (or a < 0), x increases with time t until the moment %. When the time goes across the moment £(,, the solution x experiences a sudden rise (fall) process. After that sudden process, x decreases (increases). See Fig. 9.2 (a) and (b). So, it follows

Blown- Up Characteristics

of Improved Logistic Models

(a)

217

(b)

Fig. 9.1

x is a continuous evolutionary solution.

X

•

(a)

t

(b)

Fig. 9.2

x experiences sudden rise (fall).

that in this case, the evolution of x contains discontinuous singularity (-ies) and reversal transition(s). 2. If A > 0 and a > 0, the general solution of equ. (9.17) is 1

2ax - (/? + aM) - q2

— "1 ^

q2

TTi

TT^

=

—I +

O2

(9.24)

2ax - (/3 + aM) + q2

where qi = ± (/3 — aM) and C2 is the integration constant. Substituting the initial condition equ. (9.8) into the previous expression produces c

l_ l n 2axQ -((3 + aM) - q2 ~~ q2 2ax0 - (/3 + aM) + q2

= 2

(9.25)

Substituting equ. (9.25) into equ. (9.24) and solving for x provide P + aM

q3 +

Q4exp(-q2t)q2

r£ =

2a

q3 - q4 exp (-q2t) • (2a)

(9.26)

218

Nonlinearity

and Problems on Theories of Ecological

Evolutions

where q3 = 2ax0 - (0 + aM) + q2 q4 = 2ax0 - (0 + aM) - q2 By differentiating equ. (9.26), one obtains 929394 exp ( - g 2 * )

(9.27)

a [93 - 9 4 exp (-92*)]

Evidently, the time t in equ. (9.26) satisfies 93 - 94 exp (-92*6) = 0

(9.28)

or, at the time moment i = *b = - - l n ^ 92

(9.29)

94

a blown-up occurs. In order for tb to be greater than 0, both 93 and 94 must have the same sign. We now analyze the situation in three cases. (1) If 93 and 94 have the same sign, when I93I > I94I and 92 > 0, or when I93I < I94I a n d 92 < 0, the solution x is continuous and increases with time. (2) If 93 and 94 have opposite sign, from equ. (9.26), it follows that the solution a; is a smooth solution and decreases with time. (3) When equ. (9.28) is satisfied, that is, when I93I < I94I and 92 > 0, or when I93I > \q4\ and 92 < 0, from equ. (9.26), it follows that in the process of * —> tb, x increases with time. At * = *;,, a blown-up occurs. When t > tb, the solution experiences a sudden fall, then starts to increase again with time. So, in this case, blown-ups may occur. 3. If A > 0 and a < 0, the solution of equ. (9.17) is 0 + aM

9 3 + 9 4 exp (92*) 92

2a

-93 + 94 exp (92*) • (2a)

(9.30)

where 93 = -2ax0 94 = -2ax0

+ (0 + aM) + q2 + (0 + aM) - q2

,Q

^ '

By differentiating equ. (9.30), one obtains • _

929394 exp (92*) a [-93 + 94 exp (92*)]"

(9.32)

Blown- Up Characteristics

of Improved Logistic Models

219

From equ. (9.30), it follows that if t satisfies -93 + 94 exp (q2tb) = 0

(9.33)

or, at the time moment

* = tb = — In P92

(9.34)

94

a blown-up occurs. Evidently, equ. (9.34) implies that if % > 0, then 93 and 94 need to have the same sign. So, equ. (9.30) can be analyzed as follows: (1) Both 93 and 94 have the same sign. When I93I < I94I and 92 > 0, or when | \q4\ and 92 < 0, x is a smooth, continuous solution and increases with time. (2) The parameters 93 and 94 have the opposite signs. Equ. (9.32) implies that x < 0. In this case, the solution is still smooth and continuous and decreases with time. (3) Equ. (9.33) holds true. That is, when |9~3| < |9 4 | and 92 > 0, or when 1931 < I94I and 92 < 0, equ. (9.32) implies that x > 0. In this case, x increases with time until t = tb- At the moment t = %, a blown-up occurs. That is, when t > tb, the solution experiences a process of falling, and then starts to increase with time. However, x < 0 is practically meaningless, the blown-up can only a "blown-up of the model". So, only local growth has been described here. 4. If a = 0, the general solution of equ. (9.17) is (/3 + aM) x - (3x = C4 exp [(/? + aM) t\

(9.35)

where C4 is the integration constant, which is determined by the initial condition C 4 = (J3 + aM) x0 - PM

(9.36)

Substituting equ. (9.36) into equ. (9.35) produces

By differentiating equ. (9.37) with respect to t, one obtains x = l(x0 -M)j3

+ ax0M] exp [(/? + aM) t]

(9.38)

220

Nonlinearity

and Problems on Theories of Ecological

Evolutions

So, in the case of a = 0, one can only obtain two kinds of smooth solutions - continuously increasing or decreasing with time. The specifics have something to do with the parameters. And, we will not go into details here. From the previous discussions on the logistic model and its modifications, it can be seen that as the whole evolution of ecological systems, the evolution models, especially the modified models contain blown-up(s). That is, in the process of growth (decline), sudden falls (rises) occur due to stopped continuity, reflecting the characteristics of objective evolutions: "At extremes, things will develop in the opposite directions", and subtle mathematical properties of nonlinear models. The explosive rise and fall of nonlinear models cannot be seen as instability. Or, in other words, nonlinear models should not be studied as problems of stability. In terms of the comparison analysis between the logistic model and the modified logistic models, no matter whether it is about local evolutions or whole evolutions, the modified model agrees more with the objective reality. At the same time, it shows that in terms of mathematical model building or revealing the properties of the established models, one should always "develope theories according to the Tao". Evidently, if the modeling or the method of solution does not comply with the fundamental characteristics of the original problem, a paradox of the mathematical modeling and simulation has been resulted. And, the work needs to be ignored completely.

9.4

References

The presentation of this chapter is mainly based on the works of J. C. Frauenthal (1980), J. Hofbauer and K. Sigmund (1998), G. D. Ness, W. D. Drake and S. Brechin (1993), C. Li, S. C. OuYang and Y. Lin (1998), S. C. OuYang (1994), R. H. Rainey (1967), Y. M. Shong (1992), R. Thomlinson (1976), Y. Q. Zan (1988), Y. Q. Zan (1988a), Y. Q. Zan (1989). For more details, please consult with these references.

Chapter 10

Nonlinearity and the Blown-Up Theory of Economic Evolution Systems

In this chapter, we analyze why systematic studies of economics has been a quite recent event and why applications of mathematics in economics have not produced as desirable progress as the scientific community has wanted. Based on the blown-up theory, it is argued that due to the fact of thinking participants in all economic systems, as George Soros has pointed out, neither calculus-based nor statistics-based methods will work and produce needed results. More specifically, we emphasize on the analysis of the evolution problem in merchandise pricing of a free-competition market, and the evolution problem about competitions between economic sectors and individual enterprises. In terms of merchandise pricing, it is shown mathematically that the price of an arbitrarily chosen good or service will have to experience periodic rise and fall. The study of merchandise pricing is about nonlinear evolutionary transitions instead of a stability problem at an ideal demandsupply equilibrium. In terms of competitions between economic sectors and individual enterprises, it is shown that the whole evolution of economic competitions is analogous to that of the logistic model developed for population evolutions. Since blown-ups represent non-equilibrium mutual structural reactions, involving many factors, such as quality, product variations, functions, efficiency, convenience, appearance, human emotion, etc., the discipline of economics is neither an old-fashioned inflexible branch of the natural sciences nor a science under the slaving effect of the first push system. Only when instantaneous equilibria of different levels can be well predicted and understood, one can theoretically guarantee a long lasting economic growth. 221

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and the Blown- Up Theory of Economic Evolution

Systems

Even though we have not produced a practical procedure here to analyze a given economic system, it is expected that our analysis and discussion will play the role of a brick which has been thrown out, to attract many beautiful gems, when combined with George Soros's theories organically.

10.1

The Start of Economics and Inherent Difficulties

The appearance of theories of economics is not incidental. It was a product of the history of various economic developments and activities, consisting of a long period of natural economics, gradual appearance of merchandise exchanges, introduction of currencies, specialized productions, and more large scale collaborations. However, the systematic research of economics only started quite recently, when compared to such a long history of knowledge accumulation. It was Adam Smith (1723 - 1790) who first published his influential work The Wealth of Nations more than two hundred years ago. Due to the invention of currencies, the variables, necessary for the development of economics, had been unified. So, it is natural for economics to set its foot on mathematical analysis. However, not only has mathematics been employed in studies of economics quite recently, but also the applications of mathematics in economics been very limited. In terms of mathematics, the studies of economics of the eighteenth century was limited to general reasoning and comparison analysis. In the 70s of the nineteenth century, there appeared scholars who attempted to apply calculus in the studies of economics. Even so, no obvious effect was seen in the consequent development of economics. Based on this fact, when Engels summarized the situation of mathematics and its applications in the nineteenth century (Natural Dialectics), he did not mention anything about economics based on mathematics. Only in the twentieth century, more modern mathematical methods had been more widely and more formally applied in studies of economics with branches of learning, such as mathematical economics and quantitative economics, officially established. Because of the development of statistics and computers in the twentieth century, especially the heat waves of mathematical modeling and simulation, studies of economics have been naturally touched upon. As a consequence and/or fashion, almost all recent publications in the area of theoretical economics, appearing in major journals, contain mathematical formulas. The current situation has been

The Start of Economics

and Inherent

Difficulties

223

developed to such a degree that without mathematical formulas, no theoretical approach to economics can be seen as a "theory". What's interesting is that in the area of economics, different economists might have different opinions, and that two opposing economic theories can be granted Nobel prizes at the same time. So, contrary to mathematics and all natural sciences, the area of economics possesses the characteristic of "no boundary" existing between what's right and what's wrong. What needs to be pointed out is that the current situation of employing mathematics in economics is simply fitting available mathematical methods in various economic situations. The available mathematical methods are only quantitative, formal analysis of the history and/or the present. Even as a branch of natural sciences, these methods have not touched on the problem of revealing the underlying structural mechanisms. So, accordingly, these methods have not and will not reveal the underlying, if any, structural mechanisms behind economic phenomena. What's more important is that the calculus system is originated from the formal analysis of the inequal-quantitative effects of the first push, and that the mathematical formal analysis for economic growth and/or declination is, in form, very similar to the wave theory of the particle mechanics. However, in terms of natural sciences, the wave theory is only about consequences instead of the underlying physical mechanisms. It is because any flow of materials does not contain morphological changes of the materials. The concept of demand and supply, as studied in economics, is in fact a problem of mutual restrictions or mutual reactions under equal quantitative effects. So, if mathematics is employed, one has to face the current problem of mathematical nonlinearity. It can be said that the problem of mathematical nonlinearity, even in the community of mathematics, is still an unsettled problem. Now, with the introduction of blown-up theory, it has been shown that the problem of mathematical nonlinearity is about structures. In other words, in order to understand nonlinearity, the quantitative formal analysis of the first push can no longer be employed without major revisions. That end explains not only the reason why the application of mathematics in economics has been a quite recent and restricted event, but also the fact that the traditional mathematical methods cannot solve the problem of evolutions in economics. Since the traditional mathematical methods have been shown to be not much use in front of evolution problems encountered in the natural sciences, OuYang proposed the idea and a practical procedure, based on the vorticity of materials, to resolve evolution problems encoun-

224

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and the Blown- Up Theory of Economic Evolution

Systems

tered in the natural sciences. His method is, more specifically, developed on classifications of materials' structures according to their vectorities and has been proven effective in solving practical forecasting problems. However, in the studies of economic problems, the situation on materials' rotations, are not as clear as in natural sciences, even though some scholars and practitioners, such as George Soros, have succeeded in identifying "materials" rotations in economic development. For example, demand and supply are problems of equilibrium of different levels with multi-relational restrictions imposed on both the demand and the supply. The demand and the supply are different with varied capabilities. They reflect the multiplicity of formal quantities and structures. In general, principles of the nature have been stated simply. However, as soon as people are involved in the process of concern, the situation becomes very complicated. The objectivity of the economics is different of that of the natural sciences. So, no method of the natural sciences should be employed directly in economics without revision. In this sense, studies of economics should develop and establish its own methods according to its own characteristics. In this chapter, we will only address problems, in terms of blown-up theory, existing in the current situation of applying mathematics in the economics studies.

10.2

Evolution Problem in Merchandise Prices

In a market place with free competitions, the price P of any chosen good is closely related to the demand D and the supply S. Assume that changes in the price P is directly proportional to the difference of the demand and supply. In symbols, dP — = k(D-S),k>0 at

(10.1)

Assume that both the demand D and the supply S of this consumer good are functions of the price P with other factors staying constant. That is, D = D ( P )

s = s(P)

(10 2) (lu 2j

-

In the following, we will analyze the relation between the price P and the demand D and the supply S.

Evolution Problem in Merchandise

225

Prices

In general, the relation between the demand and the price is linear. That is, the higher the price is, the lower the demand is. Conversely, the lower the price is, the higher the demand becomes, excluding defective products and abnormal factors. So, we can write D(P) = -\P

+ (3

(10.3)

where A is the rate of change of the demand with respect to the price, and (5 the saturation constant of the demand when the price is zero. It is assumed that A > 0 and /? > 0. Generally, the relation between the supply and the price is not linear. It is because when the price of a consumer good lowers, the number of buyers will increase and the demand consequently increases. The increased demand stimulates the production of the good so that the supply is increased. If the price gradually increases, the demand will accordingly decrease so that the supply will consequently be lowered. Since the demand and the supply are not correlated directly, when the price reaches certain height, even though the demand continues to drop, the supply might be increased because the increased price can stimulate the production (Fig. 10.1). Therefore, the relation of the supply with respect to the price is nonlinear. In terms of mathematics, we have S(P) = 6 + aP + jP2

(10.4)

where S > 0 is a constant and a and 7 are respectively the linear and nonlinear intensities of the supply, satisfying a > 0 and 7 < 0. D

S

A

(P)

D(p)

Pi

Fig. 10.1

P2

The functional relationships of the demand and the supply on the price.

226

Nonlinearity and the Blown-Up Theory of Economic Evolution Systems Substituting eqs. (10.3) and (10.4) into eq. (10.1) produces ftp

where

==- = AP2 + BP + C at A = -Ivy B=-k{\-+a)

C=

(10.5)

(10.6)

k(0-5)

So, eq. (10.5) about how the price change is affected by the supply is a quadratic nonlinear evolution equation. Since, in general, higher demand implies higher price, one has k > Oand the fact that both demand curve (10.3) and the supply curve (10.4) are located in the first quadrant. (Fig. 10.1). So, f3 — 5 < 0. Now, eq. (10.6) can be rewritten as A = -k-y > 0 B = -k(X + a) 0. The characteristics of the whole evolution is

P = - ~ 2A

l

—r Po + At

(lo.io) y '

where Po is the integration constant, determined by any given initial condition. If PQ > 0, eq. (10.10) decreases continuously with time and approaches

Evolution

Problem in Merchandise

227

Prices

the equilibrium state (—|^) If Po < 0, then when t = % — —-%, a discontinuous change occurs to the price change. At this time moment, due to a disagreement between the demand and the supply, when t < tb-, the greater demand makes the price go higher. When t = tb, the price falls discontinuously, indicating the fact that either the demand reaches its level of saturation or the supply increases drastically, causing the discontinuous change in the price. When t > tb, the price starts to rebound continuously with time and eventually approaches the demand-supply equilibrium Pi. This process indicates that through the adjustment of the market place, the price is no longer growing as blindly as during the first period of time. Instead, by taking the market situation into consideration, to keep a reasonable equilibrium between the production and the market demand, the price eventually stabilizes within a certain range. The whole evolution of price, especially the evolutionary characteristics of blown-ups in the prices, is not only more complete than the stability analysis at the equilibrium states, but also more realistic than studies of continuous evolutions when compared to the objective situations existing in the market place of free competitions. 2. When A > 0, eq. (10.5) becomes dP ~dt

4\A2

2A

(10.11)

A

integrating this equation produces p , lB

l

p,

/ I P 4 C

2 V A2

2A

A 2

I B , 1 [~B~

icF

2 A ^ 2 V A*

A

= e x p U

A

/ ^ - ^ i + P20)

(10.12)

where P20 is the integration constant. Our discussion now proceeds in two cases. (l)If | P + i f I < A yjlg - 4S, then it can be obtained from eq. (10.12) that „ P

I B

2

AC

=2-V^--X

tanh

IA

B2

\C

1

\

IB

U V ^ - ^ + 2 P 2 0 )-2A

In this case, changes in price are continuous.

/inio

(10 13

,

' )

228

Nonlinearity and the Blown- Up Theory of Economic Evolution Systems (2)If|P+If|>l/

B2 A?

