Chaos.

Making a New Science

James Gleick

Abstracts from the book

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Introduction

My son Kjell Erik, settled in Molde (Norway) showed me the book Chaos, Making a New Science by James Gleick, when visiting me in the Easter of 2002. I read a little here and there and was immediately facinated by it’s content. Later on I borrowed this book from a local library. During the first reading, words and concepts showed up, belonging to books I earlier had studied, concerning fractals and their magnificent beauty.

This book was written by H.O.Peitgen and P.H. Richter, with the title The Beauty of Fractals (1986).The first chapter, primary dealing with fractal geometri, had the title Frontiers of Chaos. The name Benoit Mandelbrot showed up, likewise Edward N. Lorentz and Mitchell Feigenbaum. Concepts like bifurcation diagrams and strange attractors showed up.

Gleick’s book has about 300 pages, and is by no way easy to read. Even though Gleick is an excellent writer, the reading is a demanding task. The conception Chaos could be a confusing objekt to anyone. Explaining such complicated tasks for ordinary people requires a lot of talent.Wheather Gleick really has succeeded or not is another matter. The topic is still difficult for me. It’s not simple to give clear definitions for unlike conceptions. Besides some of the mathematics used to describe parts of the chaos theory belongs to the cateory of nonlineary differential equations. Exact solutions are almost exclusive, but models and big computors has given researchers in different professions a better insight. The interest for the Chaos theory has increased in the years gone.

My new interest for the Chaos theory inspired me to write a resumé of Gleick’s book. This would force me to a thorough reading of the book, feeling that I got a good comprehension and understanding. Weather I achieved this or not, I really don’t know. The caos theory is said to be paradigm change in natural science.

Skien, den 20.oktober 2009

Kjell W. Tveten

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The Butterfly Effect

Edward Lorentz was especially interrested in weather prognoses. He run his metheological data on a computor and printed out the results as graphic models. He made different models, developed from the results from his computations. He made an astonishing observation. When he for the same starting point , and just for a verification repeated a computor calculation with a subsequent print out, he observed that while the two curves followed each other quite close in the beginning, they later gradually separated .

What caused this? He found out that while he had used a 6-decimal system in his database (memory), his printings were done in a 3-decimal system, just to save space. The data he had used as input were also in the 3-decimal system. Tbis was the answer for his problem. The extreme little difference slipt innto his system and gradually resulted in evident departure between his curves. Lorentz draw the conclusion that long time prognosis would be impossible when such small differences could make such great “waves” in a forecast. (p.17in the book)

Nevertheless, the developments for long time forecasts went on. Super-computors took over, and one of tmem, Control Data Cyber 205, did millions of calculations per second. While Lorentz had used 12 equations, this machine used half a million. National Meteorological Center, using this machine, performed the best forecasts in the world.

In this picture det famous Butterfy Effect showed up. Technical it is described as a “sensitive dependence on initial conditions”, but interpreted as a joke: “ A butterfly fluttering in Peking can release a storm next month in New York”.

Lorentz interest also aimed against the transition phase between laminar and turbulent streams, especially in fluids. In the hydrodynamic, this is closely tied up to what is called the Navie-Stoke equation, where fluid speed, viscosity, pressure and temperature takes place. Unfortunately, this equation is of the nonlinear kind, almost impossible to solve.exactly.

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Lorentz made his practical research by heating a liquid from the underside in a strict dimentioned vessel, and observed the creation of two quite adjacent, parallell, horational and stable rotating liquid cylinders. This gave a clarifying picture of the transition phase between laminar and turbulent streams.When the heating reached the point where turbulance startet, the cylinders got a wavelike formation in the longitudinal direction, just before the turbulence and the chaos was a fact

Lorentz also proposed what later was called The Lorentz Attractor, which in a three-dimentional coordinate system gives an illustration of what is called Chaotic motion in a dissipative system. In a point of departure this could be compared to the path (trajectory) a small planet could be thrown into when existing near two suns. This is visualized in the figure, reminding something like butterfly wings or a couple of owl eyes. The small planet’s trajectories between the two suns follows gives an illustration of what Lorentz called strange attractors.

Revolution

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Thomas S. Kluhn took a settlement with contemporary researchers and scientists. This did no solve the problems they were exposed to. “Under normal conditions the research scientist is not an innovator. but solver of puzzles, and the puzzle upon which he concentrates are just those which he believes can be both stated and solved within the exixting scientific tradition”.

This passed well for those who began studying the chaos problems. Few of them dared to step forward with their opinions, risking their prospective career. Elder professors felt more attraction to tell what they thought about it, but feared criticism from their collegues. Some felt still that a paradigm shift was coming, in reality a quite new way of thinking. The great problem was that chaos was based on a kind of mathematics , beeing unconvential and difficult.

Some institutions started “specializing” on unlinear dynamics and complex systems. Others gave their opinions on chaos an evangelical character.

”A theoretical picture of the trancition to turbulence is just beginning to emerge. The heart og chaos is mathematically accesssible. Chaos now presages the future as none will gainsay. But to accept the future, one must renounce much of the past”. (Page 39 in the book)

What interested the cahos researcher most ? Stephen Smale ( Berkeley, mathematics) gave an answer which could be interprete like this: ”I look for nonlinear oscillators, chaos oscillators and seeing things that physicists has learned not to see” .( p.45 in the book)

In 1959 Stephen (now in University of California) got a letter from a colleague who told him that Smales prediction about structural stability in differential equations could not bee quite correct.He refered to systems showing both cahos and stability at the same time.

