Applied chaos theory: a paradigm for complexity / 0121559408

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APPLIED CHAOS THEORY A PARADIGM FOR COMPLEXITY

APPLIED CHAOS THEORY A PARADIGlv\ FOR COMPLEXITY

A. B. CA1\1BEL ~

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CONTENTS

Lotka-\ oltt rra I:quation:-.:\ote:-- and Rt·ft>JT net·:-

7

The Discrete Logistic Equation 1n trod uction Th e Discrete Logistic Curve The ~lorpholog) o f the Disn·t> te Rturn ~lap:-Bifurcation Diagr.un Feigenbaum Cni,·crsal ;\umbe rs ~1 ul tivariable Equations ~ ots and Refe renc es

8

1()()

Lo~i:--til

124 l~-4

The Different Personalities of Entropy In tTorluction Is F.ntrnpy for Rt>al? Wh y ~lucid\' the Waters with Entropy? \lacroscopic Entropy Statistical Entropy Dynamic En tropit s :'-.'ote s and Re feren ces

9

Equation

107 10!1 111 1 1~ 117 121

127 129 I:) (I

1~~:~ l-41 1:'11

157

Dimensions and Scaling In trorluction Dim e nsiuns Ha us do rtf -Bes inwi tc l1 D i1nen si01 1 E. mlwrlding Dimension Scaling :'-." otes ,m d Refe rences

1G1 16-4 1():-J

167 1{)9

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11

CONTENTS

X

11 The Diagnostics and Control of Chaos Introduction Time Series Time Series Analysis Log-Normal Distribution and 1/f Noise Ising Model Chaos Diagnostics Chaos Control Start Your Own Chaos Laboratory Notes and References

193 195 199 204 207 208 215 217 220

12 Discussion Topics

223

Index of Names

235

Subject Index

241

PREFACE

Complexity is ubiquitous. It is in nature as well as in artifice. It occurs in large and in small systems. It can be tangible or intangible. To be aware of the existence of complexity can be like feeling a "presence" that virtually defies description. As yet, there is no agreed-upon explicit definition of complexity, although there are various operational descriptions. How much we frail human beings are meant to understand about complexity is nebulous. The task is both awesome and formidable. Indeed, it may be far too presumptuous to insist that we can explain everything. Perhaps all one should hope for is embodied within a statement by the guru of the art-;, sciences, humanities, and governance, the Science Advisor to Presidents Kennedy and Johnson, and President Emeritus of M. I. T. Jerome B. Wiesner: Some problems arc just too complicated for rational, logical solutions. Tlwv admit insight.-;, nut answers.

It is in this spirit that I have \\Tittcn this buok. I haw· attcrnptt~d to explore the potential applications of chaos theory to reveal "insights" into the xi

PREFACE

XII

structure and the dynamics of complex systems. Like any other methodology, chaos theory has its limitations, and is still in its formative stages. Accordingly, I have stepped outside nonlinear dynamics per se, and have included nonequilibrium thermodynamics, information theory, and fractal geometry. This is natural because complexity transcends the boundaries of traditional disciplines. I make no claim to originality. I have attempted to emphasize applications and have shied away from both erudite mathematical discussions and details of the mathematical elegance. There are outstanding books that focus on the mathematical process. I subscribe to the school of thought that a person who is familiar with a subject can learn it better. Accordingly, I have been more concerned with the potential applications, meanings, and limitations of equations. I consider this important in dealing with an interdisciplinary subject. By definition, anything that is applied and interdisciplinary loses its purity and becomes contaminated. This should not be deplored, because complexity itself is not pure; it is replete with paradoxes. I have presented the material at an introductory level so that a broader audience can take advantage of it. However, I have provided extensive references to assist the peripatetic reader in locating the original sources. I do not insist that chaos theory is the only vehicle through which light is shed on complexity. There are other approaches, such as catastrophe theory, cellular automata, or synergetics, that might be equally efficacious. Chaos theory, together with its compatriots that I have included, lends itself to the study of complexity because nonequilibrium, nonlinearity, and unpredictability are major characteristics of complex systems. 1 Professor John Guckenheimer of Cornell University, who has contributed so manifestly to the modern formulation of nonlinear dynamics, has rightfully questioned whether chaos is a science and has articulated its strengths and limitations much better than I can. On the other hand, there is much truth in the 1986 statement by Sir James Lighthill, 2 who occupied the Lucasian chair in mathematics at Cambridge University that Isaac Newton once held. Speaking before the Royal Society on predictability, he stated: ... the enthusiasm of our forebears for the man·elous achievements of Newtonian mechanics led them to make generalizations in this area of

1 ')

Guckenheimcr,J. (1991). "Chaos: Science or r\on-sciencc?"' Sonlinrar SrimrP Today, 1 (2), 6-H.

- Lighthill, J. (19R6). "The Re< ently Rt.TO!o{nizcd Failure of Predictabilitv in :\'ewtonian lhnamics," Pmr. Huyal Sor. of !~on dun . A407, :15-.10.

PREFACE

XIII

predi< t.tbilit) wh it h, indct·d, WL m.ty hmt · gt'Ilt'rall) te nckd to bdit·Vt' hdnn 1960, but which WC' now ft'('ognize were bl sc.