^§-, then eq. (10.12) can be solved produc-

ing „

P=

I B

2

AC C

2V^-T °

th

I

A

B2

AC

1

\

IB

(-2V^-T'- 2 ^ - 2 1

(1

°-14)

In this case, the price evolution contains blown-up(s). 3. When A < 0, eq. (10.5) becomes dP_ ~~dt

P+

IB YA

+

1

AC ~A

£2 A2

(10.15)

Integrating this equation produces P=l

B2 1 A^t+2P™

'

IB 2~A

(10.16)

where P30 is the integration constant, eq. (10.16) implies that at t h 2 AC B2 (§ + n-n — P30), n = 0, ± 1 , ±2, ... periodic blown-ups occur. ~A? / A That is, the price of the chosen good shows the behavior of periodic rise and fall. Obviously, our discussion above is about the problem of evolutionary transitions described by the nonlinear evolutions of mathematical models instead of the stability problem of the prices at the ideal demandsupply equilibrium. Since stabilities in price are conditional, it is likely that blown-up analysis may very well provide a new way to understand economic changes. 10.3

The Evolution Problem on Competitions between Economic Sectors and Individual Enterprises

Let us first look at the general situation for a new economic sector to appear and to develop. When a new economic sector appears along the appearance of a line of new products, the technology of producing these new products is generally quite inefficient. So, the production cost is quite high. Also, since these new products have no share and been relatively unknown in the market place, the demand is very small. So, consequently, the development in terms of massive production is slow. Even though the market is slow, the slowly and gradually increased market demand indicates that a new

The Evolution Problem on

Competitions

229

technology needs to be introduced. In the second stage of development, trend following companies start to pop up at a high speed. Consequently, the relevant investment is drastically increased and the productional level is increasing. In order to gain a winning edge in the market competition, the technology for massive production is improved and becomes more efficient with lowered costs. So, the market share of this new economic sector opens up quickly. After entering the third stage of development, the market place is gradually saturated with a more than sufficient supply. As the revenue increases, the profit elasticity of the products starts to erode. So, further advances becomes slower and more and more difficult. At the end, the whole sector stops any attempt for further development. And, the production stabilizes at the level of replacing aged or broken products purchased earlier. As a matter of fact, each real life economic sector goes through such a three stage life span with some minor variations. Clearly, the afore-mentioned evolution of economic development is a typical picture drawn on the concept of continuity. (Fig. 10.2). As a matter of fact, in the economics of free markets, the appearance of new products are accompanied by a massive creation of enterprises, leading to a saturated and then an over-supplied market. Under the competition principle that the superiors survive and the inferiors die out, the most enterprises will close one after another so that the overall production experiences a process of falling. With great efforts put forward by the surviving companies, the production level recovers gradually and reaches a dynamic equilibrium state. That is, the overall production stabilizes at the level just enough for the replacement purpose. This evolutionary scenario depicts a discontinuous nonlinear process for economic development. Y (production)

> Fig. 10.2

t

A continuous and ideal model for economic development

230

Nonlinearity

and the Blown- Up Theory of Economic Evolution

Fig. 10.3

Systems

Blown-ups appear in economic development.

If the production quantity is Y and the saturation level, that is the saturation parameter, is N, then the growth rate of the production —— is dt directly proportional to the production of Y and (N Y). In symbols, one has

£- —oo, indicating that the initial value system blows up. As a mathematical proposition, t ^ if, can only approach tb as closely as desired and the continuity of ux stops in the name of singularity. When t > tb, ux > 0. However, in physics, the time t can surely be tb or greater than tb, since the movement of our Earth is not affected by any human wishes. Evidently, changes from ux < 0 to ux > 0 can not be an automorphism of the initial value conditions. So, this change is called a blown-up, which represents a materials' transitional or reversal change from an old structure into a new structure. In this discussion, we only focused on the blown-ups of derivative functions. ux =

The Chapter on

Thoughts

241

2. For the following nonlinear deterministic solution problem in Lagrange language du

Tt

= kU

2

t = 0,u =

(A.7) UQ(X)

one has - kuot where A; is a constant. T h a t is, it is possible for one to obtain blown-ups in the original functions of interest. Similarly, other types of nonlinear equations also face the same blown-up problem. XIII T h e concept of blown-ups was initially introduced based on the actual existence of evolutions not automorphic t o t h e given initial value conditions, and on the mathematical form as discussed in XII above. According to mathematics, the concept of blow-ups is introduced when t^t\, and t —> if,So, one can see t h a t blown-ups / blow-ups. Since the realistic property of nonlinearity is eddy effects, which will surely cause break-offs in fluids or break-offs in movements, discontinuities are t h e n a t u r a l consequences. As a betrayal of t h e system of continuity, it is not a coincidence t h a t nonlinear forms of evolutions agree well with t h e results of vortical movements. XIV Equal quantitative effects possess universality with eddy motions as their characteristics. Non-equal quantitative effects are special cases of equal quantitative effects, characterized with wave motions and spray motions. This is why movements of celestial bodies can be computed accurately, while weather systems and earthquakes cannot. In other words, under equal quantitative effects, there does not exist equations! If there did not exist equal quantitative effects, there would be no puzzles and there would be no need for any one to view a n d to admire. So, "equal quantitative effects" are the most mysterious of all mysteries.

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XV Quantums (there is a need to reconsider the existence of quantums) can not be studied by simply employing particle dynamics. There exist theoretical problems in theoretical physics. It is the very reasonable for Albert Einstein not to believe in quantum mechanics. The current quantum mechanics is in fact the quantum statistical "mechanics" under influences of equal quantitative effects. Having not emphasized on equal quantitative effects is truly a mistake existing in modern scientific systems. XVI The universe is a "thing", and time possesses the property of objectivity - a flow of matters. Since flows are unidirectional, time cannot travel backward. However, time is not the same as Newton's pure continuous flows and is originated from the vibration and movement of planets. Time is born out of rotations of things and dies when the rotations of objects stop. So, the history of the universe is also an evolution history of "dynastic changes". When time and space are broken, one does not see any form but order - rotations! XVII The old saying that "one day in the heaven equals three hundred days on the Earth" came from the fact that celestial bodies rotate at different speeds. For example, the amount of time for Venus to rotate once equals 243 earth days. XVIII When the universe is not seen as a "thing", "time-space" become dimensions as introduced in mathematics. Quantities do not have structures and are not the essence of the nature. Therefore, the debate between absoluteness and relativity of "time space" is essentially the same as the debate whether or not the universe is a "thing". Standing on different positions, the debate becomes pointless. The reason why Newton fell for absolute "time and space" is because of the misleading effect of numerical quantities.

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243

t — t is the weakness of Newton; —— = d~\t I d t cannot be resolved dt ' by Einstein. Time is not a materialistic dimension instead "parasitic" to materials. N o t e 4: In Chinese language, the word "universe" sounds as yu-zhou, where yu stands for the cosmos (space) and zhou stands for time. XIX If the universe is a "thing", then it spins so that "far away will lead to coming back". The concept of infinity is established on the thinking logic of linearity which does not exist in reality. "Curvature is complete, and straightness is a distortion." If the universe is not a "thing", then it is very difficult to measure nonexistence with existing objects. In physics, one does not study things which do not exist. Therefore, the universe is a "thing", which is neither the infinity of time nor the infinity of space. The thinking logic of Einstein came at least from the property of rotation of materials. Positive mass is mass and negative mass is also mass. There is not difference between positive and negative masses. Therefore, the so-called negative mass is only distinguished by the relevant structures.

XX Newton's theory of particles was established on round shaped ideal objects. Even though there might exist continuous particles, one still cannot derive continuous media. Due to indeterminacy of irrational numbers, even though there might exist continuous media, one still cannot derive continuous particles. XXI The comprehension of macroscopic infinitesimals as quantities in the microscopic world is a western inexplicability. The western foundation of scientific theories is based on void and belongs to mysteries.

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XXII It is time to end Aristotle's belief: "all objects are continuous". The only part, which can still be kept, is his methodology, which holds true under relative conditions. This remaining part does not constitute an epistemology. XXIII It is also time to end the era of the computational scheme - calculus, developed in the system of continuity. It is because calculus cannot be the ultimate tool powerful enough to solve all problems under the heaven. Other than flows of particles, the problems, which can be resolved by employing calculus, include mainly effects of elastic pressures. Calculus provides an approximate computational scheme under non-equal quantitative effects. XXIV The system of particle dynamics belongs to the methodology of computational schemes of non-equal quantitative effects so that it cannot be generalized to cases of equal quantitative effects. XXV Heisenberg's inaccuracy of measurement should not be seen as a principle of uncertainty. It is natural to understand the old saying that "when observing objects through my eyes, the objects must be limited by my sensing ability; when an object is observed through an object, there is not way for one to distinguish the objects". The conclusion of S. Hawking (1988), derived based on this understanding, that "God plays dice" should be seen as steps backward in the development history of science. XXVI "Entropy and order" are two scientific mysteries. They were originally about small vortices versus big vortices. Changes in entropy should be about transformations between the heat-kinetic energies appearing in materials' rotations.

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245

XXVII Irregularity comes from the unevenness existing in materials' structures. So, irregularity is deterministic. Stochastics is a methodology instead of an epistemology. Engels' statement that "accident is born out of certainty" is correct; and Einstein's statement that "God does not play dice" is also correct. Since the claims were based on pictorial intuitive thinking logic, they have been constantly challenged by logic thinkers. In fact, determinacy is a problem about physical structures and is not a formal, mathematically quantified problem. Therefore, eddy effects not only resolve the mystery of nonlinearity, but also end the debate between determinacy and indeterminacy. In other words, the determinacy in the form of differential equations, appearing widely in the study of natural sciences, is only a part of the total physical determinacy. Learning calculus can help people follow rules and be a good citizen, which is indeed very mysterious. The reason is that calculus means regularization! XXVIII It is highly recommended to read The Book of Change. However, the reading should not simply stay on the level of reading sequences of words. Especially, scientists, who have a desire to succeed in their studies of natural sciences, should study The Book of Change. It is worth deeply thinking about the saying that know materials through their forms (epistemology), and foretell changes of the materials through the structures of the forms (methodology). XXIX It is worth for any one, especially scholars, pursuing a career in natural sciences, to read Lao Tzu. Of course, he/she should not read Lao Tzu on the level of various word combinations. Recognizing rong (structures) will lead to an understanding of objects. Knowing "common rong (structures)" will lead to the comprehension of Tao. One can surely employ formal logic. However, he/she should know its weaknesses. When forms are given, one should recognize the relevant

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structures. Pictorial thinking varies from person to person, so does logic thinking. Tao means materials and objects. Numerical quantification ignores Tao and emphasizes on minor aspects of the nature. The key is that numerical quantities cannot replace materials' structures. Big Tao has no form and is born out of things. Laws change from one thing to another and have no fixed form.

XXX It is worthy to read Aristotle's doctrines on forms and materials. Of course, the reading should not be for show, since Aristotle is the father of formal logic. Be cautious that one should not be misguided in the reading. The reading, especially when not for show, will help one to recognize its weaknesses. When being misguided, his/her career would be wasted. What a mistake any scholar could ever make! XXXI Experience comes from accumulation of forms and are ahead of any formation of thoughts. There exist both scientific truths and misleading information in experiences. So, experiences should be abstracted to the level of pictorial thinking logic with analyses of materials' structures, especially evolutionary structures, as the methodology. XXXII One should never see logic thinking as the only science. It is necessary to combine pictorial and logic thinkings together so that real benefits can be realized. Structures are the mother of numbers and science should not systematically preclude non-classical methods and approaches. The concept of blown-ups is born out of a combination of pictorial and logic thinkings. Mass is not the same as quantities of materials; and physical quantities are not the same as quantities of physics.

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Refinement

247

XXXIII Evenness is not a thing, and continuity is an assumption. There does not exist any law at extremes, and there does not exist any equation under equal quantitative effects. Through the concept of particles without any structures, Newton finished his quantification. Einstein completed his quantification by describing physical quantities through a set of numbers based on tensors or energies of fields. Therefore, the science, developed in the past 300 plus years, is a different version of the belief that "numbers are the origin of all things".

A.2

The Chapter on Search and Refinement

Refinement and description - questioning is also scientific exploration. XXXIV Modern instrument indicates that over 99% of the universe consists of fluids. In terms of volumes, it should be the case. In terms of mass, it is questionable. The sun should not a ball of gas! XXXV As for fundamental particles, it is still an open question whether or not they are finitely divisible.

XXXVI Does infinite heat allow infinitesimals? Without an universe containing "nothing", in which direction can a big explosion expand to? XXXVII Is there any dead heat? XXXVIII Can the universe be chaotic? It is an imagination and not a reality. Lao Tzu only mentioned "mixtures of things" and never proposed the idea of chaos.

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XXXIX Is it the duality of "wave and particle"? Or, is it the duality of "particles and waves"? Or, can it be said that waves are only a function or characteristic of flows of particles? Can "a probabilistic wave" be called a wave? Science should be clear with concepts and should not intentionally cause confusions of concepts, unless if we play "scientific games". XL "Quantum mechanics tells us that in fact all motions of particles are wave motions." - Hawking. "The unevenness of materials tells us that in fact all motions of particles are eddy motions." - from paragraph VIII. XLI There is a reason for a randomness to occur. So, it should not be called random. Randomness is a methodology and cannot be employed to criticize the fatalism. However, Laplace's fatalism is a determinism under non-equal quantitative effects, which does not constitute a determinism under equal quantitative effects. XLII Multiplicity and complexity are products of formal logic and quantification of eddy effects, and are the definite consequence of uneven evolutions of materials. However, in terms of vortical vectority, they become very simple. This might be the origin where the concepts of Yin and Yang of The Book of Change came from. XLIII Chaos is a conjecture of logic thinking, while disorder is the irregularity of sub-eddies. So, chaos is not the same as disorder. XLIV The doctrine of "chaos" is an artificial, computational consequence of useless error values under equal quantitative effects. It can also be said that the doctrine tends to search for an impossible dream in a wrong logic. It is not realistic. Bifurcation is the geometry of "chaos".

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249

XLV Dissipation and dispersion should be about energy transformation caused by sub-eddy motions. Linear mathematical expressions would not be a tool powerful enough to truly handle the situation. XLVI Economics = science + arts + human reasons and senses, and is a manmade culture, which is always situated in a changing environment. XLVII "HAVE" and "DON'T HAVE" exist jointly. "Estimate" is a wonder of Taoist's "unification of opposites". XLVIII There is no way to unify without considering the constraints of the structures of time and space. Such an "unification" would truly be an unification. XLIX There are two reasons why predictions have not been accurate. One is the separation of objectivity and subjectivity so that nothing can be measured correctly. The other is the computational inaccuracy under equal quantitative effects. Therefore, it is necessary to improve the collection of information and modify the methods of analysis. L For equal quantitative problems, one can employ the (generalized geometric) method based on informational pictorial structures. Abnormal structures can be applied to predict abnormal changes, especial major disasters. "One, who is good at understanding the nature, is a theorist, even though the nature seems to be difficult to be comprehended. The reason why he could understand the nature which seems to be difficult to be comprehended by others is that he sees the nature through its structures." Therefore, it is strongly recommended to ponder over Lao Tzu.

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N o t e 5: In the past 300 plus years, the thinking logic, originated in Europe, has been mainly based on quantitative analysis and left behind conclusions like that "accumulation of quantitative changes leads to qualitative changes" or the conception that "quantification is the only standard for scientificality" so that an essential difference between western and eastern thinkings has been created. That is the reason why for a long period of time up to now, the western civilization and the eastern civilization have had great difficulties to understand each other. LI Weak reactions of equal quantitative effects belong to the study of problems about "(ping pong) balls landing on the edges of the table", which are difficult to handle. However, in terms of human existence in the nature, these problems are the most significant. With this mind, one can see that what is meaningful is the prediction of major natural disasters. We should achieve at least as accurate as 70%. As of now, structural methods have helped to achieve over 80% accuracy. That is, our wishes have been granted. If the quality of information can be greatly improved along with the development of modern technology and science, structural analysis can reach even higher level of accuracy. LII Infrastructural prediction can greatly improve the overall forecasting accuracy by combining equal quantitative and non-equal quantitative effects. This method might be the future of the prediction science. LIII Thoughts can be false when compared with the reality, and theories can be incorrect. The current research of philosophy should strengthen the study of confirmation on wether or not available thoughts agree with the nature. N o t e 6: In the current practice of science, it is a custom for scientific practitioners to fit actual situations into known theories without checking the prerequisite conditions for the theories to practically work. When theoretical results are drawn, these practitioners hardly compare their theoretical results with the reality. When a disagreement is indeed noticed, it

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is often the case that the original theories are not criticized. One of the modern examples is the study of "chaos", where a nonlinearity, established in a classical field of science, leading to "chaos" is enough to have a "scientific" paper published without further considering the objective implication of the appearance of the so-called "chaos". LIV The Earth spins to the east. Where is the western wind, which is faster than the spinning velocity of the Earth, from? LV Without a stirrer, how can air be mixed? LVI The hot air layer existing in the atmosphere of the Earth is as hot as at least 1,000°C, and the temperature of the magma inside the Earth is approximately 1,000°C. How can the troposphere in between be cold? LVII Flows are unidirectional. We allow flows of materials to have a memory of the past?!! Memorized past may not be correct, because the ability of memorization is not perfect. Flows can also foretell the future, including future transitions. If the foretelling contains mistakes, it is because the past is not truly understood. LVIII Each tai-ji (extreme pole) is not a single pole! Multiple poles can constrain each other so that they can co-exist. Quasi-stability is only a form studied in science. Nonstability is also a science. The existing science ^ the scientific existence.

A.3

T h e C h a p t e r of A t t a c h m e n t

Meteorological science - where in our dream will we wake up today.

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LIX There are gold mines in the meteorological science, including an epistemology. Meteorology « philosophy. It is the theory of the theoretical physics. LX There are gold mines in the meteorological science. V. Bjerknes proved that uneven density leads to eddy effects. The significance of this result has gone way beyond the epistemological problem of the meteorological science, and is worthy for meteorologists to reconsider deeply. Similarly, the "theory of earth movements", developed by Zhang Heng, is worthy reconsideration by seismologists. LXI There are mistakes contained in the meteorological science. Rossby's long waves (1940) do not exist in reality. Without a strong desire to pursue after the truths and without the braveness of losing one's own life, one would be afraid to speak out against Rossby's concept and theory, since the theory has been treated as a classic. That is why it has been difficult to straighten up the situation. There is a need to wait for coming generations to recognize "rivers and oceans". As the first step of the reorganization, one needs to realize the fact that meteorology in general does not study regularized mathematical problems. Instead, it searches for solutions or explanations of astrophysics and chemistry problems. Climate ^ statistics and weather ^ N - S equations. One should improve the study and design of observation equipment, since meteorology is a science on figurative transformations, where observation is greater than measurement, and look is greater than calculation. The art or science of observation is the ultimate destination. In fact, weather evolutions can only be eddies instead of waves, since only vortical motions can lead to high altitude accumulation of fluid masses (water vapors), and only rolling motions (stirring) can constitute travelling clouds and planting rains, leading to torrential storms. Even though waves can also cause accumulations of masses (water vapors), the accumulated mass can not have high density and will not be rolling. So, with waves, travelling clouds and

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torrential rainfalls cannot be formed. Besides, it is because weather evolutions are vortical motions that pressure systems change slowly so that one has enough time to draw predictions in advance. LXII The meteorological science contains conceptual inaccuracies. Charney's "meteorological noises" are an inexplicability. LXIII The meteorological science is not sensitive to paradoxes. One typical example is Phillips' "mixture of waves" (1959). LXIV There exist formal logical frenzies in the meteorological science. Phillips' cr-coordinates (1957) intended to eliminate physical existence through a use of formal transformations. What an analytical dream! LXV There are treasures in the meteorological science. By applying acoordinates in the opposite fashion, one can hit two birds with one arrow. One implies that terrains are eddy effects. The other is that through the application of terrains, it becomes clear that nonlinearity contains discontinuity. LXVI Methods, such as non-dimensionalization, spectra expansion, etc., widely employed in mechanics, are essentially transforming forms (figurative structures) into numbers (numerical comparisons). They are not suitable for structural comparisons beyond comparisons of numerical quantities. Peeking at a leopard through a tiny hole can hardly provide a whole picture of the leopard. Weather evolutions are a topological problem and a problem of games. To study weather evolutions, one should transform numbers into forms and analyze vortical vectorities so that the mystery of weather evolutions can be resolved successfully.