For Smale this was to accept that cahos and unstability not was the same thing. A stable, caotic system could as well accept minor disturbances. The cahos that Lorentz had registered was in reality as stable as a (klinkekule) in a bowl. Locally it was unexpectedly, but global quite stable.

Smale scrutinized especially the concept space face, and generated his own ideas about this. Perhaps space face coueld as well be a kind of space propagation of oscillations in a two-dimentional system. Eight examples of this are shown in the figure :

But more than this . Such space phases could also be strectched in different directions, bended to what is called Smales horseshoe, and was a useful mathematical contribution to the insight of dynamic systems (p. 52 in the book)

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Jupiters “red spot”, as telescopes had shown, gave a lot of trouble. Many theories around this phenomena; a gigantic oval spot , surrounded by turbulent whirls, showed up.. Pictures from Voyager gave more details, and disclosed things that only increased the confusion.

Philip Marcus (astronomy and mathematic) in Cornell University was in 1980 one of many who tried to set up a model to simulate Jupiters red spot. He let his model simulate “the weather” on this gigantic planet, using a great computor and a system of “liquid-phase equations”. Marcus experienced to see the “formation” of something that looked like the red spot, and described it like this: “It lays there, happy as a salmon in the midst of a turbulant system, against a background full of chaos. This was a stable chaos” (p.56 in the book)

Life’s Ups and Downs

The ecologists played a special part when chaos developed to constitute a new science in the 1970-years. They utilized mathematic models, but realized their limits on this area. ( p. 59 in the book)The biologist made masterpieces of models, but these and corresponding models from ecologist, psykologist and city plan makers showed up to resemble caricatures of the realities. They attacked their problems based on the fundamental know-how in their respective professions.

The choise of equations, suitable to handle population growth, animals or human beeings, was a great and general problem. The mostly used in a periode was a modification of Malthus’ version :

Xnext = r * x ( 1 – x) , where r is the rate of growth (p. 63) They also looked for “logistic

difference equations” from the physichs. Here r represent the degree of appearance, degree of friction or something else. Shortly, one used the degree of something unlinear.

York, mathematican and philosph, found a Lorentz- document from 1963 (Deterministic Nonperiodic Flow ) This became for York a “mathematical shock” (p.63) York gave a copy to Smale, a mathematican and meterolog (p.66) Robert May, theoretical physicist, later a biolog, recommended a mathematical study of the population in an area where a “critical point” obviously existed. May still used used the

equation x next = r x ( 1 – x) ,

but increased gradually the parameter values from 2.7 to 3.0 (fish population)May brought forward a bifurcation –diagram where in a turning point came a change to a chaotic region. It came into series of numbers where the values first doubled, then a four-doubling, eight-doubling etc untill a full chaos. (p.73)

Frank Hoppenstaedt (NY University) , mathem. scientist , put a non-linear equation into his enorrmous computor, and got an unforgetable picture of something he experiencd like curious “landscapes” on his computor screen. (p.77)

Benoit Mandelbrot was born in 1924 in Warschava, and was of jewish origin. His parents came as refugees from Litauen. He grew up and got his mathematic education in France (Tulle) , but emigrated

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to USA after the war and got employment in IBM. His career in chaos problems started with studies around cotton price developments for short- and long periods, but it was his studies of the so called Cantor-effect which gave him an extended interest for the chaos phenomena. (p.91)

For IBM it had been a problem when greater data amounts should be transfered between computors by telephone lines. Spontanous noise steady occured, but could not be related to failure in equipment and transmissions, and could not be solved by the use of soldering iron and screwdrivers by “electronic people”. Mandelbrot realized the mathematical aspect in this problem, and the nearness to the Cantor effect. On principle the effect can be described like this : If one split up the time for noise in groups (clusters) , and then split further up these, then, without regard to how small these groups will be, always intervals, completely fri of disturbance.

Mandelbrot stated also two new concepts. the Noah-effect (discontinuity when a quantity changes) and Joseph-effect ( persistance; after 7 years of plenty comes 7 years of famine) He also asked the question : ”How long is the coast of Britain ?” He stated that the measuring result had to be a function of the measuring device. The shorter device, the greater the total length. He operated now with the concept “fractal shore” ( from latin fractus = to break), also “fractal geometry”, and declared that the fractal shore gives an infinite length. ”In the minds eye, a fractal is a way of seeing infinity” , said Mandelbrot (p. 98)

A quadrate with a central, inside placed quadratic hole can develop to an approximate fractal after the Koch principle. (Sierpinski) An excavated hole with gradually smaller and smaller bars and openings is achieved, and this excavation can be one indifiniterly. Set up in a three-dimentional pattern (a Menger sponge) this brings imagination to the Eiffel Tower (p.100). Here Eiffel succeded with a construction combining great mechanical strenght and acceptable weight. This principle is found in well known construvtions, such as an aircrafts fuselage and wings, but also in the nature, like wings of insects.

A Menger-sponge will outermost still look tight, but avtually it has an infinite great surface, still without any free volume !

Mandelbrot created the concept “self similarity”. Imagine a person standing between to mirrors, and what he will see. Or, something similar : A drawing that shows a big fish swallowing a smaller fish, which swallows a smaller fish etc etc.