Thcjury is still out as to whether or not chaos theory will provide solutions to complex systems. It is conceivable that a new mathematics will be developed to e xplain complexity. as has occurred in the past to describe other facades of science. Examples include the formulation of calculus by Newton and Lt:ibniz, the development of vector analysis by Josiah Willard ( ;ibbs, the discovery ofF erm i-Dirac statistics, and formulation of F eynman diagram techniques. Chaos theory is best practiced when supported by computers and appropriate software. The former can be expensive, and software development is time-consuming. But willing people should not be deprived of opportunities to probe complexity. Accordingly, I searched for modestly priced software programs written for affordable personal computers. The software programs referred to in this volume are for IBM TM or compatible PCs. II owevcr, modestly priced programs for the Macintosh T 'VI are also available . Programs may also be found in the public domain. It was in the same spirit that I did not include color plates, which require elaborate equipment and would have raised the cost of this volume. I hope that my penury will benefit solo researchers, small groups of scholars, modest institutions, and persons in on e field or another who dare to apply the new paradigm to their traditional vocations. I believe that society would prosper if we leanwd together, unshackled from disciplinary ethnicity. To be able to do that we need a common idiolect. Chaos theory is one such mode of communication. An interdisciplinary approach to chaos theory holstered by nonequilibrium thermodynamics, information tlwoiy, and fractal geometry transcends disciplinary Jines and has been found useful in multif~trious areas. Accordingly, this book is not aimed at any particular disciplinary audience, and no specific prerequisites arc necessary to understand its contents. The examples I have cited emanate from different fields that we are close to as indi\'iduals, rather than being in the domain of specialists. Alsn, I have included historical highlights because I believe that a literate person must be appreciative of the traditions that gan· momentum to the dcn·lopmcnt of the particul;u· bod y of knowledge. It will he evident frPm these remarks that the presentation is not parochial. Such an attitude is IH'< ·essary because the tcntk e nd~ all i~ quiet, and the dynamics arc steady and relatively minor except for maintenance functions. Howe\'er, during working hours there is the hustle and bustle of per~on:-. working and visiting in the building, and the communications within the building and \\ith the outside \\orld, as well as the restaurants and shops, spring into action. T he dynamics of a complex svstem may vary, and t)Vically we encounter this when during tran·l Wt' cross different time 1ones anrl our Tll)rmal biological rhythms lag or ad\'ancc. Depending on the circumstances, d~· nami c stahilit}' may bt> steady, transient, or chaoti c. Structural complexity in itself is an important charactC'ristic. It may be a rgued that an ele phant weighing one ton is more complex than a one-ton rock. As geologists will tell us, not all one-ton rocks are equally complex. In the biologi cal domain, were' tlw extinct trilobites who seem to have had compl e x structures and functions less complex than homo sapiens? \>\'as the 1937 .M ark I computer built at HarYard CniwTsity, which weighed 5 tons, had 500 miles of wiring and 3,304 electromt>chani cal relays, less complex than a present-dar, state-of:the-art desktop compute r that can do so much mort> and a t far greater speed? Is a drug like AZT more complex than peni< illin or aspir i n~ Is it its chemical structure or its kinetics in the borly that makes the difft>rence? It follows that the complexitY of a system must be' considered in tlw light of the surroundings in which tht> system finds itself. Some examples of complex problems that we arc like I) to encounter as we go about our business are traffic flows, weather changes, population dyna111ics, organit.ational beha\ior, shifts in public opinion, urban development and decay. cardiological arrhythmias, epidemics, the operation of the communications and cornputer technologies on which we rely, th e combustion processes in our automobiles, cell differentiation, immunolog\', decision making, the fracture of structures, and turbulence. Quite ob\'iousl~ these arc all rath e r diff(Te nt types ofc,·ents. This is why complexity cannot be neatly packaged into a stanrlard container, but must be dealt with hcuristicall\'. The following could expound and dictate categorically, invoking the authority of the laws ofscience! Whereas Newton had tried to relate the science of the day to the C reator, Laplace had no such interests, 7 but rather wanted to pro\'e that, like clockwork, the universe functions rationally according to the laws of mechanics. In Laplace's deterministic world there would be no uncertainty, no chance, no choice, no freedom, and no free will. Everything would be predetermined. \\"e know from personalexperienn· that this cannot be. How often han· uJwxpenecl minor cn·nts changed our so carefully laid plans? From the scientific Yie"lwint, strit't determinism must he ruled out because mcastin·ments are affe( ted by the presence of the nbsernT. F.Yen tlw !'>o-called nonim·asin· measurenwnts affect the S\stem ,1t least

8

CHAPTER 1: LIVING WITH COMPLEXITY

microscopically. We alsc5 know that the number of particles constituting any 19 system is horrendously large, about 2. 7 x 10 particles per cubic centimeter, so that their coordinates and momenta cannot be specified except statisti9 cally. Even on the scale of the world's population, namely about 5.3 X 10 , a much smaller number, we cannot tell the whereabouts, nor the activities of, individuals. There is always going to be some uncertainty. It should not be inferred that uncertainty and randomness are synonymous. Quite d1e contrary is true. In essence uncertainty is a manifestation of information, or the lack thereof. No undue randomness may be present, but we may still be uncertain in evaluating a situation. Of course, if randomness is involved, the uncertainty increases. Uncertainty also tends to increase as the system under consideration becomes more intricate. The basis of randomness lies in probability theory, while uncertainty is related to information theory.

NONLINEARITY Rarely do dynamic events follow a straight line for an extended period of time; eventually they exhibit nonlinearity. This occurs in different ways, and to provide some formal organizational mechanism it is customary to speak of oscillators, such as pendulums. To be more specific, one differentiates among different types of model oscillators such as the undamped, unforced, linear oscillator, which is the simplest, and the damped, forced, nonlinear oscillator, which is the most complicated. I shall not elaborate further on these, because I want to emphasize a different outlook. The interested reader will find an excellent descriptive comparison of the different types of oscillators in the fine volume by Thompson and Stewart. 8 Complex problems invariably involve nonlinearity. Nonlinear events may be regular or irregular. Figure 3 shows a regular nonlinear trace depicting a sine wave.

Figure 3. Sine wave .

NONLINEARITY

9

Figure 4. RcpresentatiYc trace depicting a sound waYe.