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LXVII Conservative schemes are smoothing. Smoothing is essentially linearization. LXVIII Adjusting linear diffusion parameters is equivalent to treating a lame person by trimming his longer leg. Its essence is also linearization. LXIX The theory of the meteorological science rejects real-life experiences. First, it does not recognize what experiences are. And, secondly, it also rejects itself. LXX Questionable modifications have been introduced into the meteorological science. Questionable modification number one: New information, new and actual situations, is introduced. It has been beautified with the phrase "fourdimensional assimilation". New and actual situations, existing in the life span of a new system, are essentially live reports, which is not a prediction. Questionable modification number two: Constructing parameters by employing statistical methods is essentially a forecasting based on experiences. However, the meteorological science refuses to accept real-life experiences. LXXI There exist conclusions, which make people do not know whether to laugh or to cry, in the meteorological sciences. Meteorologists don't have the practical ability to forecast weather changes, and front-line forecasters can hardly be considered meteorological scientists. LXXII There are wonders in meteorological science. The correctness of weather forecastings can be checked the next day. When a theory contains misleading conclusions, that is, when a theory is incorrect, the theory can further motivate scholars to explore why and where the mistakes are made. The

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meteorological science is indeed a paradise for all scientific explorers and pursuers of truths. "Pointing out imperfections is in fact a sign of affection of the subject; speaking out about any misleading conclusions is for the purpose of finding the ultimate truths". This is exactly the reason why "good advice often sound difficult to accept".

A.4

N o n - E n d i n g Conclusions

Each methodology comes from an epistemology, and the methodology does have ability to correct the original epistemology. This is also a situation of unification of opposites. As a summarization of our search and abstraction, the core problem is still a problem of epistemology. • Materials' structures, which one can actually see and touch, are discontinuous and discrete. This is the objective reality. The epistemology, on which calculus was established, is "continuous materials". As for its achievement, other than the development of computational schemes for even particle flows, under relative conditions, it is the discovery and numerical computation of waves under the effects of elastic pressures. This end has constituted the main tool and foundation for the study of mechanical analysis of solids. The weakness is that the methodology, developed on relative conditions, cannot be seen as an epistemology. That is, calculus cannot be effectively applied to handle problems involving discontinuity. Even though this end has been considered as the second crisis in the history of mathematics, knowing the fact is not the same as acting upon the fact. In modern scientific research works, it is often the case that scholars are wandering behind the dream of "perfecting" the system of continuity, while ignoring any practical implications. That has constituted the phenomenon in the scientific community that the main focus has been set on the development of tools instead of deepening the understanding of the nature. Especially, materials' discreteness of the microscopic level has constituted break-offs in macroscopic motions and the relative discontinuous phenomena appearing in macroscopic movements. So, there has been a fundamental inapplicability problem in the methodology, developed on the system of continuity. That is why we claim that calculus cannot be the universal tool to be employed to resolve or solve all problems under the heaven.

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• As a dynamic evolution of materials, the phenomenon t h a t "all things experience prosperity and declination, and all people must go through birth and death" is an objectively existing physical reality. So, the well-posedness of mathematical problems cannot be employed as t h e physical reality. T h e systems of initial-value (or parametric) automorphisms of mathematical problems can not describe the transitional or reversal changes of such forms as t h a t "thousands of ships pass by when one boat sinks, and a great number of plants are blossoming in front of a very sick t r e e " . Therefore, following the custom research trend of mathematical well-posedness is exactly the same as ignoring the physical reality and has constituted the "trimming of the longer leg of a lame person in order to fix his physical problem". • Since there does not exist any ideal even particle in the objective world, it clearly indicates t h e fact t h a t the universal form of materials' movements is under the so-called equal quantitative effects. T h e classical system of particle dynamics is only a special case. Because unevenness is equivalent to eddy effects, calculus-based computational schemes and particle dynamics lose their validity in practical applications. Besides, t h e fact t h a t there does not exist equations under equal quantitative effects is not purely a problem of theories. Even t h o u g h one might have a correct theory, and equations could be written formally, due to the reason of computational inaccuracy of large numbers with infinitesimal increments under equal quantitative effects, the needed procedures cannot be carried out by computers. So, under the meaning of computability, one can naturally be lead to the conclusion t h a t equations are not eternal. Therefore, the phenomenon of equal quantitative effects is an oversight of all established fields of n a t u r a l sciences. • Since each irregularity comes from vortical motions and vortical motions come from unevenness of materials' structures, irregularities are deterministic. Because calculus cannot handle irregularities, it is n a t u r a l for the methodology of t h e system of randomness t o have appeared. T h e main achievement of statistics is a betrayal of the separation of "objects and forces" and "God-made models". However, due to the stability of formal time series, as studied in statistics, truly "abnormal" irregularities have to be ignored. Evidently, limited by the condition of stability of time series, the system of randomness cannot truly describe transitions appearing in evolutions. Of course, what is more important is t h a t t h e methodology (statistics)

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of the system of randomness cannot be seen as an epistemology. • In principle, the formal logic is two-valued. It is one thinking method. Based on the state of being two valued, it is understandable why "God" needs to be introduced. However, "understandable" is not the same as what is realistic. Since "God" and all his "assistants" are unevenness, "God" himself is also rotating. Hence, there does not exist a first push but a second stir. Formal thinking logic originates from materials' structures, and structures are the mother of all the concepts, such as mass, numerical quantity, functions (force, energy, form, smell, color), forms of motion, ordering, etc., which co-exist and constrain each other. The concept of structures can be seen as generalized observations. So, the law, if any, of nature has no shape, but order. With this understanding in mind, it can be seen that quantities, forces, etc., are only pieces of information or properties about materials' structures. Figurative thinking possesses a generalized and deepened logical observation and control, and specifics. It is where experience comes from and consists of the thinking and exploration about the origin of all things. • Even though the observation systems of separate "objectivity and subjectivity" are methods through which the natural world can be understood, they cannot consist of the epistemology or the absoluteness of science. In fact, each observation instrument is a piece of material, which inevitably constitutes mutual reactions and influences of "measuring objects with objects". So, there does not exist any absolute objectivity. "Subjectivity" is also "objectivity". And, it is especially so under equal quantitative effects. Therefore, the observation systems with separate "objectivity and subjectivity" are also a methodology. Long lasting pains and long lasting fun, When will be the end of the puzzle-search in the mental sea. The mental sea is also one part of the universe, Why one has to separate "you" and "me". Even though there are patterns among scientific laws, they are not the natural Tao, Laugh at "dominance" for the next thousands of years. Thousands of evolutionary forms have the same form,

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Which is in the form of spinning vectority (Yin and Yang). And, an unification exists among oppositions. There are laws in Tao and Tao in laws, Tao takes no form and laws can be written, Tao governs the nature and is long lasting, Where eternity and Yi (change) are the same. Many springs of thinking and many autumns of search, "Prosperity and declination" and "birth and death" are analogous to flowing water. The seemingly long lasting heaven, the seemingly long lasting earth, There is the measure of years for the heaven and the earth, There are life spans in terms of time. After 30 plus years of exertion, suddenly realized, Neither the heaven nor the earth long lasts. Ask the sky above: what lasts forever? What's eternal is all things are uneven. Long time observation and long period computation, Cry into the sky with "three theories and one methodology"*. Bleeding maple leaves signal the arrival of the Autumn season. Awoke to the truth and principle of ever evolutions. It's neither a dream, nor an illusion, and nor sophistries. True theorists are good at dealing with seemingly impossible, 'Cause they can employ structures at wills. Through figurative structures, they can discern the macro and micro world. Discontinuous space and discontinuous time Brought thoughts to a different level. Transforming numbers into forms unifies methods with the Tao, Laughing at no order with the confirmation of no-form and clear order. So, after having seen spinning currents here, there and everywhere, Today is the day to leave the academia. * Blown-up, spinning current and question-answer theories, and infrastructural analysis.

Afterword A. 5

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Afterword

Professor S. C. OuYang mentioned on the last page (pp. 153) of his book Break-Offs of Moving Fluids and Several Problems on Weather Forecastings t h e problem of separate "objectivity and subjectivity" and t h e figurative thinking logic about Weather Evolutions and Structural Predictions. From what is discussed in this book, one can see the entire theory of Professor OuYang, which was not completely contained in his book mentioned above. W i t h the requests of many readers of Professor OuYang's previous book and his consent, we have organized this list of his thoughts in the areas of n a t u r a l sciences, developed in the past 30 plus years as a reference for all his students and readers who are interested in knowing more about this legendary man. Even though in form, we have only collected pieces of his thoughts, we are so sure t h a t they are all products of many, many years' real-life practice and deep thinkings. From his t h r e a d s of thoughts, one can clearly see the j u m p s of his thinking logic. At the same time, not only has the materials' uneven structural analysis of his methodology been practically proven to be effective in predicting n a t u r a l disasters, but also directly pointed to the fundamental problems existing in the currently, widely employed so-called classical systems of methodologies. T h e revolutionary consequences of his theory can be seen in all the debates s t a r t e d in t h e early 1980s. It might be due to t h e painful shocks, experienced by many, and his worry of being misunderstood, Professor OuYang has decided to retire from his prolific scientific career soon after he returned from his E u r o p e t o u r in 1997. Since his work will impact many forthcoming generations in t e r m s of how one thinks and sees the world around us, we feel obligated to finish this piece of writing as a reference for the foreseeable future.

This appendix is organized by Tiangui Xiao 1 ), Yuanxing Lei 2 ' and Yong W u 3 ' *' Chengdu University of Information Technology Chengdu 610041, T h e People's Republic of China 2 ) Free-lance writer, no permanent address

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) National Soil Bureau of Fuling District Chongqing 483000, The People's Republic of China

Appendix B

An Interview with Shoucheng OuYang

—Walk out of the past three hundred years — A Brief Account on Science and Fairy Tales, Some Final Words about Science Organized by Tianqui Xiao Chengdu University of Information technology Chengdu, 610041 The People's Republic of China

Very few scholars, especially those specialized on natural sciences, of our modern time, know or understand the significance of the statement , written in Shi Tzu appeared at least during the time of Spring and Autumn Warring states. Copernicus' "heliocentric theory" (1543) of the western civilization can help us to understand Shi Tzu. Even so, I have not seen a single colleague who became motivated by his new intelligent awakening to study or to understand the people, who had created the great Book of Change and Lao Tzu, and who tried to search for an answer on how to understand the nature and why it was not Chinese people who established "particle dynamics" and "calculus" - regularized computational schemes or games, developed on the assumptions of even particles and continuity. That is because Chinese people have known the principle that "all things are impregnated by two altering tendencies, the tendency toward completion and the tendency 261

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toward initiation, which, acting together, complete each other". (Chapter 42, Tao Teh King by Lao Tzu, Interpreted as Nature and Intelligence by A. J. Bahm, Frederick Ungar, New York, 1958). (Based on what we understand, this translation of Tao De Ching is inaccurate. The following is our new translation in the point of view of physics of the same paragraph of the original scripture: All things are discontinuous with continuity being relative and artificial, and no harmony can be achieved under the thinking logic of continuity.) In the past one hundred plus years, the scientific community has widely memorized and celebrated Giordano Bruno's (1548? 1600) personal sacrifices for his effort to spread the fact on the infinitesimal size of the earth ("the earth is very small and positioned on the right hand side") of the "heliocentric theory" ("the sun is very huge and located on the left hand side"), and the "explanation" on why apples fall to the ground by the "epoch-making" scientist Issac Newton. However, what is pity is that Newton did not know in which form apples fall to the ground or what gravitation is. Of course, because of the existence of humans, the little globe, Earth, has been brimming with lives' vigor and hopes, and with nightmares and lies. From the non-stopping praying words of ancient Greek seminary temples to the "science" of the "epoch-making" scientist's papers' form of our modern time, in terms of discovering the secrets of the nature and exploring the ultimate epistemology, "achievers" of all times have "existed" in the form of making-up various fairy tales to suit the need of their individual eras. In principle, each pursuer of truths will have to repeat Bruno's path of life with varied forms. In the history of science and technology, how many times have academic conflicts and debates been settled by non-academic means or by governmental authorities? Evidently, they are not glorious times of science and technology, since the development of science is about deepened study of epistemology and decisions on whether or not a theory is correct can only be made on the basis of how well the theory can be applied to solve practical problems. Are there "particles" as studied in the "theory of particles" ? Is continuity a fundamental characteristic of materials? In front of the fact that calculus could not really be applied to compute the land area of England or Germany precisely, is it worth the fight between Newton and Leibniz over the ownership of invention of calculus? That is why up to today, there still exist German scholars who ridicule Isaac Newton as a "rogue" or

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a "rascal" of the scientific history. Newton's reputation during his second half of his life was not very glorious. Both Newton and Leibniz did not address the following problem: What problems can calculus truly resolve or solve? For today, we need to address the question whether or not the system of particle dynamics is equivalent to epistemology and whether or not this system is a forbidden zone in the theoretical logic thinking through which transpassings are not permitted. At the same time when Homhotz complained about how he was stifled by older generations in the scientific community, he himself repeated what was done to him so that Planck established the ultimate "principle" of science: "New scientific theories are accepted not through convincing opposing colleagues. Instead, it is through the death of the opposing colleagues and the arrival of approval young generations". Thus, a real explorer of scientific truths does not need approvals and applause of the "scientists" of his time. All he needs to do is adjust himself and what he does in the process of "discovering the Tao of the nature". The phenomenon, which widely exists in the scientific history and which makes people don't know whether to laugh or cry, is that under temptations of fancy hypotheses, the majority of the scientific community in general does not question the objectivity of the theory, established on the hypotheses, even when the conclusions of the theory do not agree with the reality. Instead of questioning the feasibility and the realisticity of the hypotheses, the majority tends to continuously walk down the path of the fancy hypotheses in the hope that the hypothetical illusions can be captured eventually. As a matter of fact, the particle dynamics is nothing more than a consequence of the thinking logic of the "Hands of God" of the first push and Aristotle's "continuous materials" and "separation of objects and forces". Before people had actually seen how Newton's "particles" look like, assumptions, such as quantums, photons, ..., have appeared one after another. In essence, the concept of high velocity flows has not ended the era of Newton's thinking logic of "particles". Why should scientific explorations have to follow the path of assuming first, then discussing, and then drawing inferences, ... As a matter of fact, when people had not truly reached an agreement about the evenness and continuity of materials, the scientific community had already developed the forms of separate objects and forces into the system of equations of dynamics and worshipped this system as the ultimate theory of theoretical research. To a certain degree, this system has been employed as a synonym of any theoretical analysis. In fact, the differ-

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ential equation system, developed in the framework of particle dynamics, is local particle flows under the effects of pushing forces or the effects of elastic pressures within the range of materials' deformations. When the irregularity of uneven materials' vortical motions is concerned, this equation system loses all its validity. Evidently, this end constitutes an open problem of epistemology, which needs to be resolved as soon as possible. In other words, Engels' statement that "accident is born out of necessities" is correct, and Einstein's "God does not play dice" is reasonable. Therefore, information, except misjudged cases, is deterministic, and the system of particle dynamics is a methodology, which injures available irregular information the most. In this sense, Newton is not closely as great as Shakespeare, since Shakespeare had already known the fact that Venice merchants could never chop down a pound of meat. The "quantification of science" has been a fairy tale, which has not materialized in life, since the mathematics of variables, as of now, has only been a large probabilization, where "no inaudible, no invisible and no intangibles" can be eventually understood. So, it is natural for the scientific community to accept the development of the system of randomness. However, this system of randomness is only a methodology instead of an epistemology, since irregularity comes from the rotating movement of a wholeness and the rotations from materials' unevenness, and so irregularities are inevitable. Wu, Cheng-en created the fairy tale of "flipping clouds", which could help the Monkey King to travel ten thousand miles in one flip of his body (Journey to the West, written in the 1500s). This fairy tale is more scientific than the "science" of linear flows, as studied in the particle dynamics. However, what has been making people don't know whether to laugh or cry is that as of today, the majority of the scientific community still as before carries on the system of the particle dynamics, sees the results, derived by large probabilizing irregularities, as important theories, while ignoring the unevenness and discontinuity widely existing in eddy motions. For example, even though turbulences have been recognized as irregular flows of physics, a great many recently published scientific papers still try to regularize them in various ways. What is more ridiculous is the case where after Godel has shown the well-known incompleteness theorem in mathematics, Lorenz and his followers still initiated the mathematizations of the meteorological science and derived the so-called "nonlinear science" out of a limited tool of mathematics. Their work has been mainly represented by the "chaos" doctrine. When people realize the objective, universal existence of materials'

