The theory of Chaos

François Coppex

The theory of

Chaos

Project number two

The theory of chaos by François Coppex Project number two 1

Contents

Contents .............................................................................. .. 1

Foreword .............................................................................. .. 2

1. Introduction ....................................................................... 3

1.A. What is chaos?........................................................... 3

1.B. The study of chaos - where can it be found? ............ 3

1.C. What are the bricks of chaos? .................................. 4

2. What is chaos? ................................................................... 5

2.A. Strange attractors........................................................ 5

2.B. Bifurcations: doublings of period ............................... 7

3. The study of chaos - where can it be found? ..................... 11

3.A. Stability of Solar system ........................................... 11

3.A.1. A brief history of the study of the solar system ... 12

3.B. Weak chaos, disasters and self-organization ............ 12

4. Fractals ................................................................................. 14

4.A. Heard about dimension 2,5? ..................................... 15

4.B. Iteration of complex functions .................................. 18

4.C. Fractal pictures ........................................................ 20

Appendix ................................................................................. 22

Index ....................................................................................... 23

References .............................................................................. 25


ForewordThe theory of chaos first caught my interest oneand a half years ago when I was deep into aparticular philosophical reflexion which was toknow if the universe was entirely deterministic. Iwanted to know if it was possible that evenhuman's conscience, or spirit, could bedetermined by fatalistic laws of physics,equations. That is why I started reading scientificvulgarization about determinism, laws ofmechanics and astrophysics. In an absoluteobjective and detached spirit, I at first thoughtthat physics always told exact predictable resultstherefore that the universe was entirelydetermined by laws of physics. Human'sconscience was in my point of view absolutelydetermined by mathematical laws we still did notdiscover. Indeed, I thought the laws explainingall odd phenomenons of the universe, like life orconscience, were hidden in still unknownunderlying laws following from probabilities ofquantum physics, which still may be possible.My point of view certainly was a bit narrowbecause of the lack of knowledge I had at thattime in physics. That is why I wanted to enlargemy knowledges on this subject so that I finallycould know if science had found a path to turnthis question from a philosophical one into ascientific one. It perhaps was a bit naive but it ishard to figure out how small the scientific cultureof a high school student who does not readscientific vulgarization by himself is. Anyway, Ifinally found a strange branch of science whichwas very close to probabilities, chance,determinism. It was called chaos. I startedreading avidly and passionately all books Ifound on theory of chaos. My readings slowlyslipped into pure science for a good reason: therewas no scientific branch and texts which couldgive an answer to my philosophical question.Anyway, this led me to adopt a more flexible andenlarged way of considering this question,however my feelings tell me there is no objectivereason for humans to being an exception in ourmechanical universe. But as I said before, there is

no proven answer therefore I leave a big questionmark as answer.

Anyway, this dossier was the right occasion forme to make a synthesis of some nice scientificconcepts I learned during this year and a half. Ithought writing it in English would improve mywritten ability, which turned out to be really thecase. I first of all did this dossier for my personalinterest, for myself, in order to sum up my littlestudy on theory of chaos, however summarizingall what I learned - and what I will learn - wouldneed some more hundred pages. That is why theresult is quite huge compared to a project'sexigences, but microscopic compared to howwide theory of chaos is. My aim was not toimpress the reader by the great amount of pagesbut to have pleasure explaining - for myself atfirst - concepts, making graphics, learning theuse of Microsoft Word, regardless to the numberof pages. Reading this dossier requires anintellectual effort and it can not be read at thesame speed as for a novel. Some of this dossier'schapters are more technical (chapter 2.B., 4.A.,4.B.) but I made sure even mathematicaldemonstrations could be understood. Talking ofsuch a subject is impossible without explainingsome mathematical concepts if I want to give arealistic explanation of it. Anyway, I do not mindif the reader skips these chapters for any reason.Doing this dossier was my best reward: I enjoyedwriting it (however I admit I sometimes haddifficulty finding inspiration), my Englishimproved quite remarkably, I learned the use ofMicrosoft Word at the maximum of itspossibilities. I hope this dossier will have thefantastic power to make the reader ask himselfsome questions and perhaps even bring him amore contemporary view of the world.

The author,François Coppex

28th of December 1995


1. Introduction

«A very little cause, that we didn't catch, determinesa considerable effect that we can not not see, andthen we say that this effect is due to the chance.»

Henri Poincaré

Chaos, a word which sounds very trite. Everyone knows whatchaos is, or think so. «Yes, chaos is, well, let me think... it isemptiness or the big confusion before genesis, or even better, it is agigantic mess...» In a way this is true, but certainly too narrownowadays.

Chaos has become, these last decades, a new branch of science.Moreover, it is one of the most important branch of research, andhas got a promishing future. One of the big advantages of the theoryof chaos is that it has implications in a lot of branches of science, onthe opposite of the very mediatic branch of research of physics ofparticles. Therefore, chaos includes mathematics, physics,chemistry, astronomy, biology, economics, philosophy and hasalready got some applications in technology.

But now that your curiosity has been thrilled, let me shortlyexplain what this mysterious theory of chaos is, and give a globalview of it. Fasten your seatbelts.

1.A. What is chaos?A good synonym for chaos is «unpredictability». The theory of

chaos tries to characterize phenomenons straddling on chance anddeterminism. This means that such a phenomenon should bedeterministic, but can not be predicted a long time in advance. Inother words, it is a way to create complex phenomenons with simplestarting conditions. A chaotic system is unpredictable, but isperfectly described by easy deterministic differential equations. Thelink between these two paradoxal terms, determinism andunpredictability, is the characteristic of the initial statessubordination: two initial states which are similar can lead to verydifferent states of a same system.