Scholz, who in the starting point investigated earthquakes, was strongly interested in Mandelbrot’s fractal geometri. He favoured the surface properties, and how these interfered with each other with all their uneveness and structures. His “fractal description” found application for such unlike fields like car tires contact with the road surface, electric contacts surfaces and component parts. The uneveness which always exist gives only contact to some points. Stolz was early here, and among colleagues concidered to be a “freak”.

Blood vessels branching and structure i the body tissue can be concidere like fractals, and this is a consequence of the Koch-curve. They provide all tissue with blood, albeit the occupy a volum only 5 % of our body. Mandelbrot called this ” The Merchant of Venice Syndrome” and added : Not only coul’d you take a pound of flesh without spilling blood, you can’t take a milligram” .Other parts of our body has similar astonishments. The typical inner surface area of a human lung is really greater than a tennis court. ( p.108)

About 10 years after Mandelbrot published his fractal geometri, the biolog technicans began to study this closer. Tbe bronchial branches showed up not to be expotential, while a fractal description got the pieces to fall in place. The cardiologs found out that the hearth rythm was goverend by fractal laws. (p.109)

On the practical level the fractal geomety brought e new tool for physichs, chemists, seismologies, metalurgies, probability- and psykology research.. They were convinced, and would with pleasure convince others that Mandelbrots new geometri was a property of the nature. (p.114) Particulary the physics accepted the new science, based on the chaos concept.

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Mandelbrots fractal dimensions describe things better than they explain them. He named the nature elements with fractal dimensions, such as coastline structure, leaves and rivers branching, galaxies etc, but the physichs now wanted to know more about this. (p.118)

It has been told that the great quantum theorist Werner Heiseberg on his deathbed said that he had two questions he wanted to give Our Lord : Why relativity, and why turbulence ? I believe in fact he possibly has an answer for the first”

Turbulence is a mess of disorder at all scales. It is unstable, it is highly discipative, meaning that turbulence drains energy and creates drag ! Only for some circumstanses , for example in a jet motor, where an effective combustion depends on a quick mixing of engine fuel.

When turbulence starts, all laws seems to collaps. Look for instance to the smoke from a cigarette, laying in the ashtray. In the beginning it rise stright aloft, but in the moment the climb speed comes to a certain point, and the critical speed is exceeded, there will imediately loccur a wild chaos, and “eddies” are builded. The russian Kolnikov (p.123) gave a mathematical explanation for how such eddies worked. Anogther russian, Lev D. Landau, gave a description of “fluid dynamics” which has been accepted as a standard. The transition from turbulant to laminar streams goes over several phases, and these are described as oscillatory, skewed, varicose, cross-roll, knot and zigzag. And in addition to this, they accumulate and lays upon each others.

Swinney and Gollub (City College of New York) workwd with observations on thin liquid layers between two rotating cylinders, and used laser technick to illustrate the mentioned transitions (Couette-Taylor-flow) from laminent to turbulant streams. The belgian expert on mathematic physics, David Ruelle, gave a som simplyfied wiev on the turbulance problem : ” The equations of fluid flow are nonlinear partial differencial equations, unsolvable except in special cases”, perhaps not quite news.Nevertheless, Ruelle, together with Floris Takens, introduced the important concept “ sStrenge attractors” in an article the published in 1971. When they once was asked how they had invented the concept , Takens answered : ” Did You ever ask God whether he created this damned universe !” (p.133).

A strange attractor will exist in what one could call a “phase change”. An example could be a two.-dimentional one, with an ball, rotating in a circle. Gradually the friction forces will bring the in to the central ending point, where all movements stops In this case the ending point will be “the strange attractor”. Three- og four dimentional “phase spaces are fully allowed.

Otto Tøssler studied strange attractors as a philosophic concept, ahead of any mathematical judgment. He used as an example how nature affects the movement of a wind sock in an airport. The wind creates a turbulans, the sock planish, and all seems great. ”The principle is that nature does something against its own will and, self-entanglement produces beauty ” (p. 142)It showed soon difficult to make good illustrations on strange attractors, how these could lye in a space phase system.

An approach to the problem came with a french astronom, Michael Hénson (b.1931, p. 144) His great interest was stellar nebula, espescially globular clusters. While two globes movements in the space almost had got its solution , thanks to Newton (the elliptic paths), the calculation for three globes was seemingly very difficult to accomplish, and for further globes quite impossible( stated by Poincaré)

The dynamics in such a system could still be dealt with, albeit one made sum compromises, for example taking the starting point that it always will be created many binary globe systems in close paths. The entry of a third globe could then give a kick, throwing the newcomer out of the system, out of the cluster.

In 1960 Hénson launched his theory : The kernel in such a cluster will gradually collaps, accumulating energy and seek for a state of infinite density. He did numberless of computor calculations on paths for stars in a galaxy, for a quoted time of 200 million years. He projected changes in the paths where they penetrated a fictitious wall, vertical on the path.

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The spots then formed a line pattern which highly surprised him. It always ended up with the outline of an egg, and for a given level of energy in the galax. When he now gradually increased the energy level, interesting things showed up. Inside the “egg” more paths shoved up, some of these nearly egg-formed, nut also what Hénsom describes as “loops”. Nevertheless, there was a kind of “system in order” in this, until the energy level got new hights. The the fluctations came in different directions, more and more disorder, ending up with points in full chaos, ( p.147)

Hénson felt this state laid close up to a system of ”strange attractors”. He now continued with what Lorentz had done with his two differential equations, but now chose to limit them only to time, not in space. In a system of elliptic functions he put in variable coordinates from a system that changed the paths in an intended pattern. ( X new = y + 1 – 1.4x 2 ) He handled this on his computor OBM 7040. Every eliptical curve emerged as a thin line, and he produced many thousands of these.