In contrast, Fig. 4 shows a nonlinear curve that is irregular. The above figure shows how complex even one note emanating fi·om a single musical instrument can appear. If something that simple can look so convoluted, it is easy to comprehend why so many real-life situations appear so complex. One can generate all sorts of complex traces with a sound generator. Unfortunately, there are no explicit general solutions to nonlinear mathematical problems. In the past, there was a tendency to deal with nonlinearity by considering su ch problems as aberrations and ignoring them. V\'ith increasing populations, dwindling resources, and rising expectations, we can no longer indulge in this cavalier attitude. \Yc have to face the real-life problems that confront us and learn to deal '~ith them. This is where chaos theory comes in. Among its many applications is its ability to provide insights to nonlinear phenomena that involve random aspecL'>. Consider the discrete logistic equation that we use extensively in Chapter 6. One common form of this equation follows: X

II

+ I

= ([X

II (

I -

X

II )

(I )

Here lxl is a time dependent variable. and lal is a parameter that influences the degree of nonlinearity in this equation. Actually it tells us about the rate at which increases (or decreases) take place. The subscripts denote generations or increments of time. T hus In+ II refers to the generation that lollows the nth generation (e.g., c hild), In+ 21 means the second generation after the nth generation (e.g., grand would be the vintage essay by von K~l.nmin ,9 and the volumes 10 by Beltrami, and by Jordan and Smith. 11 It is not my intent to make contributions to nonlinear mathematics. Rather I wish to explore how we might gain insights into the behavior of nonlinear dynamic phenomena. We shall consider several techniques that have been gaining considerable attcn tion. I shall cite them here only very briefly, leaving their detailed discussion to subsequent chapters. Beca11se these are independent approaches I mention them in no particular priority or chronological order. One approach is due to A.M. LyaptmO\' (1R57-1918), wherein one places the emphasis on understanding the stability of the problem. Admittedly this is more qualitative in nat11re than is an analytical solution, but it is ,·erv useful. Another approach that is proving to be very useful is the nonlinear intcgro-differen tial equation due to A.J. Lotka ( 1880-1949) and\'. Vol terra ( 18li0-l ~HO). For systems that are suspected to he chaotic, the relatively recent disco\'l·ry of the so-called strange attractors oilers exciting new vistas into nonlinear dynamics. Still another recent development that is pro\'ing hclpflrl in the study of complexity is fhlctal geometry. All of these have found wide acceptance lwca11se of the ready a\'ailabilit\' of computers. particularly desktp ·v r l ( .'i ,. ' · :\t·w \. ,r k R.u td. 1 ~ 1 i :1 '1'1 r it •d 1 h t Prtll' ' \ ,•r l.. l n It ~< !...' . .\'r·'• · l a n , g r;neful 10 Lo b :O.kt' 't'll hn •ring mg th l.' re c rt' lll t w m \· a n en un n . :!:~ . Zckhnirh. \'. \.B .. Rul lll.til..in . .- \ .. \ ... unl ~o l..tl lo i i. D. D. l ~ t! 1 But th n h .tH' otht' r d a ~ t Trni b lt• c h .u·.\l t t' ri ~tir' 'lll"h ,1, 111.1."· !t'lll x·r.Hun·. o r t'llt' r~ h-\ t' l. .\n' n f 1 lt' 't' ' .111 ht• co nsi ( en·d ,\, .I U,t' fnlmt•tric to d c, n i ' t the ' ' ' l l ' III IIIHk , ll t, icl t 1. tin n . lltt g n H rh ' \

l ll,!t'.ld r thc ~ I ll ' had to change hy increasing populations ancl tel hnological innm·ati he t amc imoiH·d: about 2t J

CHAPTER 2: META-QUANTIFICATION OF COMPLEXITY

32

Todav

Successful Technology

NGOs

20+ Years Ago

'\. COMMl 'NITIES OF INTERESTS

50+ Years Ago GOVERNMENT

I 00+ Years Ago PRODUCERS

'\. Profits

+

Economic and Security Needs

'\.

+

Environ-

+

mental and Societal Regulations

Environmental and Societal ( ;oncems

USERS

'\. Benefits

Figure 2. Evolution of interorganizational complexity according to F. A. Koomanoff. (Reproduced with the permission of the author.)

years ago it became necessary to be mindful of community interests, while today nongovernmental organizations have become influential in policy formulation and governance. Of course, each level of organization or grouping has various subsets, which have also been proliferating. Technological advancement, too, follows a hierarchical restructuring. This process appears to be guided primarily by the intelligence of technological innovation. The information content, {I), and the material content, {M ), of various manufactured goods were compared by Fritsch, 15 (:am bel, 16 and by(:ambel andFritsch, 17 who suggest that the complexity in the manufacturing sector is related to the I/ M ratio of the products. As demands increase, and resources become scarce, there is a tendency for the {I/ M) ratio to rise, which is shown in Fig. 3. In this industrial phase space one starts with handcrafts in the lower left-hand quadrant, Q-1, where the I/ M ratio is quite low. For a while the manually produced items grow in size, such as the heavy industries (Quadrant-II), but eventually in Quadrant-III an increase in information is required, such as in modern hospitals or high technologyweapon systems, both very complex. Computers with their ever more powerful microchips fall into Quadrant-IV. Of course, there is overlapping of the different levels of industry among developed and developing nations. Further, it is not necessary for any one region to remain in the same quadrant. As the

GEOMETRIC APPROACH

33

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6). The World of Mathematics, New York: Simon and Schuster. Graham, R. L., and Spencer.J. H. (1990). "Ramsey Theory," Srientific Ameriran, 263, (I). 112-117. Code!, K. ( I931). "On Formally Undecidable Propositions of Principia 1\tathematica and Related S)'Stems," (title translated), Monatshefte fiir Mathrmatik und Physik, 38, I73-I98. The uncertain tv principle in quantllm mechanics enunciated by \Verner Heisenberg in 19~7 is ronn·ptually quite different from Codel's undecidability theorem. The uncertaintY principle ~ets limits on the accuracy with which tlw positions and momenta uf partirles can be ~pccified. If either is mt"asured accurately, there will he uncertainty in the measurement of the other. The degree of this uncertainty is fixed bv Planck's constant. Chaitin, C.J. (I98~). "C6dcl's Theorem and Information," lntematlonaljoumal of Tht'llrl'licall'hy,ics, 21 (I~), ~141-9:1!.