The initial states subordination is a well-known phenomenon bybilliard players for instance. It is nearly impossible to make abilliard ball run over the same course two times. After some bounceson the sides of the pool, two balls which are released from the sameplace, and which have the same speed, have big chances to be at twodifferent places. Why? Just because it is very probable that we didnot throw them exactly in the same direction, or because we did notplace them exactly at the same position. A mistake of onethousandth of millimeter is widely enough. Each bounce on the sideof the pool multiplies the mistake, reported to the angle, by two.After the first bounces, the positions of the two balls still look thesame but they suddenly get absolutely different directions (seepicture 1.1 and 1.2).

Picture 1.1: A mistake because of the angle. The two balls {a;b} were released from thesame place but the billiard player was unable to aim exactly at the same point. The resultis an angle α bigger than zero. Therefore, the two balls have a very different locationafter three bounces on the edges of the pool.

Picture 1.2: A mistake because of the position. The two balls {a;b} were not placedexactly at the same place. On the other hand, the billiard player managed to aim exactlyat the same point. In spite of this, the two balls occupy a very different location afterthree bounces. This is because there is a distance d between the center of the balls {a;b}which is bigger than zero. Thus, they do not have the same direction at the time of theirfirst bounce. The mistake d is reported to the angle β.

But chaos in not experimented only by playing billiard.

1.B. The study of chaos - where can it befound?

The most famous chaotic system is the atmosphere, studied bymeteorologists. Some people always complain about badmeteorologistic previsions, and hope there will be consequentprogresses in meteorology so that they finally will be able to plantheir vacations following thrustworthy meteorologistic previsions.These people will have to wait a very long time, until there will notbe any atmosphere anymore on Earth...


Weather can not be predicted over one week in advance becausethere are too many factors which can not be measured formeteorologistic previsions. We will never be able to know the exactinitial states which could help forecasting exact previsions. Imaginethat you now have a giant computer which is able to handle anunlimited amount of information. Your aim is to reach perfectprevisions. You now have to collect the needed information forgetting the real initial states condition. Since it would take too muchtime, imagine that the flow of time has stopped. The universe is likefrozen so that you may travel around the universe without disturbingany atom - sorry for such a scoff at the laws of physics. The firsttask is to study the atmosphere. Each cubic centimeter of air has tobe analysed. What substances? How many molecules of each? Theirspeed and position? After that, you will have to study the ground.What kind of earth? What about the vegetation according tophotosynthesis? What about a little worm innocently digging? Abacteria decomposing organic scraps? Animals? Humans - how topredict the behaviour of humans? Not only the earth has to bestudied but also its astronomical environment. The solar wind has animportant influence on the atmosphere, hence meteorology. Owingto this, location and speed of each molecule of the sun has to beknown. And what about gravitic phenomenons? Of course, theyhave an influence on the atmosphere. And the electrons? Certainlyyes, and the most terrific thing is that they travel nearly at lightspeed in space. It means that, if you want to forecast an exactmeteorological prevision one year in advance, you have to know thespeed, location, and kind, of every particle in a radius of one lightyear. This makes really a lot of information, which is a glaringeuphemism. Moreover, a very little perturbation or mistake either oflocation or speed, of the order of the angström (10-10 meter) canlead, some months later, to radical changes. This is called the«butterfly effect», and is similar to initial states subordination. Thenames comes from a discovery made by the meteorologist EdwardLorenz in the early sixties. He tried to study weather using threedifferential equations. With the time as variable, he represented therelations between the three equations on a graphic with three axis.The result, his famous «strange attractor», did look like the twowings of a butterfly, whence the name «butterfly effect». Do notworry if you do not understand all these concepts, I will explainthem in a more exhaustive way later on. There is a nice little popularpoem which explains the «butterfly effect» very well:

For want of nail, we lost the horseshoe;For want of horseshoe, we lost the horse;For want of horse, we lost the horseman;For want of horseman, we lost the battle;For want of battle, we lost the kingdom!

It's quite the same in meteorology. Imagine that you sneeze now.You did disturb the speed of some particles of air in front of you.These particles will disturb other ones. As the time goes on, thenumber of particles whose speed got disturbed grows on, faster andfaster. Some years later this could lead to a storm somewhere...

To sum up, a chaotic system is determined in a mathematical waybut is undetermined in a physical way.

1.C. What are the bricks of chaos?In order to get a more global view of the theory of chaos, what is

the aim of this introduction, I will try to show how wide is the fieldof its applications.

Astronomy uses it to explain the phenomenon of resonancebetween the inner planets of Solar system, the inversions of the

polarity of the earth's magnetic field, economics to try to explain therelations between State, taxes and household, catastrophes on theexchange market, biology to try to explain the irregularities of thecardiac beating of a person at rest, the evolution of populations,human physiology, Darwinism adaptation, cerebral electric activity,sociology to try to explain the changes of societies according to thenumber of inhabitants, what they are doing for a living, theirdispersion and so on, physics to study thermodynamics of structure,self-organization, plasmas, fluid mixing, lasers and quantum optics,mathematics to study the theory of numbers, theories of developedturbulence (it means the study of rough fluids' flowing). 1994 didJean-Christophe Yoccoz get the Fields medail (which is theequivalent of the Nobel price for mathematics) for his work on thebranch of the theory of chaos called dynamic systems.