His great surprice was that firstly it looked like a simple line, split up to more lines as the calculations continued, and always in a particular pattern.(P.150), and this went on and on, “ad infinitum”

In this manner, the strange attractors seemed to be fractals, and diso rer or chaos was canalized in a peculiar way, showing that nature actually was in a constrained condition. Perhaps would a deeper understanding of all this give the mathematics a possibility to a better understanding of the unapproachable nonlinear functions.

When this was published , Ruelle wrote:

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“I have not spoken of the estethical appeal of strange attractors. These system of curves, these clouds of points suggest sometimes fireworks or galaxies, sometimes strange and disquieting vegetal proliferations. A realm lies there of forms to explore, and harmonies to discover”.(p. 153)

Universiality

The concept ”renormalization” showed up around 1940, as part of the quantum theory. It made possible ro calculate interactions of electrons and photons. Faynman, Schwinger, Freeman, Dyson and others launched this theory (p.161) , bringing physichst and matamaticans closer to the understanding of how nature gives their own signals by great mass changing by transport in a compact medium, like magnetizing, polymerization and the change to boilingan or turbulenz etc.

Feigenbaum discussed how we experience colour in our existance, and what Goethe had said abut this. While Newton analyzed the spectrum of light and used mathematic to explain the things, Goethe emphasized our subjetive perception of light and colour, and the balance this could create in our consciousnes. (p.164)

While Newton used a prism to split the light up in its unlike colours, Gorthe used a prism to look through and rejoy the sight of the beautiful, coloured contours of white clouds he saw in sky. While Newtom found the basic, physical explanations for the nature of light, Goethe studied the splendour of flowers and paintings in seeking for a greater, universal explanation for the nature of light.

Feigenbaum wrote the parabel equation Y = r ( X – X2 ), which gives an expression for population increase. Doing this he felt that this could bring him closer to the understanding of nonlinear equations, but he soon experienced that it steady becom difficult to interpret the results. It became always too complex, seemed almost frightening. Albeit this selected equation was a very simpel one in this context. He then took up the global climate changes over periods, with mean values for temperature, pressure, rainfall etc etc, and what was needed for a radical change in the weather forecasts. Long time prognoses use computor programs to predict the changes. In their models lies a possibility to create at least one dramatic “different equlibrium” . In reality, something like this has never occured, but the climatologs imagine that if something like this happens, can a “white earth” be the result, with an earth covered with snow and ice. 70% of all sunlight will be reflected, and such a state could last forever. The klimatologs wonder why something like this not already has occured (p.170) Is this only a question of an accident, or simply just at random ? It is most likely that only a strong, external “ huge kick” could give a White Earth.

The concept “almost intransitivity” (almkost wihout a direct object) was launched by Lorentz, and reflect a state whwre the weather oscillate betweeen certain mean values for a longer space of time. Then suddenly. for one reason or other, the weather changes and new mean values occur. The theory of the outher “kick” was also here strengthened. One of the conclusions was that the ice ages through the worlds history happened just because of the outher kick , perhaps was it a chaos product.

In his further work with the matematical possibilities lying in the quadratic differential equation Feigenbaum had only a hand calculator of thr HP-65 type. His work was enormusly time consuming, but it gave Feigenbaum an advantage. He could evaluate the number columns showing up, and gradually he saw an unexpected regularity in the answers he got, because the numbers had a geometric convergens. The degree of this could give him a numerical expression, when he from his computor steadily got a certain factor, namely 4.669

He started the study of the period-doubling in the equation Xt + 1 = r * sin π t, based on the fact that others earlier had seen that such sinus equations could create similar series of numbers as his parabel function did. And quite right, also this function gave a geometric convergens, and the degree

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of this could be expressed by the number 4.669. Quite unbelievable ! Two such different ones gave the same result .

With uplifted mind he repeated this with other possible functions, via bifurcation, looking for chaos. They all gave the same answer. He guickly learned Fortran, and now the number became 4.66920.Over the night he found out how to double the precision , and the number he now observed showed to be 4.6692016090 (p. 174) Seemingly it was without importance if the choosed equation was a sinus, parabel, or any other function . If this only was the top of an iceberg, and what could lay under this. Was it a law of nature ?

Feigenbaum now went for new and demanding problems. Frankly he should not find some matehmatical evidence, but study numbers and functions , hoping to reveal something which could bring him further, as he expressed , create intensions.

Now came the need for a more custom made computor, but this was no simple mattter to achieve. Unlike US-departments goverended the use of such computors. The work loaded his health, hawing daily up to 22 hours working. For periods he mostly lived on a diet of cigarettes and redvine. His friends believed that Feigenbaum got his vitamins through his cigarettes ! At last the doctor had to prescribe valium and a vacation, but then Feigenbaum already had succeded in creating his universal theory. Now it looked like universality would constitute the differens between the beautiful and the useful in oue existance.

Feigenbaum realized that his new theory could became a natural law in systems beeing in the border district between order and turbulans. Turbulans always created different frequencis, but where came these frequencis from? Now one could suddenly see that such things came in sequencis, and also beeing measurable.(p.180) In the picture of what chaos really is, the interest increased for science disicipline increased, but far from all cheered Feigenbaum as the great creator. He held his lectures in unlike universities, and from this forum the spin-offs started.