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unevenness and discontinuity, the particle dynamics and its methodology have practically been lowered to a special method, which holds true under a set of very limiting conditions. Similar to all other branches of mathematics about variables, this special method cannot resolve or solve any of the problems involving transformations of extrema of small probability. Since the system of the particle dynamics cannot resolve problems about overall vortical movements, the "chaos" doctrine, consequently, becomes a scientific fairy tale of our time. As a matter of fact, there exists the problem of scientificality in the science itself developed in the past 300 plus years. In other words, it can be said that in the modern epistemology, the concept of the first push still dominates the entire spectrum of natural sciences. This fact is practically evidenced by the theory of wave motions under the effects of elastic pressures, from microcosmic duality of particles and waves (in fact, where is a wave motion from without a flow of particles? Can "probabilistic waves" be called waves? Scientists have been known for having made concepts clear. However, they, at the same time, create conceptual confusions.) to the "long wave" theory of macrocosmic fluids. Is there truly a difference between micro- and macrocosms in fluids? Are scientists allowed to explain existing inexplicabilities by non-existing inexplicabilities? In terms of methodology, the spectral method has been becoming the only universal method suitable to treat all "diseases". As a specific method, can the spectral method be seen as a method of dynamics? Why is it the case that every time when theoretical results do not agree well with the reality, the statistical method is always brought forward and the practical situation of interest is handled with stochastic "uncertainty" ? As a matter of fact, the statistical method is only a scheme allowing a usage of more abnormal information. However, it still operates and employs available information within the framework of the thinking logic of "trimming the longer leg" of a lame person in order to cure his physical unfitness. Its essence is still under the control of the "God's hands" of the first push. When people prattle about celestial evolutions, they tend to ignore the question on how the Earth, on which they live, evolve. How are the "movements" of the Earth evolved into "quakes" of the Earth? The difference of one word clearly reflects the fundamental concepts in use. Here, "quakes" represent the belief of wave motions. Even when we don't talk about the feasibility of all theories of seismology, in terms of wave motions, a wave of the earthly crust can travel around the Earth four times in 15 seconds. That is, even though some of the theories of modern seismology are cor-

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rect, they don't have any practical value in terms of real-life earthquake predictions. Evidently, quakes (or waves) are post-effects of movements. So, the current study of earth moves (quakes) has been based on the comprehension of the post-effects of waves, and does not have any practical significance. Hence, only from the name "earthquake", it can be seen that the current, relevant studies are modern fairy tales, developed on the form of forms without touching on the true essence of the matter at all. The concept of earth moves, as proposed by Zhang, Heng of the Han dynasty, is worthy considering by our modern seismologists. Is the essence of fluid movements wave motions? Is the theory of atmospheric movements a theory of wave motions? Who has seen a wave, which propagates unidirectionally? When hydrologic surveyors know that unidirectionality is the characteristic of currents and rocks in rivers can cause foams and rotations in the currents, Rossby's "long wave theory" and "lee wave theory" of topographic effects can only be man-made fairy tales. Is the so-called theory of thermodynamics about convections or vortices? Is the magma inside the Earth situated in convections or vortices? Are continental drifting and plate squeezing horizontal pushing? How is Indian Oceanic plate pushed northward? Is the Himalayas a consequence of pushing? Can the land surface on Mars without any ocean drift? Are there convections in the earthly troposphere? How is the air in the earthly atmosphere mixed? Not only does the concept of thermodynamic convections flood the research area of geoscience, but also has the concept spread into the study of cosmos, leading to the belief that the universe originated from the "heat death" of some ancient nebulae. How can a hot place be so calm as "died"? A place, which can be so calm as "died", must be situated in a constant temperature without any need for the concepts of hot and cold. Are there any "heat death" and "cold death" ? Is "heat" a representation of molecular motions? Are mathematical automorphisms a physical reality? Is the phenomenon that "all things experience prosperity and declination, and all people must go through growth and death" a continuous extrapolation of a given initial value? When we know even the heaven and the Earth cannot exist indefinitely, what is the difference between automorphisms (mathematical) and "living forever"? Bjerknes' circulation theorem (1898) pointed out the fact that the unevenness of materials causes materials' rotations. Essentially, this result has formed a fatal blow to the system of particle dynamics, developed since

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Newton's time. However, what's pity is that the community of the meteorological science and even the entire community of natural sciences has not recognized the significance of Bjerknes' work. Consequently, the theory of thermodynamic convections and the theory of particles of the first push have not been placed in their appropriate historical positions. At the same time, no one has found the important mistake, contained in the circulation theorem, either - circulations are closed. Similarly, at the same time when the scientific history has been eulogizing Copernicus' "heliocentric theory" and Bruno's braveness shown for his desire of pursuing after scientific truths, Copernicus also made a serious mistake: He imagined the orbits of planets as perfect and closed ellipses. In forms, his mistake is only a small mathematical error. However, its consequence was the production of misguided generations for the following nearly five hundred years, which had led to the introduction of the theory of nebulae, the theory of absolute flows, big bang theory, continental drifting, and all fundamental particles follow the rules in scientific theories. Science has always been mocking fairy tales in the entire human history. However, when science itself is fabricating its own fairy tales, it is still not clear what scientists themselves are thinking about in general. Individually, many scholars will surely continue their fun plays of scientific games with the accepted scientific fairy tales of their times, since science is definitely a means for them to make a comfortable living. "Earthquakes" are in fact heavenly quakes, and torrential rains are heavenly rains. The studies of natural disasters and the meteorological science should belong to the categories of astrophysics and chemistry, and should not be mathematical large probability problems from the atmospheric science and earth science. Where are humans from, and where are they going to? Where is the Earth from, and where is the Earth going to? Where is the Sun from, and where is the Sun going to? Where is the universe from, and where is the universe going to? As soon as a person is born, what is waiting for him is the death. The appearance of a new star means that it will disappear, When the Earth is calmly and spinningly moving closer to the Sun at the speed of 0.5 meter per year, we, the human species, do not need to be afraid, since the eternity is the same as death and being born again. Come in the form of rotations, leave also in the form of spinnings, and

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unevenness is exactly the eternity of materials' evolutions, existing in the universe. Let us hope that all newly born, epoch-making scientific theories of the future would not appear in the form of fabricating different fairy tales of their individual times.

This writing is based on a conversation with Professor Shoucheng OuYang after his book manuscript, entitled "...BreaJc-offs ..." was just finished. January, 1994.

Appendix C

Four Main Flaws and Ten Major Doubtful Cases of the Past 300 Years

Shoucheng OuYang Chengdu University of Information Technology Chengdu, 610041 The People's Republic of China

C.l

The Four Main Flaws of the Past 300 Years

Flaw # 1 : The variable mathematics is in fact a regularized computational scheme with the geometry of morphology excluded. It has not truly touched on the irregularity of materials' structures. And, it has not resolved singularities of the quantitative analysis. As the physical reality, singular irregularities are in fact the blown-ups, which was "born after first being put to death". The statement that there does not exist any law at extremes is a greatest law. Flaw # 2 : The doctrine of continuous particles is in fact a non-existing illusion. It is because no continuous media can be produced out of spherical objects. Conversely, because of the inseparability of irrational numbers, out of continuous medias, there is no way to cut continuous particles. Therefore, the doctrine of particles can only be, at most, a relative methodological system, and can only be applied to inequal-quantitative effects of the first push. When the second stir of equal quantitative effects is concerned with, it loses its validity. Flaw # 3 : The method of linearization is a means to fit whatever under consideration to the system of inequal-quantitative effects. So, linearization 269

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Four Main Flaws and Ten Major Doubtful Cases of the Past 300 Years

is also a method dealing with non-structures. Flaw # 4 : Non-dimensionalization is a method of the quantitative analysis. Since numerical quantities do not possess structures, pure quantitative analysis cannot constitute an epistemology. Quantities are only one of the attributes of materials. Without materials, how can one talk about quantities.

C.2

The Ten Major Doubtful Cases of the Past 300 Years

Case # 1 : The case of nonlinearity. Nonlinearity is a problem about materials' structures instead of a problem about quantitative formality. The earliest, and also the best, answer to this problem is the "unity of the Heaven and people" of the ancient China, written over three thousand years ago. The second best answer is V. Bjerknes's Circulation Theorem, proven over a hundred years ago. The "unity of the Heaven and people" speels out exactly the mutual reactions between the acting and the reacting objects. These mutual reactions are nonlinear eddy sources, which naturally cause eddy motions. So, in general, no wave motions of the first push are seen here. Case # 2 : The case of "God". "God" is also a synonym for forces and is the first push of the modern scientific system. However, each realistic force is originated from the unevenness of the acting object. Since when a movement is formed, the "object and forces" cannot be separated, the acting and the reacting objects must mutually react on each other. When materials are uneven, (in fact, there does not exist any even object), a stirring motion is resulted. Since the unevenness of materials comes first, we call the stirring motions as the second stir. And, both the acting and the reacting objects are rotating together with the "God" included. Therefore, the "unity of the Heaven and people" stands for eddy sources. Case # 3 : The case of order and orderless. Each order is a local regularization of large eddies. And, orderless stands for sub-eddies when large eddies are the focus. If one focuses on sub-eddies, then all eddies are orderly. Case # 4 : The case of entropy. Entropies are the non-automorphic irreversible flows existing in the process of large eddies being melted down into sub-eddies.

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Case # 5 : The case of quantification. It is an inevitable consequence of the quantitative analysis. However, each quantification is relative instead of absolute. So, quantitative comparability cannot be applied as the only standard for scientificality. Case # 6 : The case of determinacy and indeterminacy. The so-called determinacy is a problem about materials' structures instead of the problem of irregularity and regularity of the quantitative formal calculations. Irregularities are an attribute of materials. So, they are determinant. One should not treat quantitative inseparables as indeterminant. Case •#!: The case of earthquake and meteorology. Each quake is a post-effect of movement. Its cause is movement. Each wave is a post-effect of flows. Its cause is flows. The modern science is, in fact, about 20-20 hind sights. The first push has not and will not resolve the problem of predicting of earthquakes and meteorology. Case # 8 : The case of dead heat. Infinite heat will not allow infinitely small volumes. Infinities here are a concept of the quantitative analysis instead of one about materials' structures. Case # 9 : The case of "time and space". Absolute "time and space" are a misleading consequence of Newton's quantitative analysis. In terms of the materialism, both time and space are inseparable from each other. Time and space are relative and the universe also renews itself from one era into another. Case # 1 0 : The case of evolutions. Each structural state of materials is different of any old ones, constituting an evolution of the materials, which is characterized by non-uniform vectorities of materials' structures. Because of the non-uniformity, the vectorities complement each other and restrict each other. Complementation leads to the birth of new structures and restriction causes changes. Changes are complemented, leading to the creation of newer structures. New boms are restricted, causing further changes, . . . So, there appear never-ending evolutions. However, neither new borns and old boras nor new changes and old changes belong to initial value automorphisms.

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Appendix D

Structural Rolling Currents and Disastrous Weather Forecasting

Shoucheng OuYang and Yi Lin In this appendix, we summarize a nontraditional method for practical weather forecasting, specialized in major, large-scale disastrous systems. This method has been developed on the basis of irregularities and discontinuities, summarizes empirical experiences of the front-line forecasters, and has been practically shown to be more effective than currently and commercially applied methods. After a brief account on how the phenomenon of ultra-low temperatures was initially found, practical forecasting procedures on how to employ blown-up charts and V-30 charts are explained using reallife examples. It is expected that this method will find more wide-ranging applications in commercial and daily weather forecasting practices.

D.l

The Beginning of the Whole Story

Even though the first author graduated from college with a degree in meteorology, he questioned the validity of the traditional theories of meteorology, especially Rossby's waves. Consequently, he experienced personal conflicts with several professors. As a natural consequence of the unpleasant conflicts, he left the profession of meteorology upon his graduation from college. However, due to historical reasons in China, leaving the profession did not mean that he had successfully altered his fate from predicting weather events. In the year of 1963, there appeared a historically rarely seen torrential storm with precipitation > 200 mm in 24 hours in Northern China. So, he was required to analyze the mechanism of the storm and whether or 273

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not similar disastrous storms could occur at other locations. In his analysis, it was found that above the troposphere (300 - 100 hPa or about 9,000 - 12,000 meters above the sea level), there existed the phenomenon of ultra-low temperatures and that this phenomenon had a great correlation with the torrential rain in question. After this initial discovery was made, over 1,000 real-life cases were studied. It was found that if such a phenomenon did not exist, relatively severe weather conditions would not appear. These so-called severe weather conditions include hails, tornadoes, strong winds, sand storms, extra-ordinary torrential rains, etc. This ultra-low temperature appears at the level of 300 - 100 hPa and can be lower than the multi-year average temperature by as much as 10° ~ 25° C. This low temperature is the key for the water vapor below the low temperature layer to condense. It is similar to the situation of a plot of water being heated on a stove. The reason why there appear water drops on the inside surface of the lid is because the lid temperature is lower than that of the water. In general, it is a common knowledge that the sun heats the earth. Now, the appearance of the ultra-low temperature indicates that at the time when the sun heats the earth, it also cools the earth so that the troposphere is formed. That is how the sun has been rearing the myriad of the life forms on the earth.

D.2

The Discovery of the Ultra-Low Temperature

Even though graduated from a meteorology major, the first author did not really acquired any effective method to conduct real-life weather forecasting. So, he landed on a job in the Ministry of Water Conservancy and Hydraulic Power, studying problems on frozen rivers. In the year of 1963, after a historically rarely seen, large scale torrential rain appeared along the reaches of Hai-River of Northern China, the administration of a relevant department of the ministry mandated the first author to study whether or not such storm systems could appear on the upper reaches of Song-Hua River. There were two reasons to warrant the study. The first was that there would be a hydraulic dam in the future to be located in the midreaches of the river. If such large scale, disastrous torrential rains would appear on the upper reaches of Song-Hua River, the future dam would face the risk of being destroyed. The second reason was that the administration thought and believed that being a graduate from a top ranked university,

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OuYang should have had acquired the necessary knowledge to accomplish such an "easy" assignment. However, after searching through all available literature, it was found that there did not exist any established studies or methods available to solve such a problem. The reason was that the traditional method, based on weather maps, could not distinguish the kind of disastrous torrential storms in question from ordinary storms. To accomplish the assigned task, OuYang was forced to analyze all aspects of this special storm in question. In the analysis of the vertical atmospheric structure of the storm, he applied the available information up to 100 hPa, in contrast to the custom of only up to 500 hPa. As a consequence of this more intensive study, he discovered an important characteristic of disastrous storms. During the time period between one to seven days before the arrival of a severe storm (> 100 mm / 24 hours) or an extra-ordinary storm (> 200 mm / 24 hours), there appears the phenomenon of drastic temperature drops at the altitude of 300 - 100 hPa in the atmosphere. That was why such cooling process has been called as ultra-low temperatures. During the next three years, OuYang analyzed over 100 cases of major and disastrous storms having appeared in the past with needed information. All and each one of these cases had showed that the so-called ultra-low temperatures had occurred one to seven days ahead of the arrival of the storms. And, in comparison, ordinary storms ( « 50 mm / 24 hours) were not accompanied by the phenomenon of ultra-low temperatures. As a byproduct, he realized that strong convective conditions, including hails, tornadoes, strong winds (> 16 meters / second), sand storms, are all accompanied by ultra-low temperatures in advance. Especially, the ultra-low temperatures, which lead to the appearance of hails and tornadoes, are more intensive and severe. In these much thicker layer of ultra-low temperature appears prior to the arrival of the disastrous weather systems. What needs to be emphasized here is that the phenomenon of ultra-low temperatures does not necessarily lead to disastrous weather conditions. However, without an ultra-low temperature, it can be certain that no major disastrous weather will appear. This observed fact motivated OuYang to think differently from a person with a "scientific" mind. So, he employed this discovery to test-predict disastrous weathers during these three years of research and found that it was actually very effective. In the year of 1967, OuYang was assigned to conduct weather forecast for a special location for the purpose of a large-scale military operation. However, the appearance of

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an ultra-low temperature, accompanied by a good amount of water vapor in the atmosphere, was not followed by an expected disastrous torrential storm. Instead, a major and dense foggy condition was formed. Faced with the failure of his forecasting method, OuYang went back to check what had gone wrong with his approach and thinking logic and found the fact that the appearance of clockwise rolling currents, meaning rolling (vertically) to the east, is a necessary condition for the weather to change from a more pleasant situation to a bad situation, such as the changes from sunny and clear to cloudy, from cloudy to rainy, etc. If the vertical currents roll to the west, only a foggy condition will be formed. These two conditions hold true when all other conditions for a rainfall are already in place, such as high concentration of water vapor, etc. This new discovery has not only helped to improve the accuracy of practical weather forecasting, but also improved our understanding of the traditional methodologies, theories, and OuYang's thinking logic. After many years of deep thinking and analysis, it was finally realized that pure quantities are only tools for the mankind to better understand the nature, and the methodology of pure quantitative analysis is not the only way to understand and to modify the nature. In other words, what's hidden behind quantitative analysis is the whole structure of the materialistic world. So, as an epistemology, structural analysis is more fundamental and straight to the point than quantitative analysis. From this moment of discovery on, the direction of eddies has become the essence of all our works and has been constantly strengthened by more scientific and historical facts. For example, the quest for a full understanding of fluid motions is still an unfulfilled dream of the mechanical system developed in the past 300 plus years since the time of Newton. And, in his "Natural Dialectics", F. Engels also pointed out that the application of the 19th century mathematics had only been successful in the area of solids. In other words, the mathematical system, established in the past 300 plus years, has not recognized the importance of the direction in which fluids spin. What's interesting is that no matter how complicated a fluid motion may seem to be, based on the concept of directions of eddy motions, no matter what sizes eddies may have, accompanied with whatever irregularities, the directions of fluid eddies can have only two possibilities, no matter what positioning system is employed: clockwise and counter clockwise. This end coincides with the idea of the Book of Change: No mater how complicated the myriad of all things can be, they can all be explained by two opposite and complementary elements:

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Yin and Yang.