There are other branches of research which are based on thetheory of chaos, but it would not be of any use to list them. Ofcourse, the fact that chaos is the foundation for numerous researchesimplies the knowledge of very complex mathematical tools andtheories that non specialist people never heard about. If you want tounderstand the theory of chaos in an exhaustive way, you first willhave to learn ergodic theory, K.A.M. theory, geodesics, topology,algorithmic complexity, boolean algebra, non euclidian geometry,and all the «easy traditional stuff» like differential equations,quantum physics, relativity, and so on.

What a nightmare. The theory of chaos is normally taught in thelast year of physics at university, and only in a superficial way. Butyou may follow a specialisation in «non linear behaviours andchaos».

There is no need for me to say that I did not understand allaspects of the theory of chaos. The aim of this project was to explainall the subjects of this theory proportionnally to their importance.But I did abandon some subjects because it was impossible tounderstand them without very advanced mathemeatics. For instanceK.A.M. theory, which is an important aspect of chaos, which wassolved in 1967 by A. N. Kolmogorov, V. I. Arnold and J. Moser, orthe study of chaotic trajectories of ions in plasmas. That is why Iwill focus on a more accessible and less abstract branch of thetheory of chaos, which is the linearisation of a dynamic system. Inless technical terms, it is the fractals.

The choice of a relative simple subject as main subject, howeverit is not a very important part of theory of chaos, allows me toexplain it in a very exhaustive way. Moreover, it allows me toinclude beautiful pictures of fractals...


2. What is chaos?

Physicists tend to think that the only questions tosolve are of the type: «Those are the actualconditions, what is going to happen now?»

Richard P. Feynman

The first men thought the world was completely chaotic. But,generations after generations, the regularity of the solar day, themoon month, the sideral year, taught humans the notion ofpredictability and order. This became a very exhilarating concept, atthe point that scientists and philosophers thought the world wasentirely predictable. This was the case of Laplace (1749-1827) whothought everything in the world was deterministic. It means that pastand future states of every dynamic system is determined by its actualstate. This led to great philosophical debates: does human beinghave the power to make its own choices? Does it not exclude theexistence of God? In a scientific point of view, the world seemedentirely deterministic, until the discovery of quantum physics andthe famous «principle of uncertainty» of W. Heisenberg in 1926.Quantum physics allow the use of probabilities, and could allow aswell the possibility of an universe governed by chance. In spite ofthis, Albert Einstein, who received the Nobel price for his work onquantum physics, did never admit that the world could be governedby chance. Who does not know Einstein's famous sentence: «Goddoes not play dice.». In fact, as paradoxal as it seems, the modernman goes back to its starting point: he tend to think that the world ischaotic, as well as for the caveman...

More and more chaotic dynamic systems are being discovered.The study of these systems needs an input of information which is asbig as the output of information. It means that our calculations donot predict anything real because these systems are so chaotic,unpredictable and uncertain, that the entry information becomes asimple copy of the output information. As example think at howmuch information has to be collected in order to get exactmeteorological previsions a long time in advance.

We will now dive into the theory of chaos and examine some ofits most important mechanisms.

2.A. Strange attractorsAs I said in introduction, a dynamic system - a temporal

evolution - is characterized by the location and speed of itsmembers. For instance, what does characterize the state of Solarsystem at a time t ? Nothing else than the location and speed of itscelestial bodies. In a more general way, the state of a dynamicsystem is described by a certain number of quantities which are timedependent: x1(t), x2(t),..., xn(t). x1, x2,..., xn are functions which havetime as variable. For instance the function x(t) of a stone in avertical free fall is: r = 1

2 gt2 + V0t + r0 ,where t (time) is thevariable.

A way to study a dynamic system is to report the location of itsmembers onto only one graphic. Let is imagine a race held in astadium between the tortoise and the hare of the fable (see picture2.1).

Picture 2.1: The stadium. The tortoise and the hare are going to run one loop. We willfocus on the Ox axis only.

The hare lets the tortoise go while he has some fun at totallydifferent activities. Then he rushes forward but the tortoise reachesthe arrival before him.

Let is locate the position of each runner on Ox axis of thestadium. The state of the dynamic system (tortoise and hare) is thendescribed by two graphics: the first one represents the location ofthe tortoise as a function of time; the other one represents thelocation of the hare as a function of time (see picture 2.2).

Picture 2.2: The race between the tortoise and the hare is represented by two graphicscorresponding to the locations of the toroise and the hare (reported to the Ox axis of thestadium) with time as variable. The tortoise moves at constant speed, the graphic whichrepresents her location on the Ox axis has a maximum when she passes by the first curveof the stadium, and a minimum when she passes by the second curve. The hare is at restand start running too late: the tortoise arrives first at the end of the race.

Instead of studying n variables separately, physicists like torepresent dynamic systems by a lone point in a space of ndimensions. Thus in our two variable example (the tortoise and thehare) we consider a two dimension graphic which represents thelocation of the tortoise in function of the position of the hare (seepicture 2.3).