Now everybody would calculate and evaluate the theory in thei own spheres (p.183). In 1977 the first conference on the theme “Chaos, a science “ in Italy. One of the initiators declared afterwards : “F or the first time we got a clear model which we all could understand”.

The Experimenter

Albert Libchaber, like Benoit Mandelbrot, survived the war by hiding in the country. He was a jew, and many of his family were arrested by the Nazi. After finishing his eduction in France he was characterized to be an excellent experimenter, where elegance counted more than results.In his cryogenic experiments with liquid helium he developed his famous “Helium in a small box”, a little masterpiece of miniature components in selected materials. His intentions behind the experiment was to yield measurable convections in liquid helium, only 4 degrees over the absolute zero point, an advanced variant of the model that Lorentz earliger had worked with.

Libchaber did not know Lorentz and Feigenbaums interests and works. He was particulary interested i the shape of things, and wondered why so few forms could exist in the nature. There should be an infiniy amount and variants. Concerning streams it existed many forms, including also such unlike variants as economics and history. He gave a general sight on his opinion about such streams by saying: ” First it may be laminar, then bifurcating to more complicated state, perhaps with oscillation. Then it may be chaotic ” (p. 195)

Like Feigenbaum, he admired Goethe, but not for Goethes colour doctrine. It was Goethes ” On the transformation of Plants” he applauded.

Libchaber’s great thought was to create a convection in his small resvoir of liquid helium and find out how vibrations or other forms for disturbance would affect a nonlinear flow, perhaps reveal that such a

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system var self-.stabalizing. In spite of his scientific and sober kind of working, Libchaber was by no means negative to new philosophical streams, expressed by people like Rudolf Steiner and Theodor Schwenk. He admired them as much as a physicist could manage . Schwenk published a little book about the streaming of water, with the title “Das sensible Chaos” in 1965. It was sold out and reprinted several times.

Water streams was quite central for d’Arcy Thompson, particulary falling water in cascades and drops meeting water (splasches) (P.197)

Libchaber selcted liuid helium becaue it has an extreme low viscosity, and would react for minimal impulses and correspondingly sensitive for temperature variations. When he selected to keep the dimentions on his apparatus as small as he did, this was also a concequence of his thoroughly planned tactic, when simulating all imaginable sources of error.(P. 203) “So thoroughly concidered was his technical design that Libchaber, with this, should succeed in tricking the nature “.

Now and finally he could start his researches. It showed up intricate pattern of turbulens he never had dreamed about . Long series of period doublings visualized, the first bifurcation showed up, and det already well known phenomena with two contrary rotating liquid cylinders showed up, just like Lorentz had demonstrated in his experimnts. Oscillations who changed rhytmic in frequence, and stepwise with periode-doubling . He managed to get spectrum diagrams which clearly showed the periode doublings that the thory had predicted.

And not only this. The movements that appear in a system like this can pass over more, many more dimensions, and as movements happening inside movements etc etc. Such appearance would claim for very complicated differensial equations to describe.

Libchaber and Feigenbaum finally met each other in Paris. Libchaber could demonstrate his small apparatus, and Feigenbaum told about his theory.

In the meantime wheels had begun to roll in unlike laboratories and research institutions. The physics got great computors to simulate the quantitative experiments. Franceshini in Italy used five differential equations to display attractors and periode-duobling, without knowing anything of Feigenbaum’s work. Albeit the mathematic confimation of the theory, Libchaber’s experiment was entirely unik. It had yilded results by imaginable combination that differensial equations not could have given. (P. 210)

Images of Chaos

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Michael Barnsley met Feigenbaum at a conference on Korsika in 1979. Barnsley wondered where Feigenbaum’s sequencises 2, 4, 8, 16 really came from. Did they come from some mathematic emptiness, or were they a shade of something laying deep and hidden for insight. He started the problem by putting in complex nambers in a two-.dimental coodinate system when he on his computor worked with Feigenbaums equations, and experienced that phanthastic formes and constallations showed up. In a dizzeling joy he sent a report to a recognized mathematic periodical in France, but came bruptly down to the earth when the editor gave him the following message : “Michael, you’re talking about Julia-sets. Get in touch with Mandelbrot “

John Hubbard, an american mathematics, got an other angle of incidence on the chaos problem. He studied Newtons method for solving second, third and higher degreeds of equations, and how complex numbers sould be managed in such an operation. He found that even the simplest equation X3 – 1 =

0 , in addition to the obviously solution X = 1 had two complex solutions, namely X1 = - ½ + i * sqr (3/2) and X2 = - ½ +-i * sqr (3/2) (P.218)

Hubbard put this in his in a computor program and started studying how unlike starting values influenced the result. The three mentioned solutions for the equation was given the colours red, green and blue, and then he started. And now things started on the screen ! A marvellous spectacle of forms and colours in indescribeable patterns showed up. Never two different colours came in contact, always a third colour turned up and intervened. Hobbard interpreted the regions that was different was attractors, which could be leading back to each of the three colours. He felt he had got a small insight in the “family” of pictures and patterns, reflecting forces in in a real world.Barnsley on his side saw other “members” of the same family, but it was Benoit Mandelbrot who discovered the “grand father for this family” (p.220) What later got the name Mandelbroy set is “the most complicated in mathematics” the admirers liked to say. With it’s coloured spirals and threads , curling and twisting inn and out of each other in a facinating , three-dimentional way it coult take the breath out of those who sees this for the first time.