D.3

Weather Situational and Factor Predictions Based on Structural Currents

In this section, we will look at how our structural prediction model, which is designed to melting numbers back to geometric shapes, works with real-life examples. D.3.1

Weather

Situational

Predictions

Fig. D.l is the weather situation map at 500 hPa on June 2, 1994, at 8:00 hour (Beijing time) of the location 20° - 150° E. It can be seen that along 45° - 60° N region, from west to the east, there are a low pressure at 30° E. a ridge at 40° - 70° E, 70° - 90° E a trough, around 100° E a weak ridge, 110° E a trough, around 120° - 130° E a ridge, around 140° E a trough. Along 25° - 30° N, around 60° E, there is a high pressure, around 90° E a trough. Especially, the region around (20° N, 110° E) is controlled by a subtropical high pressure. The development and non-development of the weather situation, as described in Fig. D.l, can be studied by employing the concept of structural rolling currents. Now, let us see relevant details. Fig. D.2 is the blown-up chart, representing the structural rolling currents as described in Fig. D.l, of the same time moment. First, let us explain the concept of structural rolling currents and all the meanings of

Fig. D.l 150° E.

The 500 hPa weather map at 8:00 hour on June 2, 1994 for the region 20° -

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Fig. D.2 The blown-up chart for the 500 hPa weather map at 8:00 hour on June 2, 1994 for the region 20° - 150° E.

notations and symbols appearing in Fig. D.2. Based on the vertical atmospheric structure of the information about each significant levels, observed at sounding stations, with the vertical altitudinal range from the surface atmospheric pressure to 100 hPa (about 9,000 - 12,000 meters above the sea level), vertical rolling currents are constructed. In Fig. D.2, the shaded regions with dotted boundaries are the regions of clockwise rolling currents (see definitions below), computed based on each station's vertical wind directions and wind speed distributions. The non-shaded regions with dotted boundaries are the corresponding regions of counter clockwise rolling currents (see definitions below). The specific computations are carried out by the following formula: Ci = k(Vup-Vdown)

(D.l)

where Cj stands for the direction of the rolling current, k a quantitative parameter, indicating direction changes of the rolling current, and Vup and Vdown are the weighted averages of the wind speeds in the upper and lower layers with weights Ap, which are determined as follows: A. When the winds in the vertical range from the ground level to 100 hPa are not blowing in the same direction, the discontinuous layer, existing in the wind field, is employed as the dividing boundary of the upper level Vup and the lower level VdOWn- If the upper level Vup consists of northern winds and the lower level Vdown southern winds, including the case that the upper level Vup consists of western winds and the lower level Vdown eastern winds, (for the situation of the Southern hemisphere, the orders of wind

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directions are exactly reversed), then C» is considered a clockwise rolling current. If Vup consists of southern winds and Vdown of northern winds, (respectively, Vup consists of eastern winds and VdOWn western winds), then Ci is considered a counter clockwise rolling current. B. If there are two or more discontinuous boundaries in the vertical range from the ground level to 100 hPa, each continuous layer of same directional winds is decomposed into x-, (-x)-, y- and (-y)-directions. Then, computer weighted means of the same directional x-, (-x)-, y- and (-y)- wind speeds with weights Ap = difference of atmospheric pressures at the upper and the lower boundaries of each wind layer. Now, the greatest x- and (-x)- weighted means and the greatest y- and (-y)- weighted means will be employed to determine the discontinuous boundary of Vup and Vd0Wn- Now, different wind directional value of C, is determined according to Formula (D.l). C. If the winds in the vertical range from the ground level to 100 hPa are in the same direction, the speeds of the winds will definitely vary. Consequently, a rolling current will be formed, which is determined by the strongest winds. In this case, the center of the rolling current will be employed as the dividing boundary of the upper level Vup and the lower level 'down-

D. The value of C; = the difference of Vup and Vdown and is written at the location of each sounding station. Here, the + signs in Fig. D.2 are located at positions of the maximal Ci values in counter clockwise rolling current regains. The - signs in Fig. D.2 are located at the positions of the maximal Ct values in the regions of clockwise rolling currents. Our Fig. here is too small to contain all detailed labels. When the rolling current is clockwise, C, takes negative values. When the rolling current is counter clockwise, C, takes positive values (see Fig. D.2 for more details). The contour lines in Fig. D.2 (solid curves) stand for the degrees of unevenness in the structure of humidity, temperature and pressure in the vertical range from the ground level to 100 hPa. The specific values are computed based on the following formula: p * _ ffup ~ Idown u

uP

/-p. „N

"down

where C* represents the unevenness of the atmospheric vertical structure, called a blown-up parameter, since when 8up changes from < to > 6down,

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an extrema transformation occurs, the numerator of eq. (D.2) represents the atmospheric humidity difference. Especially, when the humidity of the lower level is relatively high, eq. (D.2) sufficiently reveals the unevenness existing in the vertical humidity distribution. T h a t is, the larger q~down the smaller C* (its algebraic value). T h e denominator of eq. (D.2) stands for t h e vertical difference of potential temperatures. And, R_ 0

= T

(**\

C

P

(D.3)

where T is t h e t e m p e r a t u r e , po the pressure at the sea level, p the pressure of a chosen altitude, R the gas constant, and Cp the specific heat under the specific pressure, 6up — Odown stands for the vertical potential t e m p e r a t u r e difference , which implicitly reveals the unevenness existing in the vertical distributions of t h e t e m p e r a t u r e and pressure. T h e design of eq. (D.2) is mainly done on the consideration of the ultra-low t e m p e r a t u r e structures, existing in the upper levels of the troposphere (300 - 100 h P a ) , a n d the uneven concentration distribution of water vapor in the lower levels of the atmosphere. This structure represents exactly t h e typical characteristic of a forthcoming disastrous, large scale torrential rain, or a characteristic for a major forthcoming or a new weather system transition, where t h e stratifications for qup, qdowm a n d #u P , 6 down are done according to paragraphs A - C above so t h a t when eqs. (D.l) and (D.2) are employed jointly, at the moment when a rolling current occurs, major weather systems' movement and transitions can be revealed in advance. Since together, eqs. (D.l) and (D.2) can intelligently reveal transitional changes of weather systems, the charts, generated by these equations, are called blown-up charts. These charts can also reveal the maintenance of existing weather systems. W h a t ' s interesting is t h a t blown-up charts describe 3-D information through 2-D displays. Here, in Fig. D.2, the G and S signs represent the sounding stations with t h e greatest and the smallest C*-values. W h a t needs t o be especially pointed out is t h a t without applying any 4-dimensional assimilation of new information, weather systems transitions can be forecasted in advance by applying our blown-up charts. Even for large-scale weather systems, such as those covering across several continents or t h e entire northern hemisphere, speedy personal computers will b e more t h a n adequate to conduct weather situational forecasts. If t h e P C ' s of the 586 generation are concerned, our computations involving more t h a n 50

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atmospheric layers of as large scale weather systems as described above, can be completed in about 10 - 15 minutes. If regions of S and - coincide with those of low values on the isobaric surface, then the regions of low values will maintain or develop, meaning the low value systems will not disappear at the same locations. If the regions of S and - coincide with those of high values on the isobaric surface, then blown-ups will occur at the regions of high values. That is, the high value systems will disappear at their original locations. That means that the high value systems will either move away or a new low value system is born. If regions of G and + coincide with those of high value systems on the isobaric surface, then the original high value systems will maintain or develop. Conversely, if the regions of G and + coincide with those of low value systems on the isobaric surface, then blown-ups will occur at the regions of low value systems. That is, these low value systems will disappear. Here, the problem of weather systems' development can be analyzed by employing the values of all contour lines in the relevant blown-up charts. Now, based on Fig. D.2, we can predict the evolution of all major weather systems as described in Fig. 1. Before going into more details, let us first be aware of that positive combinations of "+, G" and high pressures or "-, S" and low pressures indicate that the current system will stay the same. Otherwise, one would have a negative combination, which indicates that a systems transition - blown-up - will occur soon. In Fig. D.l, along the zone of 60° N from the west to the east, we have a low pressure trough at 30°E, a high pressure ridge at 60°E, a low pressure trough at 80°E, a weak ridge at 100°E, a trough at 110°E, a ridge at 120°130°, a low trough at 140° with a trough tail reaching (30°N, 120°E). The low latitude 584 line is located on the south of 30°N. In Fig. D.2, the region west of 120°E shows a negative combination, which indicates a upcoming systemic transitional change - blown-up. Here, a trough will change to a ridge and a ridge to a trough. The eastern sides of 130°E and 140°E are marked with positive combinations, indicating that the current system will maintain. In the region of 110°E -130°E and 30°N, there appear labels "+, G" so that one can predict that the 584 line will move northward up to 30°N. All these predictions are backed up on Fig. D.3, the actual event of the next day. What needs to be pointed out is that in this example, the currently traditional method cannot predict the disappearance of the weak ridge at

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Structural Rolling Currents and Disastrous

G0°

Fig. D.3 150° E

70°

80°

90°

100°

Weather

Forecasting

lltf E

The 500 hPa weather map at 8:00 hour on June 3, 1994 for the region 20° -

100° E and 50° - 60° N, the maintenance of the ridge at the east of 140° E and 50° - 60° N, the maintenance and development of the low value system at around 50° N and 140° E, the maintenance and eastward movement of the high value system at 30° N and 60° E, and the northward jump of the subtropical high at 110° E and 20° N. This example indicates that blown-up charts can be applied to predict both transitional changes and non-transitional changes. What needs to be added here is that dramatic weather phenomena appear in areas near dotted lines. However, to predict the dramatic levels or the relevant weather phenomena, one needs to employ V-30 charts (see next subsection for details). In V-30 charts, all irregular information has been fully applied. And, these charts have been successfully applied to predict special severe convective weather conditions, including torrential rains, fogs, strong winds, sand storms, etc. This so-called blown-up method cannot be employed to effectively forecast convective weather situations. It is because smoothings, such as computing averages, as applied in the development of the blown-up charts, have injured the spatial structure of the ultra-low temperatures. Our next example will show how to overcome this problem. The success of our method is based on how to understand fluid motions, the fundamental form of which is eddies, and how to employ available meteorological information to represent relevant atmospheric eddies. Or, in other words, our methods have successfully applied vertical vortices and horizontal vorticities, and sufficiently made use of the effects of irregular information and bravely applied

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the available information of the significant levels, which has been ignored by all current and widely applied prediction models. D.3.2

Factor Predictions

Based on V-36 Rolling

Currents

What is different of the traditional method is that our approach is designed on the belief that the fundamental property of materials is their structural unevenness, which is derived on Albert Eiinstein's "uneven time and space". And, the materials' structural unevenness is the essential cause of the irregularities existing in eddy motions. So, the key of our approach is to make use of the available irregular information as much as possible. To successfully produce factor predictions, our key is to avoid any possible injury of the available information, which have often happened in the process of smoothing. Especially, we have designed the so-called V-30 charts using those meteorological data of the significant levels, which have never been used by the traditional methods, developed on calculus-based theories. The vertical axis of Fig. D.4 stands for atmospheric pressure, which is different of lnp, as currently and widely applied. The horizontal axis is P (hPa)

Fig. D.4

•'

i'

'

'

280

290

300

310

I 3 2

Q

i

i

i

330

340

350

1 — — 360

370

r 380

The V-30 structure over Wuhan City at 08 hour on July 20, 1998

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Fig. D.5

T, the absolute temperature. The reason why the vertical axis is taken as p is that we like to more directly reveal each irregularity existing in the atmosphere, while lnp, to a great extent, has smoothed out many possible irregularities. In Fig. D.4, the following symbols have appeared: The symbol on the left stands for wind direction and speed existing in the atmosphere, where the bottom, horizontal bar stands for the wind direction, blowing from the end with vertical bars to the other end. Each taller vertical bar stands for 4 meters / second, and the shorter bar 2 meters / second. The triangle in the symbol on the right represents 10 meters /second. Each of these wind vectors stands for the horizontal winds at their individual altitudes indicated by their wind head positions. The letter V in the name V-30 is used to represent these wind vectors. The 3-curves in Fig. D.4 respectively represent the following: (1) The solid curve stands for the changes of 9 with altitude. That is, this curve depicts the vertical structure of the potential temperature. (2) The dotted curve stands for the changes in 9se, the quasi-equal potential temperature. That is, this curve depicts the vertical structure of 9se. Here 9se is defined to be the potential temperature containing relevant water vapor. Its value is computed on the assumption that the water vapor has condensed and the atmospheric temperature has been accordingly risen. (3) The broken curve is the 9se, the quasi-equal potential temperature, under the assumption that the water vapor is in the saturated state, labeled as 9*. These three curves are called the 30 in the name V-3#. Now, a factor prediction is done based on the structure of atmospheric rolling currents as indicated by the wind vectors in such as V-30 and the structure of the 30 curves. Now, in terms of short-term (within 24 hours) disastrous weather systems prediction, we have developed the following criteria: 1. High intensity rainfalls (small area, about 1 0 - 2 0 km 2 ). Such weather systems can be forecasted by observing the following on our V-30 charts:

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An ultra-low temperature in the range of 300 - 100 hPa, clockwise rolling currents, 9se and 6* curves are quasi-parallel with a difference about 10° and form an obtuse angle with the T-axis. Here, an obtuse angle, formed by 9* and T-axis, implies a higher than normal ground temperature. 2. Strong (severe) convections, including hails, tornados, etc. In this case, an as thick layer of ultra-low temperature as at least 100 - 150 hPa must exist, which is about 1,000 - 2,000 meters thick, clockwise rolling currents in the atmosphere from the ground level to 100 hPa. The overall concentration of water vapor in the mid- and lower levels of the atmosphere is not sufficient for rainfalls except at a certain altitude, where the concentration is near the saturation. The 9* curve and T-axis form an obtuse angle. 3. Disastrous torrential rains (large area, covering over 50 - 100 km 2 ). For such a weather system to occur, one needs to observe: (1) The ultralow temperature at the altitude range of 300 - 100 hPa; (2) Clockwise rolling currents; (3) The 9se and 9* curves are quasi-parallel and are quasiperpendicular to the T-axis; (4) The quasi-parallel portion of the 9se and 9* curves reaches at least the altitude of 500 - 400 hPa. In this case, the daily precipitation would be at least 100 mm. 4. Strong winds. For this case, all structures are similar to those of severe convections, except the following: (1) The curves of 9se and 9* curves always coincide, including a lack of water vapor in the mid- and lower levels of the atmosphere; (2) The angle formed by the #-curve and T-axis is > 90°; (3) The angle, formed by 9* and T-axis, is obtuse. 5. For a thick fog to form, the V-30 chart should show the features similar to those of disastrous torrential rains except: (1) The phenomenon of ultra-low temperature may not exist; (2) Instead of clockwise, the rolling currents in the atmosphere are counter clockwise. 6. High temperature, heat disasters can be observed in advance by the existence of eastern or northeastern winds in the entire atmosphere (< 100 hPa), forming counter clockwise rolling currents (Northern hemisphere). Fig. D.4 is a typical V-30 structure for high intensity rainfall weathers. It indicates that within the next 24 hours, the daily precipitation can reach at least 159 mm. To forecast the end of an ongoing disastrous weather system within the next 6 - 1 2 hours, one only needs to observe the formations of an acute angle between the 30 curves and T-axis and counter clockwise rolling currents in the atmosphere (< 100 hPa). Our practical applications

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of well over 1,000 real-life forecasts indicate that this criterion has not been violated even once. D.3.3

Forecasting

the

Precipitation

For the magnitude of a forthcoming precipitation, one can consider the following empirical experience as references: (1) If there exists an anticyclone current field under the altitude of 850 hPa located on the east of the area of interest, that is, 850 hPa is the area high pressures, then light to medium rains can be predicted. As for the forecasting of a specific observational station, this rule needs to be adjusted according to the situation of the current field of the station. (2) If the altitude of the anticyclone current field on the east of the observation station can reach 700 hPa, then medium to heavy rains can be predicted. (3) If the altitude of the anticyclone current field on the east of the observation station can reach 500 hPa, and the area on the blown-up chart, controlled by the anticyclone current field, shows "+" and "G" values, then major storms can be predicted. If the area controlled by the anticyclone current field is as big as 5 - 10 longitudinal and latitudinal distances, one should consider predicting severe storms. (4) As for extraordinary storms, if the 300 hPa synoptic charts are not available, one should make use of the V-30 charts. Obtain the vertical infrastructure of an area of 5 - 10 longitudinal and latitudinal distances located on the east of the observation station. If the vertical infrastructure is quasi-even, combined with either counter clockwise rolling currents or no rolling currents, then extraordinary storms can be predicted. As for situations of super-extra-ordinary storms, there exist ultra-low temperatures in the upper space. What needs to be explained is that the so-called anticyclone current field, as mentioned above, is exactly the traditional formulation of the situation with east highs and west lows as the conditions for storms. The reason why we emphasize on the current field is because the current field is more effective in applications than the altitude field. It can be seen from the previous empirical references that the problem of forecasting precipitation is determined by the effects of clockwise rolling currents existing in

Some Final

Words

287

the space over the area of interest, and the capability of water-vapor transportation of the clockwise horizontal eddies located on the east of the area. If the clockwise eddies in the vertical direction are called a "rain machine", then the horizontal clockwise eddies on the east would be called a "field of supplies". In this way, the problem of predicting rainfalls in practical applications is one about anticyclones.

D.4

Some Final Words

Our method and thinking logic presented in this appendix and this book have been applied in our real-life practical forecasts of major disastrous weather systems in the last twenty plus years with much improved rate of accuracy compared to commercial practices. What's more important is that this method does not require much background knowledge from the front-line forecasters. Our experience has shown that a college student with about two weeks of training will be competent enough to conduct real-life forecasts. Also, our successful practice has led us and many colleagues to question some of the believes in currently chaos research on predictability of weathers. To conclude this appendix, let us include the following table for the reader to see for him/herself the difference between our method and the traditional and currently commercially available methods.

D.5

References

The presentation of this chapter is mainly based on the works of A. Einstein (1997), Y. Lin (1998), E. N. Lorenz (1993), K. Marx and F. Engels (1987), S. C. OuYang, Y. Lin and Y. Wu (2000), S. C. OuYang, J. H. Miao, Y. Wu, Y. Lin, T. Y. Peng and T. G. Xiao (2000), T. G. Xiao (1999). For more details, please consult with these references.