Picture 2.3: The two graphics of picture 4 can be represented onto a single graphic,where a point represents the location of both competitors. The location of the tortoise isrepresented on Ox axis, whereas the one of the hare is represented on Oy axis. Thus, atthe beginning, the way done by the tortoise grows up very fast whereas the one of thehare does not grow up much. Later on the hare runs faster than the tortoise. But thetortoise stops because she has won the race. If the two competitors repeat their race aninfinite number of times, the representative point of the dynamic system will run throughthe same loop an infinite number of times as well, therefore it is a periodic dynamicsystem. This graphic has two dimensions because it is described by two variables. Allperiodic dynamic systems are represented by a loop, or a periodic orbit, in a space of ndimensions, where n is the number of variables (competitors).

This way of proceeding has its advantages. The hare cathes up thetortoise at the arrival, but a bit too late: the graphic is closed in thespace of phases (space of phases is the name given to such arepresentation of a dynamic system). If the two competitors repeattheir race indefinitely, then the representative point of the dynamicsystem will run through the same loop an infinite number of times.We then call this loop a periodic orbit.

If now a fox joins the race, the representative point of thedynamic system will follow a periodic orbit in a three dimensionspace (as long as the fox does not gobble the hare). In a moregeneral way, if n runners periodically pass over the same place, thedynamic system is described by a periodic orbit in a n dimensionspace. It is not much harder to conceive a loop in a 100 dimensionspace than in our 3 dimension space. But it is only a bit moredifficult to read.

A dynamic system made of three variables like the one of thetortoise, hare and fox has a periodic orbit, therefore it is adeterministic system. But there are other systems whose orbit is notperiodic but almost periodic. These systems are chaotic and their ndimension graphic is called «strange attractor». In this case, thequantities x1(t), x2(t),..., xn(t) which describe such a system are noteasy equations like: r = 12 gt2 + V0t + r0 anymore. A chaotic systemis described by non linear differential equations, or non linearrecurrent relations. A non linear equation does not expressproportional relations. For instance the resistance of the air whichgrows in a nearly exponential way as the speed increases. Anotherexample is the gravitation which grows following Newton's well-known non proportional relation. On the contrary, a linear equationis represented by a straight on a graphic.

In the early sixties, the meteorologist Edward Lorenz, then atM.I.T. (Massachusetts Institute of Technology), tried a newapproach to study Earth's atmosphere. He studied the movement of agas or a liquid under the influence of heat, a phenomenon calledthermic convection, which is responsible for the flow of warm andcold masses of air in atmosphere. Lorenz studied the specific case ofRayleigh-Bénard thermic convection. He took its three non lineardifferential equations and simplified them a bit in order to get

equations which his computer could handle. At that time computerswere not as powerful as today. It explains why he had to simplifythe Rayleigh-Bénard thermic convection equations. A thermicconvection is when a liquid or a gas is heated from its lower partand tend to get a cylindric movement (see picture 2.4).

Picture 2.4: Thermic convection happens when a liquid or a gas is heated from its lowerpart and tend to get a cylindrical movement. The heated fluid goes up in the middle andlooses heat. When it has lost some of its heat it goes down on the sides. When the heatreaches a certain limit, the flow becomes messy and turbulent, in a word: chaotic.

Edward Lorenz did use the following three variables, one for eachaxis:

dxdt = σ(Y-X)

dydt = -XZ + rX - Y

dzdt = XY - bZ

where t is the time, σ is the number of Prandtl (proportion between the viscosity of thefluid and its thermical diffusiveness), r is the vertical gradient of temperature(proportionnal to the number of Rayleigh), b is the vector of convective wave.

Edward Lorenz tried out some values for {r; σ; b} and noticed howimportant the variable r was. Indeed, when r grows up to a criticalvalue, the Rayleigh-Bénard solution becomes unstable: a convectiveeffect appears. If r keeps on growing the convective rolls loose theirstability as well: a chaotic system appears. Thus when r < 1 thesolutions of the Lorenz system tend to the point (0;0;0) of itsgraphic, which corresponds to the stability of the Rayleigh-Bénardsolution. When r > 1 , the original Rayleigh-Bénard solutionbecomes unstable but there appear new stable solutions thatcorrespond to convective movements. When r keeps on growing,these convective solutions loose their stability as well (for r > 25).Lorenz used the values r = 28, σ = 10 and b = 8

3 . The three nonlinear differential equations then become:

dxdt = 10(Y-X)

dydt = -XZ + 28X - Y

dzdt = XY -

83 Z

What Lorenz did is to solve these three differential equations with t(time) as variable using his little computer. Then he represented theseries of solutions on a three dimension graphic, a bit the way wedid before with the tortoise and the hare. The result is the famous«strange attractor» of Lorenz (see picture 2.5).


Picture 2.5: Lorenz's strange attractor. A chaotic system like this one shows initial statessubordination. A very little gap from the initial point brings about a change of trajectory,which will grow faster as the time goes on. In reality, the passing from one «wing» of thestrange attractor to the other corresponds to the inversion of the rotating direction of thefluid in convection.

The graphic looks like the two wings of a butterfly, whence the«butterfly effect» of meteorology I mentioned in the introduction.This is a typical chaotic system and a quick look at the graphicshows it to us: the solution of the three equations can only berepresented by an infinite number of points. Therefore, this graphicdoes not show us the exhaustive solution of the three equations, butonly the solution for a relative short period of time. The reason is anesthetical one because if we want an exhaustive graphic it wouldlook like a big stain of colour.

There are plenty of other dynamic systems which lead to astrange attractor. For instance, let is take Hénon's attractor (seepicture 2.6).

Picture 2.6: Hénon's strange attractor. Contrarily to Lorenz's strange attractor, this one isrepresented in two dimensions, therefore has two equations, or variables.