Such distinctive pictures are usually called Julia-sets. One should believe that such pictures should demand an infinite of information, but believe it ore not, it needs only a few dozens of code signs over a transformation line to get such pictures reproduced. The secret behind this that Mandelbrot used an iteration process for quadratic equations with complex numbers. In its most simple form his sysyet, looks like this :

Z = Z ² + C , which means ” chose a number . multiply with it self and add the original number” (P227)

H.O.Peitgan and P.H. Richter, the first matematican ande the other physcist, wrote the magnificiant book “The Beauty of Fractals”. Their carriere was based on what is called the mandelbrot-sets. For them this constituted a whole universe of new ideas, a modern art philosophy, a justifying of a quite new way to tumble about mathematics, and bring complex systems out to greater groups of people.

Richter started with chemistry, then biochemistry, and studied oscillations in biological developments. He found relations in immunity systems as well as in the conversion of sugar in the fermentation

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process. Peitgen thought it was quite OK that he, beeing a mathematican, run his function experiments on a computor and looked at the virtually pictures on his screen, instead of searching for strongly matematic proofs for what he could expect.

James Yorke took a better look at the well known playing machine Pin Ball, where a ball is given a certain impuls, shooting up along an inclined plane, either right back or left back if the impuls is too small or too strong. Provided the impuls is of right size, the ball could get many unlike tracées on its way downwards, steadily changing direction after hittings with variuos hindrances, for at last to end in one or another premium hole. He thought of a system where the staring pulse could be variated with very small, exactly alike and fitted gradients, and he tried to simulate this. He meant that such a system would give a fractal development, where on some places only was small response for an altered impuls, but on other places came great and unforeseen “economic response “ for such altered impuls, and that this showed up on certain points in a scale from left to right.

Michael Barnsley chose a quite other way. He experimented with what is called Julia-sets, and the form other organisms could take. P.236) He used a some peculiar technique to build up what could be called a fractal picture of a fern frond, and gave a description of the procedure. He called this “the global construction of fractal by means of iterated functions”, and sometimes he called it “ the Chaos Game”.To play this game quick is a computor with a graphic screen necessary, but it can also be done on a piece of graduated paper. One can easily set up a pair of rules connected to crown – coin throwing, and mark a point on a random place on the paper. Then could for example coin mean “set a new point 5 cm northeast”. A crone means then “set a new point 25% closer to the center.” And what happened ? Yes, all points plotted startet by and by to create a figure with a clear outline, beeing more and more sharper and sharper, the longer one continued. The final result depended on the rules set up. The more and complicated rules , the more complicated the picture.

Barnsley presented an incredible thing, namely to get a complete fern frond drawn ,using a small desktop computor, and he described the fern as “a staggering image, correct in any aspect . No biologist wouls have any trouble identifying it.( P. 238)

William Burke by Santa Cruz University was cosmolog, and very interested in Einsteins theory about gravitation waves. He read with interest Robert May’s article in Nature, where May advertized for a better education to the solution of simple, nonlinear equations.

Then Lorentz-attractors showed up, and Burke set up three dfferential equations. Could these be handledby an analog computor ? On Santa Cruz there was only an old and lumpy one, having evident weaknesses, but it could be tuned and justified for variables and parameters to this quite special task in the solution of such differential equations.

Burke really experienced that Lorentz attractors with their characteristic “Owl masks” came and disappeared on his screen. Pictures came and disappeard but could not be reproduced, because they were very sensible for “the initial conditions, but what he really saw made a deep impression.

A group of new enthusiasts formed around this projekt, calling itself ” The Dynamic System Collectors”, some times only “The Chaos Cabal”. (P. 248) Thiese were especiially yong students witg

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different technical connections to Santa Cruz, but with a special interest in finding the absolute truth in the outstanding world of the science. Concerning the question of the importance of determinism for inteligent development and biological selection, they often stood in a strong oposition to their respektive professors.

The mathematical education was brought into focus. Why was this syllabus always ended with the unlinear equations, where the teacher suggested a conversion to linear functions which only could give an appromimative solution ? What really missed, getting further in our scientific search for the truth ?

Joseph Ford was a champion for the chaos theory, and operated at Georgia Institue of Technlogy. He had long since realized that nonlinear dynamics was rhe future for the physics, especially within astronomy and perticle physics. He invited all interested to send in articles about nonlinear connections, about strange attractors, opinions about chaos etc etc. Ford collected this in a list, published it as abstracts and felt that he gradually got an overview on the complex.

Among the big problems, the question about the unforeseen still laid there. An answer to this seemed to lay in the russian Lyapunov’s exponent . This is a yardstick for what is happening in the previuos mentioned “phase space of an attractor”, and different degrees of “folding” such systems can undergo before an attractor has been created.