288

Structural

Rolling Currents and Disastrous

Weather

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Appendix E

Afterword

This book is one part of our research plan, which Professor Shoucheng OuYang and I made in the summer of 1997. As a matter of fact, this book does not have to be written in this format. However, considering the fact that in the past 300 plus years, the world of learning has been used to the quantitative analysis of the first push system, or, in other words, to a certain degree, the quantitative analysis has been seen as an unshakable foundation for all scientific inquires of new knowledge, we finally decided to explain the problems existing in the quantitative analysis in the form of the quantitative analysis. It is our hope that by doing so, we can make our colleagues thinking and will be helpful in future scientific explorations. Of course, what's important here is that this work touches on potential conceptual changes. Since Newton introduced the concept of non-structural particles of materials, he had successfully and implicitly transformed the principle that "numbers are the origin of all things" of the Pythagoras' school into the "scientific law" that "numbers are the reasons of all objects", leading to the creation of his Mathematical Principles of Natural Philosophy. The reason why we use the word "implicit transformation" is that the principle that "numbers are the origin of all things" is difficult for people to accept. After Newton introduced the assumption of particles, he could have successfully got around the problem of materials' structures. In the form of non-structural particles of ideally, infinitesimally small, he had successfully gained acceptance of the Pythagoras' school's principle by the entire scientific community. It can also be said that his ideal particles have pointed the scientific attention to a different place. Consequently, Newton not only 289

Afterword

290

completed Aristotle's first push system, but also established a formal, quantitative analysis method. Due to the successes of the quantitative formal logic in the studies of elastic effects, the non-realisticity of his assumption of particles and problems existing in the quantitative analysis have been covered up. The concept of quantities is human abstractions originated from materials. No quantity is the same as a realistic object. So, it is natural to see why the scientific community could not and will not accept the Pythagoras' principle that "numbers are the origin of all things". However, after people were introduced with the assumption of particles, they unconsciously accepted the "law" that "numbers are the reasons of all objects". Since Newton introduced quantities into the studies of philosophy, it naturally made the study of epistemology be trapped in the belief that "numbers are the origin of all things".

E.l

Existence of Numerical Quantities

As an exploration of epistemological concepts, we first face the problem of whether or not numerical quantities actually exist. To this end, OuYang has summarized the quantitative analysis into the following five main problems: 1. Uncertainty of Real Numbers Irrational numbers are representatives of unascertained numbers. For example, the irrational number ir, relevant to circles, is an unascertained number. It is a non-repeating and non-terminating real number. So, in applications, the actual value of IT can never be accurately obtained. If a statistical analysis is applied to analyze the binary representation of the numerical value of ir, it can be seen that 7r is similar to a random variable with the greatest unascertainty. Of course, out of this end, the number 7r should not be seen as random, since it can be precisely determined by the ratio of the circumference and the diameter of a circle. At the same time, practical computations of the area (or volume) of a circular region can be done without employing numerical calculations. Similarly, \f2 is also an uncertain real number. However, when one considers the structure of the isosceles right triangle with leg length 1, \/2 equals the length of the hypotenuse and so is a very certain number. These two simple examples show that pure quantitative analysis always contains some degree of uncertainty, such as what Lao Tzu said: "If crooked, then

Existence of Numerical Quantities

291

it will be straight. Only when it curves, it will be complete", and what Zhan Yin (a scholar of the time of the Spring and Autumn Warring States) said: "Numbers cannot describe all that are out there". In other words, the quantitative analysis is not as concise and rigorous as the geometric structural analysis. What's more important here is that the uncertainty of the quantitative analysis cannot constitute any objective realistic uncertainty. This end is one of the reasons why the quantitative analysis is not complete. 2. Artificiality of Real Numbers The number zero is a quantity without any magnitude in the mathematics of real numbers. It has been seen in the history of mathematics as a great invention of the mankind. (The invention of zero has been credited to Babylonia; and some scholars claim that it was Indians who first invented the number zero). However, it is not a great discovery. In the analysis of mathematical quantities, zero processes an infinite amount of magic power. It is the beginning of all numbers and also plays the role of the end of all numbers. Since zero helps the introduction of base systems, the quantitative analysis could have been developed massively and can constitute the foundation of all numbers. So, without zero, the entire quantitative analysis would paralyze. The number zero can "dissolve" all (any number multiplied by zero equals zero), and is the killer of all numerical manipulations. (Each non-zero number divided by zero equals infinity. So, all mathematical manipulations have to stop). Thus, the number zero plays an important role in all successes and failures of the quantitative analysis. What's practically important is that the number zero does not represent an objective existence. It is because "Tao is about things", or physics does not speak about non-existence. And, it is also because in the materialistic world, there do not exist two absolutely identical objects. So, neither zero nor infinity exist. The corresponding quantities are only formal abstractions without any backing from the materialistic world. Since Newton introduced mathematics into the studies of philosophy, it is inevitable that the problems, existing in the formal mathematics, are also brought into the studies of physics and epistemology. For example, Newton's absolute "time and space "are numerical quantities, which have not naturally and clearly spelled out the properties of "time and space". With Newton, both "time and space" are continuous without any explanation on where they were from. Therefore, all the differential equations with varying time, derived

292

Afterword

on calculus, operate under the "numerical time and space". Albert Einstein criticized Newtonian "time and space" and proposed the unevenness of "time and space". However, in his operations, Einstein did not practically realize his uneven time and space. In his "eternal equations", Einstein continued to use Newtonian numerical "time and space" without even attempting to tell why "time and space are uneven". I.Prigogine believes that "time comes into being before existence". That is, time first and materials second. What Prigogine said is very straightforward. However, his "time prior to existence" is not the physical reality. OuYang has pointed out that the reason why we still cannot tell what time and space are after the mankind has entered the era of "high tech" is because of the misleading effect of the quantitative effect. Based on the conclusion that structural unevenness of materials cause rotations of the materials, OuYang derived the consequence that both "time and space" are rotating, and that time is a measure of materials' rotation. So, existence is prior to time. That is, rotating materials appear before time. Even though the time difference between the appearances of time and materials can be small, time cannot appear first. Then, he introduced the concept of "broken time and space". That is, both "time and space" are measures established on the basis of materials' discontinuity. In other words, both "time and space" are materialistic and will always be relative. The concept of absolute "time and space" was a consequence of the non-structural and non-materialistic numerical quantities. So, numerical inseparability and materials' separability constitute a contradiction of logic. The numerical zero leads to the problem of whether or not there exist evolutions without any materials. And, the absoluteness of numerical quantities still cannot clearly spell out the periodicity and non-periodicity of materials' evolutions. The introduction of zero has brought forward conveniences and puzzles for quantitative analysis so that even up to now, some implications of this number are still not very clear. However, one thing is clear: Zero is an artifact. Even though it can destroy all other numbers, it cannot eliminate the existence of a piece of sand. 3. Regularization (or Large Probabilization) of the Variable Mathematics Because of the requirement of continuity for all calculus operations and the constraint of stability of time series of statistical methods, these two currently, widely applied methodologies have to injure irregular or small

Existence

of Numerical

Quantities

293

probability information. W h a t ' s important is whether irregular or small probability information stands for any physical meaning or existence. Especially, all transitional information is either irregular or small probabilistic. It can be said t h a t irregular or small probability problems can be solved by neither calculus nor statistical methods. As an epistemology, irregular or small probability events are exactly t h e products of irregular rotations of materials' structures. In other words, each irregularity has its own special meaning of physics so t h a t irregularities and small probabilities cannot be seen as random. 4. Incomputability and Non-existence of Equations under Equal Quantitative Effects In Section 6 of C h a p t e r 3, we have talked about the problem of equalquantitative effects. Even though it seems to be a common sense problem, due to the reason of pursuing after pure quantifications in the past 300 plus years, scholars have made a long list of mistakes. Since there do not exist absolutely identical objects in the objective world, the so-called equal quantities stand for mutual reactions of quasi-equal objects. W h a t ' s described by Newton's second law of mechanics is inequal quantitative effects. T h a t is, it is a slaving (or "being b e a t e n " ) law under the condition t h a t the object, being acted upon, is much smaller t h a n the acting object. However, Newton's third law and the law of "universal gravitation" have touched upon t h e m u t u a l reactions of quasi-equalities. It can be said t h a t as long as t h e object, being acted up on, has its own structure (not an ideal particle), the problem of concern is about m u t u a l reactions. And, since the mutual reaction of uneven materials, in fact, all materials are uneven, is a rotation motion, it led OuYang t o propose the concept of the second stir. So, the current "nonlinear science" is a science of the second stir instead of studies of the problems of Newton's second law of the first push. Since the second stir is originated from materials' structures, we call nonlinear problems as structural problems of materials' mutual reactions and Newton's second law of the first push as a quantitative problem of passive actions. Evidently, as an epistemology, t h e concept of the second stir constitutes an important scientific discovery of the 300 plus years after Newton and an important scientific step forward of the twentieth century. So, the eddy motions, representing structural mutual reactions of materials, will be named OuYang law. Since Newton's third law contains the mutual reacting, quasiequal quantitative histority, (However, Newton himself did not realize the

294

Afterword

vorticity of movements caused by uneven structures), such an expanded law can be named as Newton-OuYang Law. One important implication of OuYang Law is that it points out the incompleteness of the quantitative analysis, developed in the past three hundred plus years, that nonlinearity is no longer a problem of Newton's second law of the first push, and that problems of Newton's "universal gravitation" cannot be resolved by employing Newton's second law. Since the unevenness and discontinuity of materials will surely lead to uneven and discontinuous rotations, both "time and space" are not only "uneven" or "curved", but also rotationally "broken" - "discontinuous". In other words, since the "unevenness or curvature of time and space" come from the vorticity of materials, the essence of multi-dimensional "time and space" is the multiplicity of materials' rotations. This end is where OuYang's concept of a "multiple rotating universe" comes from. 5. The Problem of Equal Quantitative Non-Isomorphisms It can be said that the aim of the scientific system, developed in the past 300 plus years, has been a science of "quantification", constituting the origin where the practice, that quantitative comparability is the only standard for scientificality, is from. This scientific aim can also be said to be the core of Newtonian particle mechanic system. From the time when Newton assumed that the concept of mass is a measure co-produced by the density and volume, to that the concept of mass is a "quantity for materials", to the method of non-dimensionalization widely employed in mechanical analysis, all these concepts and methods can be seen as artifacts developed for the purpose of quantification. There is no doubt that quantification has become the base of the foundation of the modern science, which has been developed to such a degree that without employing the quantitative analysis, no study can enter the door of science. However, the foundation of the quantitative science, developed since the time of Newton, has been very weak. When we discussed this problem with Professor OuYang, he showed us his two hands. Evidently, the pure quantitative analysis can never tell the right and the left hand apart. This is a simple case of the so-called "equal quantitative non-isomorphic" problem. In other words, through the pure quantitative analysis, the understanding of the world can never be complete with the serious problem of missing physical essences. So, the quantitative comparability should not be employed as the only standard for scientificality.

Unification of Units

E.2

295

Unification of Units

In the current quantitative analysis system, it is the results, derived by using the method of non-dimensionalization (or called unification of units), that have been completely missing the underlying physical essences, that have been treated highly as scientific theories. For example, if the method of non-dimensionalization were not applied, then the idea of differential equations would not have gone very far. It would have gone at most as far as ordinary differential equations. It is because, in terms of systems of partial differential equations, the physical meaning of each equation is different. Without applying the method of non-dimensionalization, these partial differential equations would not be simplified to ordinary differential equations so that the consequent solutions would not be obtained. Each introduction of non-dimensionalization naturally ignores the physical essences of the original equations, just as in the case of apples and pears. In the beginning, the apples and pears cannot be added. After unifying the units of apples and pears to the level of fruits, the numbers of apples can be added to the number of pears in the unit of fruits. However, the result of addition does not tell how many apples and how many pears are there any more. With this example in place, Professor OuYang has jokingly called Newton's Mathematical Principles of Natural Philosophy as an "apple and pear" philosophy. Even though that is a joke, it indeed points to the problems existing in the quantitative analysis. In other words, one should never treat the quantitative analysis as the ultimate goal of science. Or, it can also be said that the concept of equal quantitative effects has not only completely revealed the weaknesses of the quantitative analysis, but also represented a commonly existing objective phenomenon. It can be employed not only in the natural sciences, but also in areas like economics, military science, political science, and other areas of social sciences. Evidently, problems of equal-quantities cannot and should not be continually considered on the basis of the assumption of particles. Instead, they should be treated as problems of non-particle-like structures. What's important is that OuYang has not only deepened the epistemology about materials through vorticities, but also established the method of infrastructural analysis based on distinctions of materials' structural characteristics and vectorities (the directions of rotation). What counts here is that his method can be and have been successfully applied to resolve the problem of predicting transitions appearing in evolutions. This problem

296

Afterword

has been a difficult strong hold in all traditional methods a n d theories. Even though this method was originally designed for the analysis of fluids, it can be applied to study and analyze break-off problems of solids so t h a t it has been employed in predictions of earthquakes and moves. T h e central idea of the method of the informational infrastructural analysis consists of (1) to "melt numbers into shapes" instead of the Newtonian system of "abstracting" shapes into numbers; (2) Introducing the concepts of rotational directions and discontinuities into t h e geometry of transformations so t h a t the m e t h o d of topology has been enriched and can be practically applied. In other words, the method of t h e so-called informational infrastructure has constituted as new system of methodology independent of the traditional system of differential equations and the m e t h o d of statistics. At the same time, it also develops and enriches t h e age-old qualitative analysis m e t h o d based on practical experiences. T h a t is, the informational infrastructural analysis applies approximate quantification analysis only after the underlying properties of materials are known. T h a t is, each quantitative analysis should be employed conditionally instead of unconditionally. More specifically, the informational infrastructural analysis does not support the idea of adding apples with pears. Instead, it insists on adding apples to apples and pears to pears. In other words, in each application of the informational infrastructural analysis, structures are always the first with quantities being the second. W i t h the underlying structures known or under the condition t h a t the underlying structures are not going to be distorted, the relevant quantities are not required to be identical. For example, if the "pliers"-like structure of a h a n d between t h e t h u m b and t h e other four fingers is of t h e number one importance, t h e n in this term, t h e t o t a l numbers of t h u m b s a n d the fingers do not change t h e "pliers"-like structure with the condition t h a t neither the total number of thumb(s) nor t h a t of fingers can be less t h a n one. T h a t is the place where "melting numbers into shapes" differs from "abstracting shapes into numbers". At this junction, w h a t ' s important t o note is t h a t the "quantified" science, developed in the past 300 plus years, has never achieved the claimed numerical accuracy. Due to the reason why it has been the case, no absolute quantification will ever be achieved in t h e future, either. Therefore, numerical comparison and quantitative comparability, as the only s t a n d a r d for scientificality, have never been and will never be complete and ideal. In the previous paragraphs, it has been argued t h a t the position of t h e modern science, established a n d developed in the past 300 plus years, is in

Materialism of "Time and Space"

297

its quantitative forms. So, due to all the fundamental problems, existing in the quantitative analysis, it can be seen that such a science is still not the science which can reveal the laws governing the operations of all things. The first-push system, of which Newton's second law plays the central role, has only provided a rough or an approximate quantitative analysis for the movements of objects under inequal quantitative effects - when the acting objects are much, much greater than the objects being acted upon. That is the very reason why the first push system has not and will not resolve problems about fluids and fluid motions. And, that is the reason why OuYang pointed out that nonlinearity does not belong to the category of problems solvable by Newton's second law of mechanics. So, Newton could not tell in the quantitative form how the mutually reacting objects, as described in his third law, would behave, what the meaning of the "universality" and what the "gravitation", as mentioned in his law of universal gravitation, are. This end has led to a series of problems. What's most important here is the fact that the first push system cannot provide a plausible explanation for the commonly seen phenomenon of "birth, growth, disappearance, and death" existing in materials' evolutions and cannot illustrate the periodicity and non-periodicity of evolutions.

E.3

Materialism of "Time and Space"

Aiming at the weakness of the first push system, as described above, OuYang specifically emphasized on the concept of materialism of "time and space". He pointed out that both "time and space" are measures for the movements of materials. That is because even when an object is not moving, the object still rotates with the carrying object, making time a measure for materials' rotation. As for the relevant objects, due to their structural unevenness, the consequent mutual reactions lead to spinnings of the objects so that rotations within rotations of different scales appear. That is, the multiplicity of rotations within rotations constitutes the origin from which the concept of vorticity and relativity of the "time and space" are introduced. Therefore, in terms of the materialism, time can only appear after existence. The corresponding evolutions of materials and the evolution science would represent: 1) The fundamental characteristic of materials is the infrastructural unevenness so that any distinction of materials is based on different structural

298

Afterword

unevenness. 2) The structural unevenness of materials causes rotations or rolling movements of materials, forming evolutions. That is, the true meaning of evolutions is "rolling forward". 3) Because of materials' rotations, there appears the phenomenon of periodicity. Because of the duality of rotations, since clockwise and counter clockwise rotations must co-exist, there exists non-uniformity in rotating materials, which leads to the creation of sub-eddies. Therefore, the duality of rotations is not only the origin of the multiplicity of eddies, but also the cause for the formation of non-periodicities. The duality is also where evolutionary symmetries and asymmetries come from. 4) Since evolutions are about the vorticity of materials and the nonuniformity of multiple rotations, in the form of sub- and sub-sub-eddies, through materials' rotations, heat-kinetic energy transformations are completed so that the death of materials in certain structural form appears and the birth of the materials in a different structural form is created. That is the transitionality - blow-n-ups of evolutions. And, that is how the foreverlasting evolution of "birth, growth, disappearance and death" of materials is created. 5) Evolutions are about the future. So, the key of the studies on evolutions is the prediction of the future. In other words, based on the past and the present, one should be able to tell about the future. Otherwise, no evolution science is within the human discourse and there is no need to mention any prediction. Even when an evolution repeats the past (periodicity), it is still about the future and about "rolling forward". What's important here is evolutionary transitions. That is, based on the past and the current rotational, multiple non-uniformity of materials' rotations, the true evolution science must be able to foretell the future transitional changes different of those of the past and the present. Only in this way, one can talk about true predictions. With this in mind, OuYang defined the evolution science as systematic studies about materials' rotations (and rolling movements) and about materials' structural changes appearing in rotations. What should be emphasized is that evolutions must be about the future (moving forward). It should not be a science of the past. 6) The evolution science is deterministic and is characterized by structural distinguishability. This is a natural consequence of the fact of the physical process of evolutions with clear layer structures. The concept of

How Does Our Book Come

About?