Michel Hénon was an astronomer at Nice observatory in southernFrance. He came to this strange attractor because of investigationsof the orbits of astronomical objects. This strange attractor, linked

with Hénon's name, is an example of a very simple dynamic systemwhich exhibits strange behaviour, and who has only two equations,or variables. Therefore, this strange attractor is represented on a twodimension graphic. The equations of Hénon's strange attractor arerecurrent relations (see chapter 2.B.) but not differential equationsas it was the case for Lorenz's strange attractor. Here are the twoequations:

Xn+1 = Yn - aXn 2 + 1Yn+1 = bXn

The parameters for picture 2.6 are: a = 1.4; b = 3; x0 = 0; y0 = 0.Another example of a dynamic system which leads to a strange

attractor is a well studied oscillating chemical reaction discovered in1950: the Belousov-Zhabotinsky (BZ) reaction. It is the oxydationof the malonic acid CH2(COOH)2 by the bromate Br03

1- in presenceof metallic ions, like cerium. During the reaction, the ceriumoscillates between the ion Ce3+ and Ce4+ of the catalysing redox pairCe3+ / Ce4+. This reaction is realized in a cell in which reagants areinjected at constant speed. Thus, with the help of a colouredindicator, which is in this case the catalyser cerium, it is possible toobserve the periodical variations of concentration of the redox pair.Therefore the variations of colour - in this case from colourless topale yellow - are observed in function of the time. According to thevalue of the parameters (injection speed of reagants, temperature,and so on), the reaction is whether oscilliating or chaotic. Theinterest of the BZ reaction is that it illustrates lots of important ideasabout chaos, like determinism, attraction, initial states subordinationand other ones which are more abstract. Unfortunately it becomestoo difficult for me to understand these concepts therefore I will notbe able to develop the theories following from the BZ reaction.Nowadays we know about thirty chemical reactions which generateperiodic variations of concentration.

Once more, a lot of other dynamic systems lead to strangeattractors. It is the case of a compass brought under the influence oftwo magnetic fields, quantum optics, the inversion of the polarity ofthe earth's magnetic field (it means that during the periods ofinversed polarity North magnetic pole was in South...) and a lotmore. As the time goes on more and more strange attractors arebeing discovered, in various branches of research.

2.B. Bifuractions: doubling of periodMitchell Feigenbaum found a job in research at Los Alamos in

1973. He was supposed to study non linear behaviours, so did he.Moreover he did so well that he discovered an new iterativeproperty of a logistical series called «doubling of period». At thattime, pocket calculators first appeared. Feigenbaum owned a HP-65calculator, with the help of which he studied doublings of period in1975. In short, Feigenbaum found out a new path to chaos usingiterative process.

An iteration is when the solution of a mathematical function istaken as new starting value and so on. In other words, it describesthe behaviour of the series x, ƒ(x), ƒ(ƒ(x)), ƒ(ƒ(ƒ(x))), ..., where ƒ isa particular function. When the series tend to one particular point,we say that the series converges, and the point of convergence iscalled an attractor. On the contrary, if the value tend to be infinite ortend to zero, the funcion diverges and has no attractor. This makes alot of new concepts, therefore I will show one example. Let isconsider the mathematical function ƒ(x) = x2. We give x the arbitraryvalue 2. We now iterate this function:

ƒ(2) = 22 = 4ƒ(ƒ(2)) = ƒ(4) = 42 = 16ƒ(ƒ(ƒ(2))) = ƒ(16) = 162 = 256


ƒ(ƒ(ƒ(ƒ(2)))) = ƒ(256) = 2562 = 65536After only 4 iterations, the value of the function is 65536. It isobvious that when the number of iterations increases the function'svalue increases as well. In this case there is no attractor because thefunction's value tend to be infinite. Another way to represent aniterated function is to write it in a recurrent form. Thus ƒ(x) = x2

becomes: Xn+1 = xn 2, which means the same as x, ƒ(x), ƒ(ƒ(x)),ƒ(ƒ(ƒ(x))), ..., where ƒ appears n times. It is easier to understand aniterative process with the help of a graphic (see picture 2.7).

Picture 2.7: An iterative process. Iterations allow to find solutions of functions orequations. We first choose an arbitrary value for x0 . Then we trace a vertical straightuntil we reach function ƒ. The reached value is ƒ(x0). Then we report ƒ(x0) to the straightof equation Y = X in order to obtain x1. The first iteration is done. For the second one wetrace a vertical straight from the previously obtained point to function ƒ. The reachedvalue is ƒ(ƒ(x0)), or ƒ(x1). Next we report ƒ(x1) to the straight Y = X in order to obtain x2.The second iteration is done. And so on. Accordingly to the function's value, iterativeprocess will show after some iterations either divergent behaviour - when the value tendto be infinite or zero - or convergent behaviour - when the value tend to a real tumber. Ifconvergence appears, we say that the real number, which is xn on picture 2.7, to whichiterative process tend is an «attractor».

Feigenbaum did study a particular kind of equation called«logistical equation». This one is well-known by biologists becauseof applications in growth of populations. Everybody knows thatwhen a population, for instance foxes, reaches a certain limit value,its number of members stops growing because there is a saturationphenomenon. We consider two species living on a island. There arefoxes and hares. If the population of hares suddenly grows up to anabnormally high number, it will become easier for the foxes to cathand gobble them. On the other hand, if there are too many foxesthere will not be enough food - in this case hares - to feed the wholepopulation therefore their number will fall down. Such a behaviouris expressed in the logistical series which is: Xn+1 = λ Xn(1- Xn). Theexpression (1- Xn), which introduces the non linear term Xn+1 = λXn- λXn 2, allows saturation phenomenons. The logistical equation isused in quantum optics and other domains as well. Feigenbaum didstudy the logistical series' iterative behaviour for values of λ whichare between 0 and 4.