:1.

tl.

7. H. 9. I 0. II. 12. 13. 14. 15. IG. I7.

18. I9. 20.

21. 2~.

2~~.

24.

25. 26.

27.

28.

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NOTES ANO REFERENCES

:HJ. I.

1·1.

1: • ,a/ ( , 1 , · , Ur 1, "-~"" \ , ., k_ !."" II ofstadtcr, D. R. ( I ~~7~1 ) L utf, I "-'"'"'- lla Book\. Chaitin, C: . .J. ( l!JH ~J . ojJ. Chaitin, ( •. J. 1 l!•HH). "R~mdnnm t · s ~ in \ nt hlllt lit .. .), II J• I m 259 t I I . N l-~ ~~ Turing , .\. t l 1l 17 1. "( In Computab le :\u1nhc rs wi th a n \pph , ui u n to tht' Er•t,.- IH' hlu n " ' Problem," l'ro I. •n d on .\la llu ma/i , a/ Sr., .. -12, ~ . \ 0--:!6 "> . Sint t' lurin g ' ~ nngiu a l work Ill 1 ~1.\7 •i/J el l . \ , ,tlllllll lt' r n t a!1 , 11 r V 1 and l:'n,L,rill .rri n~. :'o: t·w York: \ ':11 1 :'\o str.md Rt inl lt~ ld l • lll l ' ~ \1 1 1 Rm kt l R 1:-( 7 .If, Tool.l, BostOII : lloughton :\lifllill c .).; /.mt'k . \\'. II. I tl'hl ). ('n u !In I I· II\ " /'h1'1in ojlll{orma/lon, Rt" ading. :\L\: \ddison-\\'csl t·l·l'u hli, hi n g l ' Rasband,S. 0: . ( l~l!l!l ) . Chaolir lly t {1 1/1 / r • t•f -'"• •llli lll' I s,- ,,,. I I :\ew\ .. , k jtohll \\ It- .\.: ~ c b 1) f chance and nece ssity daily around us, and we noti< c that dittcrent e ntities are .lfft·cted ditferentlv bv similar circumstances. Thus it is not uncommon for a stable svstem to mon· into an unstable state, whi ch in turn can lead to a new stablt· unler. lt is instructive to differentiate between "stable" and "unstable" equilibrium. If every initial state in phase space is close to equilibriulll and mon·s on to nther sta tes that abo are close to equilibrium, we sa}· that the system is stable. But wh a t docs "dose to equilibrium" mean? After all, systems in tlw equilibrium state may he so in varying degrees. Accordingly, we difkrentiate between e quilibria that are "weakl\' stable" and "strungl} stable." If for anv initial displacement x (0), ,\: (0), we have x (I)

--1

0, .\· (I)

--1

0, while I

--1 oo

ha,·e as.pnplolir \labiliiJ. \\'e shall e11counter this when we di st uss the logistic curYe. \\'c knnw from experience that a flow cannot occur unless there is an imbalance of dridng f(,r ces; there must be a potential differun' or lack of equilibrium. A term related to the equilibrium state is the so-called steady f1ow situation. Fnr the case when mass is conser\'ed, we writ a Liouville-t~-pc master equation as follows:

\\t ·

dp · - + V' · (p V) = 0 dl

(7)

where Ipi is the dcn~ity and IV I is the ba1~·n · ntri c ,·clo< it~, which 1s a weighted an·rage of .tppmpriate refere nce n·locities. fhe interpretation of thi~ equation can ,·ary depending on whether the Eulerian ,·iewpointwhcre the observer is fi xed in space-or the Lagrangian appn•ach-whcrlutio n that \\'C :1rc now i11 is helping its predecessor. the industri:1l n·nlhttion. to sun in· b\' making it mon· product in·. :-\cw technologies :1lso h:n c the good fortune orlwing ;1blc to till new markets. CmH erning the nunH•r,unr o f the engineering p;tr:Hnctcr. the time for conmrerc iali1ation depends grcath· o n how well those in the R&D bhor:ttotT work with the m;lrketing tun c . thercll\· :tttr:1cting constllncrs. Further. the issue depends grcath o n t n ~t i­ tutional i11centin·s and h:1rricrs tLHion:tlh' as well ,ts in tc r n a ti o n,tlh . T h ere is another matter th:tt must he considered. n:mll'h. mcasur in r t h e time fo t collllltt'rt iali1:ltion. \\'here dm·:- one st:lrt ~ For cxamplt'. ho w I n n ~ .lid it t:1kc for ci,·ili;m IIHclcar po\\cr to lwcomc < olllllltTCi:tlucd · ri te fir ... t t · . ~ . nucle:tr reactor f'or the ~cner: Hion of clcct ri, 1•o\\'t' t " -'' o pc tJ k to. 2 . .Jackson, F.. A. (I \lH!I). I't>npl'(tivi'J 011 \'~ynnmi r.> , \'ol. I .. Cambndgc lug-land:

:t I. -,, t) .

7.

H.

\l.

I 0. II. 1!!.

n.

C.tmbridgc l ' ni\. PIe s~. Scvdel, R. ( I!IHH). Fmm },'quilibrium to ( h os. !\:cw \r•tk: J11< >mi,--Tcchnolo¢cal l'rr >hlt'llb," S }'llngrtin - Fmm Jl u ro,rnpir I n 1\1a rro1 w jJi r Ordn, (pp. llH-I9G), E. Frehland, c d. Berlin: :-;pringct -\ 'crlag. :\ote: For stimulating discussions rcgdrding these paramc tt r ~. I am indcbtr •d tufli ng t ~ 1 u ; Ition wi tho u t the fo n sslcrattractor in three-dimensional space. Depending on how long oJH·kts the program run, there will he rnorc or less crowding of I lw tr;~jectorics. Figure 12 shows tlw so-called Poincare sectioJJ or map. This powerful tool is obtained by slicing through the tr;uenories in phase space with a stllfan· having a dimension one less than the phase space. Thus, points divide lines, lines divide surfaces, and so on. Sections can he obtained in the other planes as well. I .ast hut JJot least, we consider the Lorenz equations, which arc continuous and thrc·cl that this is nature's way of ensuring sustainable growth. The Golden Mean

The infinite series or the Fibonacci numbers leads to anothe r interesting observation. \\'e divide sucn·ssivc Fibonan i numbers In· one ,mother, i.e. ,

88

CHAPTER 5: RAPID GROWTH

Figure 3. Photograph of a sunflower seed head. (Copyright (©) by Derek Fell. Reproduced with permission.)