With the information theory, many new aspects turned up, invented by a researcher by Bell Telephone Laboratories , who’s name was Claude Shannon (P. 255) He gave it the name “ The mathematical Theory of Communication”

He emphasized the obvious redundant of words we use for our communication, which letters that was often repeated in the languag, and what relations this created. His theories was used by cryptologs during the war, based on the statistical pattern the use of certain letters revealed.Communication experts developed a techniqe to comprimating data to giving more space by transmission and storing. (P.257)

Shannons information theory brought in a new discussion about entropy as a concept. in this theory. Entropy is a consept from the 2. law of thermodynamic, explaining the state of increasing disorder

“ Divide a swimming basin in two equal parts and fill it up with water in the one, and ink in the other. . Then remove the division wall, and look what happens Yes, a gradually equalization , due to the molecyles movements” Never such a process could go the opposite way !Could one develop methods with analyses of water samples which could give basis for calculating the time as a function of the degree of dilution ? Could the information theory explain such a problem (P.267)The Santa Cruz group continued their untiring works with strange attractors, chaotic states and information theor. Especially Shaw, Farmer, Crutchfielc, Pacard and Spiegel continued. When they

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met the most queer subjects was debated seriously. They met often on a “Coffehouse” and played a kind of quiz where the questions came in rapid succession: “How far away is the nearest attractor ?” “Why is the leaves on threes fluttering ?” “What does it really mean that a loose bumper rattle?”

Burke was convinced about that the slack speedometer in his car rattled in an unlinear way. Shaw had a predilection for a dripping water faucet. In the moment the drop gets off, a complicated three-dimentional process is released, where surface tension and hydrdynamic are strongly implicated. Here gives plenty of unlinear, partially differential equations, having only approximate solutions.(P. 263)Shaw studied the droplet problem very intimately, and did the experience that he would need three differential equations to describe a model of the drops condition in the moment it gets off, a kind of minimum to describe chaos, like Poincaré and Lorentz had shown.

Then Shaw came out with a theory that perhaps was the Santa Cruz Group*s smartest and practical contribution to the enterprice in the chaos subject For the dripping crane Shaw set up a two-dimentional graf where the x-axis .represented time intevals between two drops, and the y-axis the next times interval. The measurements happened with with milliseconds accuracy ( the drops braking a ray of light). On the way he observe that droplet fromation and release was extreemly sensible for all forms of noise from the environments. Even noise as foot- steps outside the laboratory or traffic noise from outside could influence.

When the water stream gradually was increased, the system went usually in a state of “periode doubling bifurcation” . The drops came in pairs, and the intervals become shorter. The end came when the system became chaotic, and the intervals was a system of compleate disorder. The graf showed only a veritable mess of points. Shaw concluded that the transition to chaos here was initiated by noise (P. 266)

The passage from beeing a rebel to a phycist went slowly for the Santa Cruz Group, and often they wondered how far they reakky had come. The faculty had no longer the great belief in their project , and was warned about that any doctorate on a not-.existing field like the chaos not could be expected. But no one gave any order to stop the Chaos experiments.

Feigenbaum once showed up, but his lecture was mathematically hard to understand, and his “renorrmalization was only for the initiated. Any general treatment of nonlimear equations.seemed almost impossible. Every single treatment or field had to be handled individually, and any form of “universiality” could not be adapted.

(P.268)

Much aligned when Shaw got the opportunity to present a cross section of the material that the Group had builded up in a “meeting” in Laguna Beach” in 1979. Here he brilliant told about “attractors in phase place “ and “fixed points”. He threw light on Lorentz attractors and his dripping crane, he explained the geomerty in phase space and what this could mean for the information theory. (P.269)

The lecture was a triumph, but the team could not live for ever. By and by they left the group. Crutchfield was the first, then followed Forman and Pacard, but the spinn-off effect of their work did

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not wait for. The climatologists now discussed seriously the Chaos effect, the economists studied the stock market to find attractors in dimensions from 3.7 to 5.3. A lot of intricate mathematics was classified and proposed. Fractal dimensions, Hausdorffs dimensions, Lyapunovs dimensions and informatics as well.

The brillians in the measuring of chaotic systems was best explained by Farmer and Yorke, systems going on the “degree of predictability”, “information amounts” and “mixing problematic”. Some scientists really found out they could find signs of deterministicd chaos through all our physical litterature. Many phenomena we so far had noe physical explanation for, and was often described as an effect of chaos.

Inner Rhythms.

Huberman had coopereted with the Santa Cruz Group, he was a biolog and special interested in schizofrenics. In a lecture about the problems that schifronetics met when trying to follow the movements of a pendelum with their eyes, he told his publick about the model experiments he had done.In the model was a circular trough hanged up like a pendelum, and up in the traugh was a ball. When the trough pendeled, this created a movement that hindered the free movement for the ball, and forced down to the bottom of the through. Hauberman had rided this model for hours on a computor, changed parameters and set up grafes for the movements. He found both order and chaos in this system. When the degree of unlinearity increased, the system passed through a quick periode of doubling sequencies, and created a type of disorder which was unmistakeably like what had been reported in medical litterature about shizofrenism.

His public represented many different fields, there war pure mathematicans, physicists, psyciatrics ans biologs. It came up unlike, critical reactions after Hubermans lecture, particulary aimed against what was called a “ too simple model”. “The reality of such medical phenomena is far more complicated, and so should the right model also be.” (P. 278)

Such contentions was difficult to oppose for Huberman, but he got a good support for his simplification from a psyktiatrist Arnold Mandell, who for a long time had been interested in the chaos problems. A complicated model wouls perhaps never disclose chaos, something that more simpel models obviously easely had attained.

The well known Gaia hypotesis with white-and black daysis , covering the surface on the whole earth was by Ralph Abraham mentioned as a defence for simple models. (P. 279)

In the 1980- years the chaos concept gave a new life to the phsycology. Our body seemed to be a system of both movements and oscillations, and it had rhytms which never could be revealed by microscoping or blood testing. Cancer was now evaluated as a consequence of periodisity and abnormality in the cells. Particulary the interets concentrated around the functions of the heart, where the heart rhytm strongly govern our life and death. (P.280)

Ruelle had studied chaos in the heart regions: “a dynamic system of vital interest to everyone of us”The normal heart rhytm is periodical , but a vemtricular fibrillation can be the cause of death.