We now have the required mathematical knowledges to followFeigenbaum's way of finding out chaos. At first we will iterate thelogistical function for a value of λ which is under 1. In order to stepinto the iterative process we have to decide of a first arbitrary valuefor x0. The choice of this value does not have any influence oniterative convergence, as it is obvious on a graphic. Let is take λ =0.5; x0 = 0.1. The logistical series then becomes:

x1 = λx0(1-x0) = 0.5 ⋅ 0.1 (1 - 0.1) = 0.045x2 = λx1(1-x1) = 0.5 ⋅ 0.045 (1 - 0.045) = 0.0214875x3 = λx2(1-x2) = 0.5 ⋅ 0.0214875 (1 - 0.0214875) = 0.010512x4 = λx3(1-x3) = 0.5 ⋅ 0.010512 (1 - 0.010512) = 0.005201

It is obvious that there is no attractor because the successiveiterations tend to zero (see picture 2.8).

Picture 2.8: An iteration of the logistical series with λ = 0.5. We choose an input valuex0, which is 0.1 on this graphic. The output value is obtained by tracing a vertical straightuntil it reaches the logistical equation's graphic. Next, we report this value to the straightof equation Y=X. Then we have to use this result as input value X1. We obtain X2 bytracing a vertical straight from the obtained value X1 until we reach the logisticalequation's graphic, then the obtained value has to be reported to the straight Y=X. Thisprocess has to be repeated an infinite number of times to gain precision. Nothinginteresting appears for a value of λ under 1. Moreover there is no attractor because thesuccessive iterative values tend to zero.

Now let is take the case when λ has a value between 1 and 3; forexample λ = 2.5. The logistical series then becomes:

x1 = λx0(1-x0) = 2.5 ⋅ 0.1 (1 - 0.1) = 0.1875x2 = λx1(1-x1) = 2.5 ⋅ 0.1875 (1 - 0.1875) = 0.380859x3 = λx2(1-x2) = 2.5 ⋅ 0.380859 (1 - 0.380859) = 0.589513x4 = λx3(1-x3) = 2.5 ⋅ 0.589513 (1 - 0.589513) = 0.604968x5 = λx4(1-x4) = 2.5 ⋅ 0.604968 (1 - 0.604968) = 0.597454x6 = λx5(1-x5) = 2.5 ⋅ 0.597454 (1 - 0.597454) = 0.6012567

This time an attractor x = 0.60... appears. It means that if we keep oniterating the logistical series an infinite number of times its solutionwill tend to 0.60... (see picture 2.9).

Picture 2.9: An iteration of the logistical series with λ = 2.5. An attractor at x = 0.60...appears.

Now serious things are pointing out when λ is bigger than 3. Butfirst let is iterate the function with λ = 3.2.

x1 = λx0(1-x0) = 3.2 ⋅ 0.1 (1 - 0.1) = 0.288x2 = λx1(1-x1) = 3.2 ⋅ 0.288 (1 - 0.288) = 0.6561792x3 = λx2(1-x2) = 3.2 ⋅ 0.6561792 (1 - 0.6561792) = 0.721945


x4 = λx3(1-x3) = 3.2 ⋅ 0.721945 (1 - 0.721945) = 0.642368x5 = λx4(1-x4) = 3.2 ⋅ 0.642368 (1 - 0.642368) = 0.73514

It seems that the obtained results are oscillating between two valuesinstead of the lone value of picture 2.9. Thus there are not oneattractor but two. This phenomenon is called «doubling of period»and shows that the equation now admits two solutions (see picture2.10).

Picture 2.10: When λ is bigger than 3, two attractors appear, hence doubling of periodphenomenon. In fact the iterative process is like a kid who can not choose between avanilla ice cream or a bananasplit. He hesitates, makes one step toward up to the vanillaice cream, stops and thinks he would prefer a bananasplit, then makes one stepbackwards, stops, haunted by the wanting of vanilla ice cream. It is quite the same withthe case of the logistical equation. It (she) is attracted by the alluring pectorals of mister0.63, but once it (she) has them under hand, a sudden thought at mister 0.75's remarkablebiceps makes it (she) leave toward him until nostalgy of mister 0.63's pectorals strikesback. To get serious, the doubling of period, or bifurcation, makes the iterative processattracted by two attractors instead of only one.

It is realistic to think that if a doubling of period occurred when λ >3, there must be other ones as λ gets closer to 4. In order to illustratethis hypothesis, let is see what happens when λ = 3.95.

x1 = λx0(1-x0) = 3.95 ⋅ 0.1 (1 - 0.1) = 0.3555x2 = λx1(1-x1) = 3.95 ⋅ 0.3555 (1 -0.3555) = 0.905023x3 = λx2(1-x2) = 3.95 ⋅ 0.905023 (1 -0.905023) = 0.339527619x4 = λx3(1-x3) = 3.95 ⋅ 0.3395276 (1 - 0.3395276) = 0.885782x5 = λx4(1-x4) = 3.95 ⋅ 0.885782 (1 - 0.885782) = 0.39963029x6 = λx5(1-x5) = 3.95 ⋅ 0.3996302 (1 - 0.3996302) = 0.947707x7 = λx6(1-x6) = 3.95 ⋅ 0.947707 (1 - 0.947707) = 0.19575446x8 = λx7(1-x7) = 3.95 ⋅ 0.1957544 (1 - 0.1957544) = 0.621866x9 = λx8(1-x8) = 3.95 ⋅ 0.621866 (1 - 0.621866) = 0.92883642x10 = λx9(1-x9) = 3.95 ⋅ 0.9288364 (1 - 0.9288364) = 0.26109

It looks really strange, as if there was no attractor. Nevertheless,there is an infinite number of attractors, which inhibits orbitalbehaviour. That is to say that the iterating has become chaotic (seepicture 2.11).