N 11 + 1/N,1' forming a continued fraction expansion. If we evaluate these ratios we find that they converge. For example, 1/1 = 1, 2/ 1 = 2, 3/2 = 1.5, 5/3 = 1.667, R/5 = 1.6, 13/8 = 1.625. 21 I 13 = 1.615, 34/21 = 1.619, 5.')/34 = 1.618, 89/55 = 1.618, 144/89 = 1.618 .... Conversely, we can perform the successive divisions N 11 / }vrn+l, which approach the irrational n umber 4 (.Y:'> - 1)/2 = 0.61RO;~ .... These values are Yery close to the \'alue of the

NOTES AND REFERENCES

89

Fi!,'l.ITC 4. Rectang-le for golden nJe,\11 .

tt'iebra tcd dh,ine ratio, or goldrn mran, !¢1, named afte r Phidi {:xpt:IHiiturl':- wt· rc t{'( !mit ally ineffectin·, nonlinear terms they contain, Eqs. 5a and b h>ive oscillcuorv solution s .ts one might expect from tlw earlier phenomenological discussion. I·.quations 5a and b can be gennalizcd to tlw ca~c of n-specics ( 0 there is exponential growth, while when k i < 0 there is exponential decay. It is the second term on the right side, namely the nonlinear quadratic term, that introduces the interactive complexities. Here {Ni~) is the number of possible binary encounters between species {i) and {j). The competition coefficients {au) describe the rate of the encounters among the species and, depending on the situation, they may be positive, negative, or zero. The latter indicates that while the species do coexist, they do not interact. Under certain circumstances this situation may be modified by appropriate brokerage if interaction is desirable. The Lotka-Volterra equations are applied widely to a host of different problems in biology, economics, neural networks, physics, and even adversarial relations among nations. I shall briefly review two basic models formulated by Beltrami. 16 We start with what he calls the "competition model." Here two populations, {1} and {2}, vie for the same food supply, although it could just as well be two organizations that are after the same resource. Even indirect interaction will interfere with the growth of both populations. One can write:

(7a)

(7b) In these equations ~nax-1 and Nma:x-2 are the carrying capacities for the two populations, respectively, while the constants A1 and A- 2 are indicative of the relative competitiveness of the two populations. The second model of Beltrami is the "combat model." Simply interpreted, combat is a type of competition. The frame work of this model benefits from 17 the early work of F. W. Lanchester, which deals with the dependence of various combat variables on the force strength. For example, losses suffered by one side are proportional to the size of the forces of the other side.

(Sa)

(Sb)

LOTKA-VOL TERRA EQUAl IONS

105

In Eqs. ~band b, all constants arl' !HJsltl\t". \Ill IIllis stgn i~ pl.ll e d 111 tro tll of' each term bet ausc l ombatconstitutes attrition. T h e case of compt'tition Iwn,·een three species has been furmubtcd 1 by ~Ia\ a11d Leonard. tl f"hcY st.trt \dth the quad ra tit alh tlt•nlincar (;;mssl .utka-\ 'oltcrra equation s for {n l competing specie s:

(9) f hn in H> J.:.c I h e S\"lllm etry :tSSUill pti on r1 -= 1'_! = 1:1 = r, and insist t h :t t th e populations affet tone another in thc same manner. Thus a 1:? = a~: 1 = a:11 = a. and u~ 1 = C~ :l:? = a~:1 =B. An ordingl~·.

( lOb )

d.\"1 -a.\'1 - 1-' AX,- - \ .1] rI I = Xdl .

(I Oc)

l =' ing this a pp•·uat has a pattern, one can write the equat ions when then· a1 e even more competing species. The problem, ofnntrse, is haYing reliable in format ion for 1he ,·arious coellicients. The foregoing constitutes a rt'\"icw of the continuous form of the logisti c equation, but i11 no waY exhausts the sul~jerl. T he logistic equation can be particularly useful i11 deali11g with problems where experience is lac king. Ideally. when confronted with a pro hlem one tries to ide11 1ifY the a pprupriate lundanH.'ntal S(ientilic law withi11 the ptii"Yiew of which the problem would he likelY to bll. This is not alwa\·s possible. There m:\\" be 110 applicable fundanwntallaw, or there may simp!~ not he sullicicnt information. In such cases it is useful to look lor a suitable distribution lunctiotl. For example, one might explore a Caussian or 11ormal distribution function and soln· for two monw11ts, or one might try the Poisson function and soln· for the first lltollH'lll. lfthesc attempts do not prme to be useful, the otlwr hand, if two clilfcrt>nt \'altws •ll the indept>tuicnt variable an· kttar T r.wsformations." Joumal o(Stati.1timl J>hysirs, 19, ~ 5 -.1)2_ Rasband S. :\" _ ( 1!1911 ). ojJ. rit. Ben habib. J, and Dav. R. II. (19M2). ",\ Charartcrintion of Erratic D\'nami cs in the (h'erlapping \' its su ccessiVl' stvlization:- from Bach to Schoenllpt of temperature, and this is manifested to us in the form of the length ot the meniscus in a thermometer outfitted with a scale given in degrees Fahrenheit or ( :elsius ( cen tigracle). \Ye cannot place the concept calle d "temper~lturc" into a b(JX f1>r saf{_·kce ping. \\l1ik it is a propcrtv of svstcms, it is sort of ephemeral. This transitory n a ture of t em pcrat u rc dncs not bot her us, because we arc used to tem pcra tu rc changes "·hcth c r it be by listening to a weather person on the news, or hearing the nurse munnur 1\l the plnsician as we hllpe fulh n· r nn·r f1 11111 illness. Clearly. t~ · mpcr~tture is not as tangible as is a pie( c of ro ' k. Let u..; he a lit tic more curious and ask: \\h e n is a hoch 1t·;dh· lh •ttt 'J' r lt ,m another? \nd the .mswer is: \\lwn its ,·onstitucnt p .trtil II s . 111 · m o n · h 1h h h·