299

whole evolutions can be summarized as follows: Materials' structures —> rotations (constituting periodicity) —>• duality of rotations (symmetry and asymmetry —»•) non-uniformity of rotations (nonperiodicity) —> sub-eddies (periodicity of sub-eddies) —> duality of subeddies —> non-uniformity of sub-eddies (non-periodicity of sub-eddies) —>• sub-sub-eddies —>•... —> Completion of heat-kinetic energy transformations -» disappearance of the materials' old structures —> transitions (blown-ups) —¥ birth of new structures —»•... There is no doubt t h a t the previous concept of whole evolutions and the claim t h a t time is a measure of materials' rotations are an important step forward in the development of epistemology. T h a t implies t h a t each variable of time implicitly contains an assurance of rotating materials' structures. We believe t h a t such a point of view has satisfactorily resolved t h e problems of materialism of "time and space" and of t h e dialectical logic of evolutions. T h e fundamental science of the first push system, developed in the past 300 plus years, has met difficulties in its foundation. To this end, OuYang pointed out t h a t Newton's work, Mathematical Principles of the Natural Philosophy, still cannot constitute the evolutionary principle of t h e n a t u r a l philosophy.

E.4

H o w D o e s Our Book C o m e A b o u t ?

W h a t I like to mention at point of writing is t h a t I had discussed all the fundamental problems, touched upon in this book, with Professor OuYang during the summer of 1997, when I was a visiting professor at G M D Instit u t e for C o m p u t e r Architecture and Software Technology of the German National Research Center for Information Technology, Berlin, Germany, and Professor OuYang was there to a t t e n d t h e 15th IMACS (International Association of Mathematical and Computer Simulation) World Congress, as one of my invited speakers. As a consequence of our over-the-night discussion, I sensed t h e need and desire to read such a book. However, at the time, I did not know in which format to write this book. In the summer of 1998, along my world lecture tour, I visited Professor OuYang in Chengdu, China, and discussed on how to carry out t h e idea of a book project. Mr. Yong Wu was also invited to come t o Chengdu. At the special time moment, Professor OuYang was physically ill and was tied up by his exploration of long-term disastrous weather predictions. Consider-

300

Afterword

ing the fact that it would be difficult to change any accustomed concepts and believes, we decided to explain and illustrate the fundamental concepts and positions of evolutions and the evolution science on the background of problems and difficulties existing in the modern science so that we can, hopefully, create a thinking space for the readers and ourselves. At the end, Yong Wu and I decided to name our book as it is entitled now and to carry out the entire writing. In form, it seems that we tried to "challenge" the first push system of the past 300 plus years. However, the true spirit of our book here is about how to resolve practical problems with generality and deepened theoretical understandings. Besides, when one faces practical difficulties, it is natural for him to question the validity of the theory behind his methods. Evidently, theories are originated in human endeavors to gain deeper understanding about of the nature. And, all theories at the end have to touch on the underlying philosophical point of view of the world. Currently, it seems that all people sense the huge and obvious difference between the eastern and western civilizations. However, what is the intension of the difference? I believe that the difference originates from the different underlying epistemology and that the two civilizations complement each other well. Especially, OuYang has dug up the fact that the epistemology of materials' vorticities itself is also a methodology so that one can see that the eastern philosophy has more generality and coverage. Roughly, the basis of the western civilization was established on Christianity of the Hebrew civilization, which has been well represented by the book of Genesis of the holy bible and the slaving philosophy of the ancient Greece - "Forces exist independently outside of objects". On the other hand, the basis of the eastern civilization was developed on materials' figurative and structural constraints, as well described in the Book of Change, representing a philosophy of mutual slaving and limiting. Therefore, the world-view of the quantitative form of the pure slaving philosophy has been difficult for the eastern civilization to accept. And, it has also been difficult for the western civilization to understand the structural point of view of the world under mutual slaving. That might be the reason why the western and eastern civilizations have followed their individual and different thinking logics and methodologies. As a central problem for scientific explorations, it is also a problem of epistemology for us to ask: What has been indeed described by the scientific system of the past 300 plus years? As a consideration of this problem, what's discussed in this book has clearly shown that the scientific system

How Does Our Book Come

About?

301

of the past 300 plus years has essentially described a quantitative analysis for quasi-stability and quasi-equilibrium without providing the causes of t h e quasi-stability and quasi-equilibrium and without touching on, of course, the problems of instability and noon-equilibrium. So, in principle, t h e scientific system of the past 300 plus years has not dealt with problems of evolutions. W i t h the concept of particles, Newton and his followers completed the slaving quantification. Based on the approach of employing a

set of numbers to describe some relevant physical quantities, Albert Einstein a n d others completed another quantification through tensors or the energies of fields. In form, their quantification stands for mutual reactions between fields. However, in practical operations, they did not avoid the slaving effect. It is still an open question whether or not quantified propagation of field energies has described m u t u a l reactions between fields. As a communication between concepts, the main focus of our discussion here in this book is t o point out the problem of m u t u a l reactions. This problem has not been treated ^""^ssfully by t h e means of t h e slaving effect and should not be excessively expanded. T h a t might be the reason why "the current science is not the same as t h e scientific reality". W h e n Professor OuYang finished reading this book manuscript, he gave us an interesting comment. Let us quote the comment here as our conclusion remark. "The 'God' has told us who he is t h r o u g h a most straightforward (structure), the convenient (stirring forces), and the most beautiful (eddy motions) fashion. However, t h e mankind has applied t h e most abstract (quantitative), t h e most inconvenient (pushing forces), and t h e most inaccurate manner to understand him."

YiLin (also known as Jeffrey Yi-Lin Forrest) January 12,2001 President and Director International Institute for General Systems Studies, Inc. 23 Kings Lane Grove City, PA 16127, USA http://www.iigss.net Jeffrey, forrest @sru. edu

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Index

fi-plane, 104

Aristotle's model, 81 assemblage flow, 163 assemblage of particle flows, 163 assumption of particles, 19 assumption of positive pressure, 103 asymmetric tensor, 182 asymptotic stability, 34 atmospheric density, 100 atmospheric density pressure, 89 atmospheric discontinuity, 94 atmospheric dynamics, 94 atmospheric evolution, 94, 99 atmospheric long wave, 93, 102 atmospheric pressure, 77 atmospheric refrigerator, 116 atmospheric state, 92 atmospheric temperature, 114 atom, 6

aabular representation, 23 Abelian gauge field, 179 absolute maximum, 153 absolute minimum, 153 absolute temperature index, 82 absorption of light, 157 accelerating circulation, 77 accuracy rate, 94 adjustment of market place, 227 advection equation, 44, 48, 96 affinitive potential, 147 analytic manipulation, 21 analytic representation, 22 ancient Babylonia, 1 ancient Chinese, 1 ancient Egyptian, 1 angular velocity, 101 anticyclone, 80 antiderivative, 30 applicable information, 9 approximation approach, 46 approximation expansion, 149 approximation method, 94 approximation of nonlinearity, 45 approximationsof nonlinearity, 88 Archimedes, 92 Aristotelian formal logic, 4 Aristotelian thinking logic, 4

atomic line width, 174 atomic reversal relaxation time, 174 atomic structure, 87 automorphic behavior, 151 autonomic equation, 57 autonomic system, 32 Backlund transformation, 112 background radiation, 82 Bernoulli equation, 19, 40 Bernoulli, J., 19 Berry, M., 168 311

312

Bessel integration, 48 Bianchi identity, 185 bifurcation phenomenon, 121 bifurcation solution, 70 bifurcation theory, 120, 172 big bang, 82 big bang explosion, 83 big river culture, 8 biology, 121 birth rate, 130, 213 birth-death exchange, 88 birth-death-like evolution, 102 Bjerknes' Circulation Theorem, 163 Bjerknes's circulation theorem, 78 Bjerknes, V., 6, 76 black body radiation, 3, 157 blood dynamics, 94 blow-up, 55 blown-up, 55 blown-up characteristics of fluid, 95 blown-up effect, 145 blown-up evolution, 88 blown-up of model, 219 blown-up solution, 55 blown-up system, 11 blown-up theory, 54 Bohr principle, 87 Bohr's Principle of Correspondence, 167 Bohr, N., 87 Boltzmann constant, 137 Boltzmann formula, 137 Boltzmann, L., 137 Book of Change, 8 Born, M., 158, 160 boundary condition, 55 bounded function, 23 bounded movement, 60 Boussinesq equation, 111 branching chain reaction, 201 broken space, 7 broken time, 7 broken wave, 86 broken wave length, 86

Index broken zone, 99 Burgers equation, 107, 113 butterfly effect, 129 Carnot, S., 136 carriers of chains, 201 Cartesian coordinate space, 184 castle-like environment, 7 catastrophe by folding, 153 catastrophe theory, 93, 153 Cauchy problem, 46 Cauchy, A. L., 46 causal relation, 9 causal relationship, 164 celestial body, 188 celestial evolution, 190 celestial movement, 7, 18 chain reaction, 201 chaos in laser beam, 120 chaos technology, 120 chaos theory, 90 chaotic fluid, 120 chaotic indeterminacy, 6 characteristic curve, 48, 70 characteristic line integral, 129 characteristic of flow, 84 characteristics of laser, 169 Charney, J., 93 Charneyis filtering model, 93 chemical potential, 147 chemical process, 193 chemical reaction system, 147 chemistry, 121 chlorine gas, 201 circulation, 76 civil engineering, 121 classical mechanics, 9 classifications of materials' structures, 224 Clausius, R., 136 cloud, 53 cloud charts of Venus, 117 comparison analysis, 220 competition principle, 229

313

Index complete ionization, 116 complex function, 71 complexity, 10 computation machine, 92 computational instability, 6, 49, 93 computational scheme, 12 computational uncertainty, 89 computer simulation, 11 computer technology, 5 concept of continuity, 20, 26, 229 concept of fields, 177 concept of fluids, 91 concept of stability, 34, 47 condition of unity, 160 conjugate variable, 179 conservation of energy, 136 constraint equation, 182 continuity system, 120 continuous field, 7 continuous function, 26 continuous media, 94 continuous particle, 12, 182 continuous string, 183 continuous zone, 100 control parameter, 153 convergency of fluids, 53 convergent-divergent state, 163 converging movement, 95 cool flame, 196 cool temperature space, 115 Copernicus, N., 2 Coriolis force, 77, 101 covariance derivative of tensors, 185 curvature of light rays, 183 curvature scale, 185 curved space, 189 curved time and space, 183 curved time-space, 184 curved variable, 189 cyclone, 80 D'Alembert operator, 187 Dalton, J., 7 Dauglas, A., 6

de Broglie's relation, 160 de Broglie, L., 158 dead heat state, 138 death coefficient, 212 death of continuous growth, 214 death rate, 130, 213 defective non-transitional blown-up, 56 definite integral, 30 degree of advancement, 194 degree of chaos, 137 degree of orderless, 137 degree of reaction, 194 degree of richness, 137 degree of saturation, 230 demand and supply, 224 demand curve, 226 demand-supply equilibrium, 226 density diffusion, 147 density pressure, 79, 101 dependent variable, 22 derivative of functions, 28 Descartes, R., 17 deterministic random motion, 46 deterministic structural evolution, 46 deviation force, 89 difference equation, 132 differentiable function, 29 differential, 30 differential analysis, 21 differential equation, 31 diffusion, 140 diffusion flow, 147 discontinuity, 27 discontinuity of nonlinearity, 153 discontinuous reaction phenomenon, 193 discontinuous singularity, 217 discontinuous zone, 94 dispersive wave motion, 85, 86 dissipative structure, 93 dissipative term, 102, 127 distance distribution, 168 disturbance, 94

314

divergence equation, 95 divergence of fluids, 53 divergence vorticity theory, 93 diverging movement, 95 double spiral pattern, 130 double spiral structure, 129 duality of continuity and discontinuity, 182 duality of rotation, 83 dynamic equilibrium state, 229 dynamic meteorology, 94 dynamic point, 71 dynamic system, 5 dynamic transformation, 72, 74 dynamics, 18 dynamics of chemical reactions, 193 earthly west wind, 116 eastern civilization, 7 Eastern Mystery, 88 ecological system, 211 ecology, 212 economic crises, 53 economic sector, 228 economics of free markets, 229 eddy current, 84 eddy effect, 8, 76, 101 eddy irregularity, 89 eddy motion, 8, 81, 84 eddy source, 79 effect of elastic pressure, 101, 127 effect of floatation, 115 effective pump excitation, 174 effects of elastic pressure, 79, 178 Einstein, Albert, 3, 157 elasticity coefficient, 68 electric charge, 181 electric conduction, 140 electro-magnetic reaction, 178 electrodynamics, 187 electromagnetic equation, 177 electromagnetic field, 177 electromagnetic induction, 177 electronic diffraction, 161

Index electronic flow, 161 elementary chemical reaction, 194 elementary evolution equation, 213 elementary physical reaction, 194 elementary reaction, 194 elliptic function, 68 emission of radiation, 170 energy characteristic wave function, 168 energy density, 169 energy level, 168 energy transformation, 88 energy transportation of fields, 178 Engels, 222 entirely asymptotic stability, 35 entropy, 136 entropy H, 137 entropy of a system, 137 entropy transportation quantity, 146 epistemological mistake, 20 equal quantitative effect, 87, 167 equal quantitative movement, 87 equation determinism, 10 equation of continuity, 160 equator, 106 equilibrium condition, 138 equilibrium entropy equation, 147 equilibrium singularity, 37 equilibrium solution, 34 equilibrium state, 34, 59, 227 equilibrium wave theory, 93 error spiral, 130 error-value calculation, 46 error-value computation, 129 error-value spiral, 61 essence of irregularity, 12 Euclidean geometry, 17 Euclidean space, 184 Euler equation, 77 Euler language, 95 European Renaissance, 1 even function, 23 even material, 86 evolution, 22

Index evolution problem, 38 evolution science, 12 evolutionary continuity, 96 evolutionary essence of nonlinearity, 40 evolutionary transition, 47 evolutions in economics, 223 excitation condition, 173 excitation radiation, 169 excited emission, 173 excited photon, 170 excited state, 169 existence of chaos, 51 expansion coefficient, 124 explosive evolutionary growth, 152 explosive moving forward, 212 exponential model, 214 external control, 201 external environment, 138 external forcing, 129 external pump, 169 factor field, 94 Faraday, M., 177 fast time scale, 149 fast variable, 149 fatalism, 9 Feigenbaum, M. J., 121 Ferrel circulation, 105 Fick's Law, 120, 140 field manifold, 178 field of eddies, 80 field strength, 174 field theory, 177 figurative structure, 8, 11, 87, 90 figurative writing language, 8 filtering model, 93 first law of thermodynamics, 136 first order approximation, 186 first push, 18, 189 first push system, 230 flow function, 97 fluid convection, 121 fluid dynamics, 92

315

fluid dynamics of geophysics, 103 fluid mechanics, 77, 94 fluid microscopic particle, 159 fluid motion, 10 fluid movement, 92 fluid science, 94 fluids' resistance, 101 folding catastrophe, 153 food chain, 211 Fouier Law, 120 four-dimensional time-space, 180 Fourier Law, 140 Fourier series, 48 Fourier, J. B. J., 48 fractal, 120 free competition, 224 free particle, 159 free society, 53 free space, 187 front, 53 function, 21 function relation, 22 fundamental theorem of algebra, 64 galaxy, 80 Galileo, 18 gaseous chemical reaction, 195 gauge field, 177 gauge field's unification, 179 gauge potential, 181 gauge theory, 178 gauge transformation, 180 gem rod, 174 general evolution system, 66 general relativity theory, 82, 183 generalized observ-control, 10 geocentric theory, 2 geodynamics of fluids, 94 geometric model, 214 Glashow, S. L., 178 Goldstone's mechanism, 179 gradient of velocity field, 147 gradient operator, 100 gradient stirring force, 101, 127

316

graphical representation, 22 gravitation, 81, 188 gravitational acceleration, 77 gravitational constant, 186 gravitational external wave, 93 gravitational field, 138, 177 gravitational field equation, 186 gravitational internal wave, 93 gravitational wave, 183 Greek Civilization, 1 group, 125 group speed, 85 growth coefficient, 212 growth rate of production, 230 Guldberg, 194 Hadly Circulation, 105 Haken, H., 5, 148 Hamilton system, 121 harmonic expansion, 107 harmonic wave, 178 Hawking, S. W., 11 head-kinetic energy, 88 heat conduction, 139 heat energy, 136 heat sphere, 115 heat-conduction coefficient, 124 heat-kinetic energy transformation, 102 heat-kinetic force, 80 heaven-quake, 117 heavenly rain, 117 Heinsenberg's Principle of Uncertainty, 166 Heisenberg's principle of inaccurate measurement, 158 Heisenberg's Principle of Uncertainty, 158 Heisenberg, W., 158 heliocentric theory, 2 Henon, 121 Hermitian operator, 165 Higgs field, 179 Higgs field variable, 179

Index Higgs' mechanism, 179 high speed object, 177 Hooke's law, 19 Hooke, R., 19 horizontal eddy motion, 105 Huygens, C , 3 hydrogen atom, 157 hydrogen gas, 201 hyperbolic equation, 70 ideal demand-supply equilibrium, 228 imaginary mass, 180 implicit transformation, 71 independent variable, 22 indeterminate equation, 133 indistinguishable number, 89 inequal quantitative movement, 87 infinity, 25 information source, 137 information theory, 137 informational topology, 13 initial condition, 55 initial density, 195 initial diverging system, 96 initial field, 59 initially convergent state, 96 instability, 47 instability of stock market, 34 instable moduli, 149 instantaneous dynamics, 230 instantaneous equilibrium, 230 integration, 30 integration constant, 30 integration scheme, 61 interior component, 180 interior space, 180 internal entropy, 146 internal randomness, 6 introduction of currency, 222 invariant fluid form, 150 invisible organization, 88 ionized water molecule, 116 irregular chaotic phenomenon, 175 irregular information, 9