Picture 2.11: The iterative process has become chaotic. There is an infinite number ofattractors, which inhibits orbital behaviour. Indeed, an infinite number of iterations wouldturn this picture into a black stain.

Establishing a proof of the cahotic behaviour of this iterativeprocess using logic is not very hard. So did Feigenbaum, which ledhim to an important discovery. When λ has a value between 1 and 3,hence an interval of 2 units, it was shown that the function admitsone attractor (see picture 2.9). Then a doubling of period isobserved when λ has the value λ = 3,2 (see picture 2.10). In fact,more precise calculations, which I spare in order not to bore thereader, would show that a doubling of period occurs in the interval]3;3.46[ (it means values from 3 not included to 3.46 not included).When λ has the value λ = 3.46, a new doubling of period, orbifurcation, leads to 4 attractors. Pushing forward calculationswould show a new bifurcation when λ = 3.56, then anotherbifurcation when λ = 3.5815, and so on (see picture 2.12).

Picture 2.12: It represents bifurcations with doublings of period of the logisticalequation. On the Ox axis are represented the values for λ and the Oy axis represents thevalues of the attractors after a very big number of iterations. When λ goes beyond 3 abiperiodic system appears, until λ reaches 3.46. Upon which a new doublement of periodoccurs, which leads to four attractors. After the value 3.56 there are eight attractors.After 3.5815 there are sixteen attractors, after 3.5861 thirty-two attractors, after3.587090815 sixty-four attractors, and so on.

But Feigenbaum, who had all the needed time to think at suchproblems during his slow computer was calculating these values, didwonder if there was not a key number explaining the order ofappearance of the bifurcations. It seemed to him the intervalsbetween new bifurcations were getting smaller following a


geometrical progression. Thus he did try to find out this magicalnumber, a bit the way we are doing now.

Between 1 and 3, which means an interval of 2 units, there isonly one attractor. Between 3 and 3.46, an interval of 0.46 unit,there are two attractors. Between 3.46 and 3.56, an interval of 0.1unit, there are four attractors. Between 3.56 and 3.5815, an intervalof 0.0215 unit, there are eight attractors. «Well, he said, how does itlook if I divide the first interval by the second one, then the secondone by the third one, then the third one by the fourth one, and soon?» The intervals have following respective values: 2; 0.46; 0.1;0.0215;... . 2 divided by 0.46 makes 4.3478. 0.46 divided by 0.1makes 4.6. 0,1 divided by 0.0215 makes 4.6511. And so on.Feigenbaum was right. He pushed forward his calculations andfound the magical number, which was: 4.6692016090, which is nowcalled Feigenbaum's constant. The knowledge of this constantallows to predict when doublings of period appear and how muchattractors has a function for a value of λ situated between 1 and 4.For instance picture 2.11, which represents the iterative processwhen λ = 3.95, has billions of billions of billions... of attractors. Iwould not have enough place with a ten meter high paper pile towrite all numbers needed for giving the exact amount of attractors.In a way, there is not an infinite number of attractors but still somany that it is considered in most cases as if it was infinite. ButFeigenbaum wanted to know if his discovery was a behaviour whichonly was linked to the logistical equation. So he took once more hisHP-65 and began the same work with a trigonometric function: Xn+1= λsinπXn. He found out the same number: 4.6692016090.Feigenbaum did discover something that nobody ever found outbefore him. In fact, Feigenbaum's constant appears in all serieswhich have chaotic behaviour, which means a lot of series.Nowadays Feigenbaum's constant has found applications inhydrodynamics, turbulence, and other domains of science.

In the frame of this dossier there are some concepts followingfrom Feigenbaum's discovery that must be kept in mind. Oddyenough, they all will be useful in the context of the study of fractals.The first concept is the iteration, which will help us discovering theMandelbrot set. The second concept is the invariance of scaleshowed by Feigenbaum's constant. It teaches us that this chaoticbehaviour (see picture 2.12) has got a denominator between alldoublings of period. Imagine that you could now dive far intopicture 2.12: you would always see the same bifurcations whichalways have the same Feigenbaum's constant proportional interval.The invariance of scale is one of the characteristics which givesfractals their fascinating power.


3. The study of chaos - where can it be found?

«Becoming aware of what happened in our mindcorresponds exactly to what happens in the nature isan incomparable experience, the best thing that couldhappen to a scientist. It is impressive every time ithappens. We get the surprise to discover that aconstruction elaborated by our own mind actually canbecome true in the real world. A big shock, a big,very big joy.»

Leo Kadanoff

Imagine a lone bacteria. Then a mitosis: there are now twobacterias. But after a couple of seconds, each of them duplicatesitself during another mitosis. There are now four bacterias, whoonce more duplicate themeselves, and so on. Soon there will bethousands, millions, billions of bacterias. It's quite the same with thetheory of chaos.