A great step forward seems to lay in letting a computor work with a realistic model . For a long time one had mapped and catalogized dozens of irregular rhythms. To interpret the right signals, showing up on a electro-cardiogram could be difficult , because it showed up so many variants. (P.281) When the researches used the chaos theory they discovered that the two-dimentional cardiology created a wrong simplifying of the unregular heartbeats or rhytm disaturbances.

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Actually there was a threshold which prevented a good co-operation between physcologist and mathematican / physicist.Physicists in general don’t like mathematic, and the world fractal was not in their textbooks. The muscular contractions in the heart is goverend by electrical pulses in a three-.dimentional systemand is about impossible to simulate in a computor. To develop valves or flaps to handle with the blood stream in perfect operations, billions of times , has been an incredible technical challenge, now partly dissolved , thanks to advanced computor technology.

Heart flimmer (ventricular fibrillation) is a very serious state, often leading to death. The condition of the heart under such conditions has been compared with “a sack full of worms” (P. 283)The heart then only vibrates, it doesn’t pump blood. Remarkably this happens at the same time that the impulses controlling the ordinary pulses find their right place, and the muscle cells working correct, but the flimmer destroys this entirely. The flimmer gives a complex system of disprder To stop the flimmer, a strong electric impulse must be given, but the size of this will usually be quite empirical. For other types of unnormal heart rhytms there is today a great assortment of medicines, developed after the trial and error methode. (P. 284)

Winfree studied mathematics and biology with Cornell, John Hopkins and Princeton, and was special interested in “biological watches”. He experimented with mosquitos , who not have 24, but 23 hours built-in clock. Winfree discovered that when giving his mosquitos a carefully fitted light pulse bay midnight time, managed to destroy their biological clock (P.286).

The most refered experiment with biological clocks was done by Leon Glass, Michael Geenella and Alvin Schew. They used small lunps from the heart of seven days old chickens, and exposed them to shakings. Immediately the “heartbeates” started in these small limps, with a frequency about one per second. And not only that : When they put in extreemly, rhytmical electric signals via microeelectrode to one cell, this activated all the cells to the same rhytm. This discovery was published in Science in 1981, where Glass writes : ”many different rhythms can be established between a stimulus and a little piece of chickens heart. Using nonlinear mathematics, we can understand quite well the different rhythms and their orderings”. They reported about” period-doubling – beat patterns that would bifurcate and bifurcate again as the stimulus changed” p. 290)

Other examples of chaotic inflows and unnormalities in the heart frequence was found among older data. They showed that it was certain patterns in the distribution of clean sinus- and ectopical beatings. The last ones came always in the sequence 3 – 5 – 7, and the sinus came in the sequence 2 – 5 – 8 – 11. A kind of order in the chaos it seemingly was. (P.291)

An interesting idea emerged : One can consider chaos as a positive and health promoting factor. In a linear system will something beeing crooced or getting a push, start to bump further on.

In an unlinear system will such disturbances not be of mentionable influence, the system will quickly normalize.

Huygens, a scientist from the 17. century, once detected that nearly all of his clocks on a wall went syncronic, with their pendelum oscilations. He concluded that this was coordinated by oscillations through the back wall, where the clocks were uphanged.

It seems like the nature accxepts an increased degree of synchronisation. Likewise it seems to favourize a broad spectrum of rhytms, and the good dynamics in nature shows to have a pronounced fraktal character. (P. 293)

Mandell chritizised those who produced “psykopharmacologic medicines”. They created more disorder in the patients system than it helped the (perhaps with the exception of litium) Such medicines acted only linear, and the research and trying out which leads to such products are also linear. (P 298)In the chaos theory one apparantly has got an opportunity to study how the degree of disorder in our existence could be an important part of life.

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Chaos and Beyond

What was actually attained in those 20 years which had gone since 1970 ? Well, for a long number of professionals the interest for the chaos concept and theory had increased. Many had started to search for old data, intending to study them in new light from bifurcations and chaos. The mathematics connected to the use of nonlinear equations frightende many to go deeper into the problems.

Hao Bai Lin, a chinese physict, collected many historical articles about chaos in a reference work, and described the content like this : A kind of order without periodicity, A single expanding field og research to which mathematicians , physicists, hydrodynamicists, ecologists and many others have all made important contributions.

Arthur Winfre thought that the name chaos was too limitary. He was mostly interested in laws which described more-dimentional complexibility. Beyond that the hunting for strange attractors went further on. Einsteins famous claim that God not plays cubes with the clodes was turned upside down by Ford. but he added that God probably had used loaded cubes.

Schaffer used strange attractions to explain and foresee the epidemic development of child diseases like measles and chickenpox. He collected data from New York, Baltimore, Aberdeen, Scotland and Wales. In his dynamic model he presupposed that the start came when the children returned to school after the summer vacation (P.315), and died out after a certain time, when a kind of resistance was established. . But his model indicated unexpected difference in the behaviour to these epidemics. Chickenpox should variate in periodes, and measles come like a chaotic wave. And that was just what happened in the reality.

P.S. If you want to buy this book, contact http:// www.amazon.co.uk

Dok : K:\Chaos,making a new science\Chaos, art.compl m.bilder 3:10. 2009

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