Index irregular singular point, 37 irreversible movement, 139 irreversible process, 137, 141 irreversible system, 137 irreversible thermo-electric phenomenon, 139 isolated system, 136 isotopic spinning space, 181 iteration formula, 133 Jacobi operator, 97 jet stream, 53 KdV equation, 107, 110 Kelvin, L., 3, 136 Kepler, J., 18 Kiepel school, 93 Kiepel's filtering model, 93 kinematic equation, 33 kinematic equation of assemblages, 163 kinematics, 18 kinetic force, 141 king of all theories, 94 Korteweg and de Vries, 107 Kronecker symbol, 185 Kuchemann, 80 lackness of information, 137 Lagrange density, 179, 181 Lagrange equation, 180 Lagrange function, 180 Lagrange's energy criteria, 33 Lagrange's stability, 36 Lagrange, J. L., 33 land-ocean breeze, 78 Lao Tzu, 8 Lao Tzu's teaching, 9 Laplace, 33 Laplace's fatalism, 46 Laplace's initial value determinism, 8 Laplace, P. S., 8 laser, 169 laser evolution equation, 152

317

laser evolution model, 171 laser field equation, 169, 174 laser process, 170 laser threshold condition, 171 law of conservation of informational infrastructure, 190 law of floatation, 92 law of mass actions, 194 law of universal gravitation, 19 laws of mechanics, 18 level of saturation, 227 life evolution, 53 life span of chain carrier, 204 life span of photon, 170 light amplifier, 169 light quantum, 157 light ray spectrum, 158 limit, 24 Lin, Yi, 190 linear analysis, 70 linear deterministic movement, 126 linear dispersive wave, 86 linear dissipation, 129 linear evolution, 45 linear field, 181 linear form, 127 linear group, 126 linear heating process, 129 linear intensity of supply, 225 linear non-equilibrium thermodynamics, 139 linear push, 189 linear pushing force, 129 linear region, 148 linear transformation, 125 linear transformation group, 126 linearization, 47 linearized stability, 151 Lipschitz condition, 32 liquid reaction process, 196 local approximation, 102 local evolution, 220 local existence theorem, 32 local field, 146

318

local gravitational field, 184 local non-inertial system, 185 local strengthening of vorticity, 100 local wave frequency, 85 local wave number, 85 local weakening of vorticity, 100 logic of continuity, 17 logistic equation, 212 logistic model, 212 logistic population model, 130 long-lasting economic growth, 231 longitudinal circular circulation, 105 Lorentz's metric, 186 Lorenz model, 124 Lorenz's chaos, 46, 134 Lorenz, E. N., 5, 121 lower troposphere, 115 Lyapunov exponent, 50 Lyapunov stability, 36 Lyapunov stability theory, 33 Mach, E., 3 machine computation, 134 macro-world, 120 macroscopic movement, 120 magma flow, 115 magnetic fluid mechanics, 94 manifold, 178 market place, 224 market share, 229 massive production, 228 materialism of force, 82 materialistic evolutions of the universe, 117 materialistic time and space, 190 materials' movement, 11 mathematical analysis, 17, 222 mathematical economics, 222 mathematical modeling, 11, 222 mathematical nonlinearity, 223 mathematical physics problem, 21 mathematical quantification, 10 mathematical simulation, 222 Maxwell, 177

Index Mayan, 1 measure of energy, 137 measure of uncertainty, 137 measurement uncertainty, 87 mechanical system, 189 mechanical work, 137 media, 12 merchandise exchange, 222 Mercury's perihelion, 183 meteorology, 121 method of estimation, 68 method of linearization, 187 metric, 184 Michelson-Morley experiment, 3 micro-dynamic quantum theory, 46 micro-world, 120 microcosmic particle, 53 microscopic particle, 159 microscopic quantitative effect, 164 microscopic unit, 162 minimum entropy principle, 139 Minkowski space, 184 model of model, 128 modern theoretical physics, 177 modern topology, 12 monochromaticity, 169 monotonic function, 23 morphological change, 83 morphological characteristics, 168 morphology function, 159 movement, 22 movement intensity, 128 multi-valued function, 23 multiple bifurcation, 135 multiplicity, 10 multiplicity of formal quantity, 224 multiplicity of structures, 224 mutation moment, 190 mutational moment, 101 mutual reacting field, 138 mutual reacting wave, 86 Mutual reaction, 178 mutual reaction, 95 mutual restriction, 223

Index mystery of nonlinearity, 4, 38, 78 narrow observ-control, 71 narrow relativity theory, 183 Natural Dialectics, 11, 222 natural economics, 222 Navier, L. M. H., 92 Navier-Stokes equation, 100 Navier-Stokes system, 20, 92 nebula, 80 nebular structure, 87 negative vorticity, 98 Neptune, 117 new material, 138 Newton's Law of Viscosity, 140 Newton's second law, 81 Newton, I., 2 Newtonian gravitation, 186 Newtonian mechanics, 18, 81 Newtonian physics, 18 non-Abelian gauge field, 179 non-autonomic system, 32 non-compressible even fluid, 107 non-dimensional time, 204 non-equilibrium condition, 138 non-equilibrium mutual structural reaction, 230 non-equilibrium thermodynamics, 148 non-Euclidean geometry, 185 non-isolated system, 138 non-structural quantity, 87 non-transitional blown-up, 56, 214 non-uniform eddy motion, 102 non-uniform vortical vectority, 87 non-uniformity of eddies, 163 non-viscous fluid, 20, 92 noncommutative operator, 166 nonlinear characteristics, 93 nonlinear characteristics of movement, 167 nonlinear chemical dynamics, 193 nonlinear deterministic movement, 126 nonlinear differential equation, 32

319 nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear nonlinear

dispersive wave, 86 elasticity model, 68 evolution, 20, 45 evolution model, 54 evolution problem, 4, 6 evolutionary field, 179 feedback effect, 146 field, 181 heating process, 129 intensityof supply, 225 irreversible process, 144 mutual reaction, 211 mystery, 38 non-equilibrium

thermodynamics, 139 nonlinear numerical computation, 136 nonlinear problem, 5 nonlinear region, 148 nonlinear scienc, 38 nonlinear science, 5, 91 nonlinear stirring force, 188 nonlinear wave, 86 nonlinear whole movement, 164 nonlinearity intensity, 128 nonperiodic flow, 6, 121 nonstructural quantity, 11 normal branch solution, 56 normal non-transitional blown-up, 56 nth degree polynomial, 64 nuclear explosion, 201 numerical approximation, 46 numerical integration, 49 numerical iteration, 134 numerical scheme, 46 numerical weather forecast, 93 objective evolutions of the universe, 117 oceanic behavior, 92 oceanic dynamics, 94 odd function, 23 Ohm Law, 120 one-dimensional advection equation, 69

320

one-dimensional iteration, 132 Onsager's reciprocity relation, 141 Onsager, L., 139 open system, 211 opinion of eddy motion, 83 opinion of wave motions, 83 optics, 121 orderless state, 145 orderly nature, 138 orderly structure, 138 ordinary differential equation, 31 oscillating singular point, 45 OuYang, Shoucheng, 190 over-supplied market, 229 p- plane, 78 panrelativity theory, 87 pansystems theory, 11 parabolic equation, 70 partial differential equation, 20, 31 partial fraction, 62 particle mechanics, 10, 17, 158 particle's continuity, 85 Pascal, 92 Pascal Law, 92 periodic transitional blown-up, 56 phase plane, 35 phase space variable, 128 phenomenon of break-offs, 85 phenomenon of viscosity, 140 philosophical epistemology, 158 photochemical reaction, 115 photon production rate, 169 physical mechanism, 223 physical morphology, 86 piecewise defined function, 22 Planck constant, 159 Planck, M., 157 Planck, M. K. E. L., 3 plane wave, 160 planet-like atomic model, 157 planetary detector, 117 planetary fluid dynamics, 94 planetary west wind circulation, 117

Index plasma fluid mechanics, 94 Poincare, J. H., 3, 33 point of equilibrium, 138 polar eddy, 80 population, 53 population density, 215 population evolution, 212 population evolution equation, 211 population evolution model, 130 population model, 130, 213 population size, 130 positive vorticity, 98 potential field, 159 potential function, 78, 102 practical significance, 136 Prandtl number, 124 precessional angle, 183 prediction science, 58 pressure gradient force, 77, 101 price, 224 price evolution model, 226 price stability, 226 Prigogine's principle, 148 Prigogine's theory, 94, 148 principle of equal quantitative effects, 184 principle of increasing entropy, 136, 137 probabilistic flow, 161 probabilistic wave, 158 probabilistic world, 6 probability density, 160 problem of fluid motion, 5 problem of stability, 33 process of absorbing photons, 173 production, 225 production of laser beam, 170 productional level, 229 profit elasticity, 229 program storage, 87 projection mapping, 72 propagation of light speed, 183 propagation speed, 129 Ptolemy, C., 2

Index pump excitation source, 171 pyroelectric effect, 141 quantified comparability, 89 quantified method, 120 quantitative comparability, 9, 12 quantitative determinacy, 10 quantitative economics, 222 quantitative formal analysis, 223 quantitative irregularity, 10 quantitative logic thinking, 189 quantitative science, 90 quantitative time and space, 188 quantum, 157 quantum analysis, 157 quantum assemblage, 159 quantum behavior, 168 quantum chaos, 120, 167 quantum chaos theory, 168 quantum effect, 159 quantum mechanics, 158 quantum system, 168 quantum theory, 3 quantum-kind quantitative effect, 164 quasi-equal quantity, 129 quasi-linear equation, 96 quasi-linear hyperbolic equation, 85 quatnum mechanics, 121 radiation of light, 157 rain, 53 rainfall, 99 random event, 137 range of elasticity, 69 rate of change, 29 rate of electric conduction, 141 rate of loss, 170 Rayleigh number, 124 reaction process, 194 reaction rate, 147, 193 reaction system, 194 realistic time and space, 189 reciprocating current, 88 reciprocating movement, 83

321

reciprocity relation, 139, 165 reduction of order, 113 regular singular point, 37 regularized computational scheme, 89 regularized mathematical operation, 46 regularized mathematical quantification, 89 relativity, 180 relativity principle, 87 Ren, Zhenqiu, 190 repeller, 153 research of dynamics, 90 resonant cavity loss, 174 retarded potential, 187 reversal change, 47 reversal transition, 217 reversal weather change, 75 reversible evolution, 139 reversible system, 137 Riccati equation, 19, 42 Riccati variable transformation, 58 Ricci tensor, 185 Richardson, 92 Richardson, L. F., 6 ridge, 99 Riemann ball, 71 Riemann integration, 46 Riemann, G. F . B., 46 Riemann-Christoffel tensor, 185 role of feedback, 148 Rossby wave, 102 Rossby's long wave formula, 103 Rossby's school, 93 Rossby, C. G., 102 rotating movement, 95 rotating vortical chain, 163 rounding error, 134 Ruelle, 121 Russby's long wave, 90 Rutherford, E., 157 Salam, A., 178 Saltzman's kinematic equation, 121

322

Saltzman's model, 127 Saltzman, B., 78 saturated market, 229 saturation constant, 225 saturation level, 230 saturation parameter, 230 scalar quantity field, 177 scaleness, 121 Schlog, 198 Schlog reaction model, 198 science of God, 189 scientificality, 90 second laws of thermodynamics, 136 second order tensor, 184 second stir, 82 self-organization, 148 semi-broken state, 64 semi-classical behavior, 168 semi-classical limit, 168 sensitivity, 47, 129 separating variables without expansion, 69 series expansion, 46 severe singular point, 37 Shannon, C. E., 137 shock, 114 shock solution, 113 short wave, 86 Shrodinger equation, 159 Shrodinger, E., 158 single-valued function, 23 singular branch solution, 56 singular point, 36 slaving effect, 230 slaving principle, 149 slaving system, 18 slow kinematic equation, 150 slow time scale, 149 slow variable, 149 small vibration, 94 Smith, Adam, 222 smooth shock, 114 smoothing effect, 49 smoothing scheme, 61

Index solar radiation, 115 solar system, 80 solenoid term, 78 solitary wave, 107, 112 Soret effect, 141 Soros, George, 221 spatial dynamics, 71 spatial transformation, 74 spectral expansion, 49, 128 spectral method, 94 spectral truncation, 121 speed of light, 186 spill, 129 spindrift, 53 spinning current, 84 spinning field, 163 spinning universe, 88 spraying current, 88 stability analysis, 227 stability of chemical reactions, 33 stability of planets, 33 stability of populational changes, 34 stability of stock market, 34 stability theory, 93 stabilization, 47 stable behavior, 34 stable economic development, 230 stable moduli, 149 stable time series, 9 standing wave solution, 187 state function, 31 statics, 18 stationary state, 138, 168 statistical analysis, 70 statistical characteristics, 168 statistical method, 9 statistical physics, 137 statistics, 94 step function, 153 stirring force, 89, 188 stirring gradient force, 79 stochastic system, 9 stochastic uncertainty, 6 stock market price, 53

323

Index Stokes, G. G., 92 straight chain reaction, 201 strange attractor, 121 stratification intensity, 128 stratosphere, 106 stress-energy tensor, 186 strong reaction, 178 structural analysis, 12 structural balance, 231 structural determinacy, 11, 164 structural determinism, 168 structural evolution, 81 structural mechanism, 223 structural prediction, 80 structural unevenness, 188 structuresof vectority, 12 Sturm-Lioville Theorem, 49, 128 styrene polyreaction, 201 SU(2) gauge field, 181 Su-Shi Principle, 87 super distant effect, 177 superstring theory, 182 supply curve, 226 symmetric transitional blown-up, 56 synergetics, 93, 148 system of calculus, 18 systems movement, 120 systems of differential equations, 31 tangent space, 51 Tao of nature, 54 Taylor series, 108 temperature field, 105 temperature ratio heat flow, 147 theoretical economics, 222 theory of dead heat, 138 theory of dynamic meteorology, 93 theory of elasticity, 18 theory of entropies, 138 theory of fluid statics, 92 theory of fluids, 91 theory of gravitation, 183 thermodynamic equilibrium, 148 thermodynamic flow, 141

thermodynamic force, 141 thermodynamic potential, 147 thermodynamics, 136 thermomagnetic effect, 141 thinking logic of continuity, 189 thinking logic of linearity, 46 thinking logic of particles, 169 thinking logic on discontinuity, 190 Thorn, R., 5, 153 Thomson, 139 three-body problem, 5, 38 time evolutionary characteristics, 168 time-space distribution, 81 time-space space, 182 topographic leeward wave, 90 total density, 147 total entropy flow, 146 traditional formal analysis, 85 trajectory, 49 transformation of energy, 177 transformational characteristics of fluid, 95 transformational invariance, 181 transitional blown-up, 56 transitional change, 40, 59 transitional weather change, 75 transitional zone, 99 transportation process, 141 transportation quantity, 141 traveling wave, 48 travelling wave, 108 trend following company, 229 tropical region, 105 troposphere, 105, 115 trough, 99 truncated spectral expansion, 128 truncation error, 134 twisting force, 80 two-body dynamic problem, 19 ultra-low temperature, 12 unbounded motion, 60 uncertainty model, 87 unequal-quantitative causality, 167

324

uneven density, 80, 162 uneven material, 86 uneven rotation, 190 uneven singular field, 182 uneven space, 81 uneven time, 81 unevenness, 137 unevenness of time and space, 189 'ifnidirectionality, 84 unified field theory, 178 universal constant, 121 universal gravitation, 188 universality, 188 universe's evolution, 82 unstable development, 34 upper troposphere, 115 vectority analysis, 81 Venus, 117 vertical stratification, 107 vertical vorticity, 103 viscosity, 140 viscosity coefficient, 102, 124 viscosity force, 100 viscosity intensity, 147 viscous fluid, 92 viscous fluid motion, 20 von Neumann's Principle, 87 vortex flow, 53 vortical advection qualitative analysis, 98 vortical vectority, 88 vorticity, 97 vorticity equation, 95, 103 vorticity of materials, 223 Waage, 194 water molecule, 116 wave function, 158, 159 wave motion, 83 wave motion equation, 187 wave motions of electrons, 161 wave number, 128 wave-particle duality, 158, 160

Index weak nonlinear equation, 107 weak reaction, 178 weak singular point, 37 weather analysis, 95 weather change, 92 weather evolutions, 53 weather forecasting, 6 weather map, 99 weather predictability, 12 weather situation forecast, 93 Weinberg, S., 178 well posed problem, 36 well-posedness, 31 west wind circulation, 115 western civilization, 7 western wind circulation, 105 whole evolution, 26, 53 whole evolution of ecological systems, 220 whole evolution of systems, 133 whole evolutionary characteristics, 212 whole solution, 46 wind field, 94 Wu, Xuemou, 11 Yang-Mills field, 179 Yang-Mills gauge field, 181

BEYOND HOIISTRUCTUIinL QUHHTlTflTIVE flKHLYSIS BLOWN-UPS, SPINNING CURRENTS AND MODERN SCIENCE

This book summarizes the main scientific achievements of the blownup theory of evolution science, which was first seen in published form in 1994. It explores — using the viewpoint and methodology of the blown-up theory — possible generalizations of Newtonian particle mechanics a n d computational schemes, developed on Newton's a n d Leibniz's calculus, as well as the scientific systems a n d the corresponding epistemological propositions, introduced and polished t f t n the past three hundred years. briefly explain the fundamental concepts, then analyze .pics and problems of the current, active research widely in the natural sciences. Along the lines of the analyses, ce new points of view and the corresponding methods. • point out that the blown-up theory originated from the mutual slavings of materials' structures so that "numbers are ..nsformed into forms".This discovery reveals that nonlinearity is not problem solvable in the first-push system, and that the materials' operty of rotation is not only an epistemology but also a ifiethodology.The authors then point to the fact that nonlinearity is ' second stir of mutual slavings of materials.

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