130

CHAPTER 8: THE DIFFERENT PERSONAliTIES OF ENTROPY

agitated, namel) when their kinetic energy is higher. But it is impossible to have particles that move about at a fantastic speed wait long enough to have their temperature taken. However, we can determine their energy by simple calculations 7 that tell us that about 10,000°K is equivalent to 1.29 electron volts. Once we accept this pluralistic way of looking at matters we realize that many concepts that we take for gran ted, and ascribe a physical meaning to, are really the fruits of our imagination that we have gotten used to. Bluntly stated we have been "programmed." We could coin a definition for entropy: "Entropy is the ability to reach equilibrium." Another definition of entropy could be: "a measure of chaos." Still another could be: "an indication of transmitting information." This is why entropy, unlike energy, is not conserved. There have been occasions where persons were willing to accept the concept of entropy and ask me which one they should use. My simple answer is: "There is no one best entropy definition. Use the one best suited for your purposes. Also, please don't overlook the possibility that describing certain complex systems may require evaluating several of its entropies, not just one." The connecting link among the different forms of the entropy function is that, in one way or another, they are indicative of dissipative effects, deviation from equilibrium, information, and chaotic behavior. There is no reason why they must look alike as if they had come out of the same mold.

WHY MUDDY THE WATERS WITH ENTROPY? One of my good friends, a colleague I esteem highly for his scholarship, raises his eyebrows whenever I bring entropy into our spirited discussions concerning chaos theory. I know he is not alone, so allow me to share with you, dear reader, the reasons why one should be mindful of entropy. First of all, modern deterministic chaos generally deals with dissipative structures, not conservative systems. These are not reversible, and the entropy serves as a compass to indicate the direction away from the equilibrium state. Second, complex dynamical systems involve uncertainty-namely, incomplete or missing information. Here Shannon's entropy proves to be a powerful measuring stick. Third, complexity involves randomness. Here Boltzmann's statistical entropy is indispensable because it deals with probabilities and is a measure of chaotic conditions. Fourth, complex dynamical systems follow trajectories;R their divergence is measured by the Kolmogorov entropy. Finally, complex systems are open and dissipative, and hence in addition to the internal entropy production, they experience entropy exchanges

WHY MUDDY THE WATERS WITH ENTROPY?

131

with other svstems or their environment. T hi:-. explain~ the < I(Wremo11 to dilfert·nti,lte lwt\HTn the rnulcnrles. In the final analysis 1he second law would he ,·iolated a!icr all. and irre\'ersibility cannot be ignored. Both S;:ilard and Brillouin lwlrwd ushn in the information interpretation of entropy that Claude Shannon perfected. Thus an informational interpretation of entropv was established. Now another era is awakening. This is the geometric interpretation of the entropy stirnulated by the bizarre shapes 1hat strange at tractors ha\'t' . \\'t· shall Ia) the groundwork for this in this chapter, but elaborate on fractal dimensions in a later chapter. In describing the \'arious pllt S( icntili c;dh th a tlong-tcrlll prcdittiCtns < anll (Jt he made.

152

CHAPTER 8: THE DIFFERENT PERSONALITIES OF ENTROPY

Figure 5. Arthur Eddington (U.S. Library of Congress).

It is the second law of thermodynamics that tells us anything about the direction of time, and this accords it a unique place among the laws of physics. Twelfth-century Persian poet and algebraist Omar Khayyam 47 wrote: The Moving Finger writes; and, having writ, Moves on: not all thy Piety nor Wit

DYNAMIC ENTROPIES

153

Shall lure it hack to l an1 cl halt a I .iru . :'\or all thy Tears wash out a Word of it. Referring to the classical and modern

law~

u f phy~u ~ . Eddillgtoll rcnl.ukt·d:

The classical physicist ha~ been using without misgi\·ing .1 s\'stem of laws which do not recognize a directed time; then· is only one law olnature-thc second law ofthennodynamics-which recognit.cs a distinction between past and future ... [it] holds, I think, the supreme position among the bws of :\'ature .... Let us drI) stnns it up well. "Time is nature's way of keeping en·rnhing from happening all .tt on ce. ,\t the time I wrote this senion, a stim11bting \'olumc In· Co\'t'IH'\. a11d I Iighficld~'~ de11 ). up. rit. 10. Dvson, F. .J. (1!171 ) . "Ene rgy in th e l ' nhrrse," Srimtijir A mPriran, 255(3), 'i l-59. 11. Cardwell, D. :-i. L. ( 1971). From \ra/1/o Oaus1w , Itha ca . !\'Y: Co rnell l ' niv. Press. I:!. ~we Stodo!.r, .\. (I !J.t5). Stmm and Ca., T urbin!'>, :--1cw York: P ter Smith. 1:1. \\'ci nlwrg. S. (l!IH~). ThrFir.\ t Thrrr,\linutt'.\ , :\ewYmk: Basil Books. I-t. Shapiro, .\. II. (I ~15 3 ). Th e lrmnm in and Thenn udynamio of Comprn sih/1' Fluid F/nw. :\lew York: The Ronald Pre ss Comp.my. I i . Trihus, i\1. (I ~lfi I). 'll!nm It,, Y

NOTES AND REFERENCES

\Ia/!

·,()_

11. ·,·" · i!J.

,,;,..

H11n,;aricn, 6, 2 1' ') -:~:-t'J . Ri n yi. , \ . ( I !l()O) " On \k rk . \\ ll.l!l-c utan ,tn d t t •.

CHAPTER

9

DIMENSIONS AND SCALING

INTRODUCTION Shape and size arc important whether \\Tare buying clothing or a \ Fr .H tals and -;clf.-;imilal"it\' abound otht·J·wisc innaturc--onr mn1 circnlator\' and h ro n c hi ~1 l s\·stems arc t~-pical. Fi~ure 2 shows the ~trunnrc t)f a ( hincse cabh:1ge leaf.

164

CHAPTER 9: DIMENSIONS AND SCALING

Figure 2. Chinese cabbage leaf.

DIMENSIONS As we have noted, complex systems can exhibit similarity, and they can be scaled. Accordingly, we must establish their dimensions. For purposes of review the types of dimensions that are of primary interest to us are: (a) Euclidean dimensions, {DE}, that we are well familiar with since our early school days; (b) topological dimensions, {DT}, that derive by dividing one set by another. For example, volumes subsume areas, areas subsume lines, and lines subsume points. It follows that as a rule topological dimensions will be larger than their correspondi~g Euclidean dimension; (c) fractal dimensions, {DFI, such as the Hausdorff-Besicovitch {DH-B}, or capacity, {De}, dimensions. In general (Schaffer and Tidd 4 ): (1)

In an epochal paper, Farmer, Ott, and Yorke 5 outlined the dimension of chaotic attractors. They point out that there are three basic types of fractal dimensions. The "metric" dimensions include the HausdorffBesicovitch dimension and the capacity dimension. The metric dimensions of a strange attractor are independent of the frequency with which the attractor is visited. The second class of fractal dimensions is said to be "probabilistic" because it does depend on the frequency with which the attractor is visited. The information dimension that will be defined shortly is also a type of probabilistic dimension because it derives from Shannon's entropy.

HAUSDORFF-BESICOVITCH DIMENSION

165

HAUSDORFF-BESICOVITCH DIMENSION The different aspects of dimensionalitv and scaling make It impc r tht>m from the origin. ( 0, IIi) . -\ltc rn atiYeh·. we can stop the iteration wlwn it reachc-. a '-Pt~ c itierlnumbt' r. The number of itera tions is crucial because it determines th e deta il th a t is ren:-aled. See Fig. 9. The two images in Fig. 9 were gen e rated with the Sintar Suftw,tre program KaleidoScope. 2\l In the uppe r image the number of iterations was only I 00. \\·hile in the lowe r figure the number of iterations was 10.000. The algorithm was the same in both cases. The difff'rcnce in detail is t'\ident.

\\ ,,, ,,--:..''• ',~ ,~,, .~ .... ' --~ ·-, , ,..... 0 -. * ~1'1'\ ' . ·,:._. ' . a,.;~ • ,., ,..··-·-· ' ,,.... ." ,

.... -..... . ~

,, ..\,. ..... "!tt.•', ,, ''II'

Figure 9. l " )lll ;t g l ' ' ,- ,

l J>I'I "

I! < I

•• .• .; . \ \1

~· . •

J. ' .u

I ll

. . .. " "

·~

•< •

•t

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t

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,,

.... .

1.1 1

CHAPTER 10: GAllERY OF MONSTERS

186

Fractals are self-similar; they exhibit scale im·ariance. In other words, when we magnify repeatedly, the various images look the same regardless of size. This characteristic has implications in fields such as image compression, physiology, video-TV animation, and petroleum exploration. Fractals are made up of parts that are scale copies of the \\·hole. In other words, there is replication. This becomes particularly pronounced in fractals such as the Mandelbrot set ·where as one keeps zooming in, clones keep on appearing. However, differentjulia images appear when one gets close to different points along the rim of the l\1andelbrot set. This is demonstrated in Fig. 10, which is a fractal montage. In tum, Fig. ll is a photo montage of the Julia sets at different points inside the buds of the Mandelbrot set. Here the nondescript image at the right side is the julia set at the center of the main bud and corresponds to the point marked!+}. The figure at the top corresponds to the point marked

...

/

..-+~~* ...

Figure 10. The different ctlnfigurations of .Julia sets at different points o n th e rim of the :\landclbrot set. (Images gcnnatcd separeteh with " '· :\1. Scha!Tcr and l'. "'· Tidd' s program and then merged. )

187

figUre 11 P"--' " r·-- ·r · · ~" e f ··' · : " ' \ 1 ... , _,. I ~

+ ,, t\

- ..:: ,.,, t ~ ; r ·e- r r ~ n t

I \ \ 1. _-., ,

,I

ll L t

u

I h_· ,

' ]'

llld . \ lil t l I ,.., e ft b u d ,tn d the

tl
nard dimension on the le\el of complexity of atmospheric ph e nomena.

Chapter 70-Ca//ery of Monsters l) 2) 3) ·1) :)) () ) 7)

l'\ J

Determine the fractal dimension of a point. Determine the fractal dimension ofa straight line. Dtermine the fi·actal dimension o f a planL· surface. lktcnnine the fractal dimension of a cone. Determine the fractal dimension ofvour hand. DiscliSs how it is possible that fractal eli mcnsion s may exceed :>. Determine the fra< tal dimension of an equilateral triangle . \\'hat will the fractal dimension be if the base is rcmo\'ed so tha t it looks like a l'irnnnf1cx? \\'ill the fractal dimension change if the equilateral triangle is '>toncl on its apex? ( :ontintie the steps of the nm Koch snmd1ake in ( haptcr l tl . llow man\' additional steps did yo u manage to dra\\:. \\"hat •dla l 't'

232

CHAPTER 12: DISCUSSION TOPICS

9) 10) I1) 12)

13) 14) 15) 16) 1 7) 18)

19)

20) 21) 22) 23)

~U)

would you expect if you were to go to the limit, i.e., n ~ oo? Which increases faster: the perimeter or the area inside it? What are the elements of the array making up the IFS if the triangle is stood on its base, on one of its sides, and on its apex? Determine the fractal dimensions ofyour initials. Determine the respective IFS arrays for your initials. Develop an alternate form of the Cantor set. Specifically, remove the middle one-fifth instead of the middle one-third. Determine the dimension of this new Cantor set. Develop an alternate von Koch snowflake by adding one-quarter instead of one-third tents. Determine the dimension. What is the fractal dimension of the Peano curve? Explain the structure of the Eiffel Tower in Paris in terms of a three-dimensional Sierpinski triangle. Discuss the fractal dimension of Buckminster Fuller's geodesic stn1ctures. Confirm the dimension of the Sierpinsky triangle given in this chapter by performing the appropriate calculations. What are representative fractal dimensions of the class of molecules called "fullerrene 's," nicknamed "buckyballs," named after the celebrated Buckminster Fuller? A cylindrical rod having a diameter of 1 em is 1 m long. What is its fractal dimension? Next, assume that the rod is bent into an L shape so that one leg is 25 em and the other is 75 em. What would the fractal dimension become? What would be the fractal dimension if the legs were equal in length, i.e., 50 em each? Repeat problem 19 for the case of a bar 1 m long and having a cross section 1 em by 1 em. Repeat problem 14 for the blade of a metallic roll-up measuring tape. What generalizations follow from Problems 14, 15, and 16? Propose some applications of Cantor dust. Is talcum powder an application of Cantor dust? What about flour or powdered sugar? Is there a conflict in applying the H-B dimension to dynamic systems, considering that time is not apparent in the equation that defines the H-B dimension? Discuss how valid it is to apply the H-B dimension in analyzing time series data? In such situations might the Kolmogorov entropy recommend itself because of its dynamic nature? How about the Renyi entropy? Discuss the following statement: "All strange attractors are fractals, but not all fractals are strange attractors."

CHAPTER 12: DISCUSSION TOPICS

2:-l) 26 )

'l .U

~lust

fractals be self-simil.n :Discuss th e appli cations of dl.lus, tr ~hakcspcarc. \\'., 1."> shannon, c .. 146-14!1 Shapero, D .. ~OH :-.hapiro, ;\., I :1b ~haw, (.D., W Sha,,·, R. ~ .. 7i ~hin hrnt, T., ~I :i ~hlt>sin~t'r, \1. F., 17 ~ . ~Ot i , 21.) ~howaltcr,

K., 21 6 Sickrowich, .J. J., H Sicrpinsb, \\'., IHO Simai ka, Y. \1., 171 Simon, II., 2 Smith, 1'., II ~oc r.l!cs, 7 ~okolofl~ D . D., I 6 solomonon·, R. J. ~;, Spano. \I. !.., 21 :i Sparrow. C., {)9 Spt·ncer, H., 7H Stamhkr, 1., 2Iti St;mlq, H. E., 1/ t) ~(('('!!, 1.. .\ ., ti :-.tcngns. I., I :~H Stt·"·;u·t. II. It, H. tiO Stm~.F. \\'.,19ti :-.ugihar.t, (;., 2(H S\\tlt,J B.. 77. ~O·L ~~ :~ Swinnn,ll. Iti7 . ~04, 21 I :-.win n II. 1... I :vd)('hcl v. \' ., ·IH Stil..td, L.. 1 :~2

t.,.

l'akt·tb, 1- . In, /tl I'C'uber, \1. l . I 71 f'lt om a.' . L. , :!!1 fhomp>uu ,.J. \1. T.. ~ . btl I'lwmson , \\'. (Lord 1\.t•hin) . 2H, 1.">1 l'idd, ( . \\·., !12, lti·l, ~~~-IK '~. 1Kt;-Jx7 , 2().1

I orna!lkv, I. ( :., 9:1 fribm, :\1., I :16 fruty, C. I. .. 72, 7:1, 71. I'uk('\ ,.J. \\'.. 200 I'urcutte, D. L.. l!iti, 171 I'uring, . \ . C)f)

u l :cda, \ ., 1>7

v \'alltT,.J., '17 van dt'l' Pol. B.. h ,-, 1an der \lerwc , .\., J:{!-l \'astano,.J. :\., 77, 2(H, 21 .1 \ '< Hl7. ~ n ·~ . \\'hitt'ht' ~HI. .\ . '\

~

•u . .!1•,

INDEX OF NAMES

240

\Yhitchead, \. N .. 2:1 Wiener, N., 140 Wiescnhahn, D. F., Ill \'\'ilkcs . .J. 0., Ill \Villiamson, l\1. II., 167 Winfree,/\., 7H, 7D Wittgcnstein, L., 41 Wolf. A., 77. 20·!, 21:1

y \'orke,J.A., Hi, 70, 76. 110, ltH, :!If>, :!lli

Yourgrau, ~ .. Yuste, M., !)~

I~~~

z Zcldovich, Y. A. B., I 6 Zeller, VV. F., 12:1 Zcrmclo, E., 141 Zhabotinsky, A. M., 7H Zimmerman, I. D., 198 Zipf. C. K., 207

SUBJECT INDEX

~caling, ~07

noiS!i arithmcti< growth, H2 \tnold'~ ell.

7"'

.ts\'tllptotic ~t.thilitv , ·19, 91. !l:~ .ltnwspheric carhon citull.idc I~Hi-1 117 , I(

(J'~Irt O I. :i ~J

7H. e n tr n pv halaiH t equation , I : 1~1 , ntn•JH exchange . 13H e ntropv filii< tion. I '!.7 c ntn•pY prndut tio n, I 1~1 ('qu.ltion ol motion, 61 equilibrium. 1.-J, 4tl-t tl t•rgodicit~. l'iO F.u< !ide an dint e n~iun, I f>·l , Euktian svstem, •Jt' lf-,inularit\', I ti~ ~ c ;tling. HI -.rattng raph. ;, St htnnpt ·tn's dock model. ~ O K s, o u]oplin ra g . ~O(i :-.n o n cl l.aw o f I h tT nHHh'll