Since the first chaotic behaviours were studied by EdwardLorenz in 1960, then by Mitchell Feigenbaum at Los Alamos in theearly seventies, the field of its applications grew in a nearlyexponential way. At the beginning of this century Henri Poincarénoticed that some dynamic systems behaved in a strange way. Buthe was unable to push forward his observations in a mathematicallanguage. Mitchell Feigenbaum made conspicuous a specific way toa chaotic behaviour which is the «doubling of period». EdwardLorenz is well-known for having found a very important facet ofchaos which is the «strange attractor». At that time Russianscientists made important discoveries on chaos as well. But it is onlysince 1975 that chaos became popular thanks to the theory offractals from Benoît Mandelbrot. Since then, chaotic behaviourssuddenly became visible in more and more dynamic systems. Forinstance seismic activity, inversions of the polarity of the earth'smagnetic field, volcanic erruptions, fluid mixing, catastrophes onthe stock exchange market, evolution of populations, Darwinismadaptation, cerebral chaos, weak chaos and disasters, biologicalsynchronisation and stability of Solar system. We now will discusssome of the above listed chaotic phenomenons.

3.A. Stability of Solar systemAs strange as it seems, some planets do not quietly stay in their

stable orbits. In other words, some planets do not run the samecourse undefinitely. It is the case of the inner planets of Solarsystem. Newton's laws of mechanic have become obsolete if wewant to study the positions of the inner planets of Solar system. Ademoniac ghost called chaos slipped into the laws of mechanicgoverning astronomy. Therefore, Newton's laws of gravitation haveto be replaced by the famous K.A.M. theory. In fact, the instabilityof the inner planets of Solar system follows from a phenomenoncalled «resonance».

A phenomenon of resonance appears when Jupiter and Saturn arevery near from each other. Thus, their gravitic influence on the innerplanets becomes stronger. When another planet, for instance Mars,gets as close as possible from the Jupiter-Saturn pair, the planetsubstains a stronger attractive power than usual (see picture 3.1).

Picture 3.1: Resonance phenomenon. When distance d1 between Jupiter and Saturn is aslittle as possible, their attractive influence on the inner planets becomes stronger thanusual. Moreover, when both distances d2 and d1 are as little as possible, the attractivepower of the Jupiter-Saturn pair diverts the orbit of the inner planet very lightly.According to our knowledge of dissipative dynamic systems - dynamic systems whichare under the law of initial state subordination - the divertion grows as the time goes on.After a long period of time (some million of years) the position of the disturbed planetwill become totally unpredictable.

The result is that the orbit of Mars changes in an extremly weakway. Its speed and direction got disturbed. However it is a very littlechange, the perturbation will grow as the time goes on. In otherwords, the inner planets of Solar system are under the influence ofthe initial states subordination. All inner planets (Mercury, Venus,Earth, Mars) are disrupted by the resonance phenomenon. Theoutter planets (Uranus, Neptun and Pluto) have a stable orbitbecause they are too far away from the Jupiter-Saturn pair. JacquesLaskar did demonstrate the chaotic behaviour of the inner planets ofSolar system using numeric methods. According to his calculations,a lack of precision of 15 meters on the Earth becomes after 100million years a lack of precision of 150 million kilometers.Moreover, orbits of Mercury and Venus were studied over a relativelong period of time because they are very near from each other.Most of the time Mercury's orbit is included into Venus' one. Butaccording to a margin of uncertainty following from their chaoticbehaviour both planets have an extremly close orbit during somethousand years. During this period both planets have a chance tocollide with each other or pass so near frome each other thatMercury could be thrown out of Solar system. Of course, it is meantto be calculations over a long period of time such as 1 million year,but not to say that such a dangerous behaviour appears everydecade.

It would be interesting to overcome this instability so we couldexplain the history of life on earth, variations of climate, periods offrost and perhaps death of dinosaurians. But there are practicalproblems following from the initial states subordination whichprevent such studies. I will now attempt a brief history of thediscovery of chaos in Solar system.

3.A.1 A brief history of the study of Solar system


Nowadays we think that Solar system has been relatively stablefor some million of years because human being did evolve to what itis now. If the Earth's orbit had changed a lot during the past millionyears, human being would not have become so developed, or evenwould not have survived. But this principle on planet's orbits wasnot always obvious. Thus Greeks thought planets were «erraticstars», until the work of Hipparque, and later on Ptolémée, showedthe idea of regular movement. Greeks then represented planet'sorbits as «perfect» circular movements, which already meant a greatstep forward.

From 1609 to 1618, Johannes Kepler (1571-1630) suggested anew theory. Following from Copernicus' idea of placing the Sun atthe center of the universe, Kepler thinks that planets have anelliptical orbit around the Sun. After one loop the planet runs overthe same loop again and again. Once more this was an importantstep forward however his theory is not absolutely true.

1687 is Isaac Newton's (1642-1727) law of universal gravitationexpressed. This will allow big steps towards chaos. Indeed, allmasses have an attracting power among them. This leads toreformulation of Kepler's obsolete theory because, for instance,Jupiter is attracted by the Sun, Saturn by the Sun but by Jupiter aswell. Thus each planet disturbs the orbit of all others so that there isno reason to think at the planet's orbits as fixed and invariableellipses. At that time Newton did not have the needed mathematicaltools for finding out the orbit's disruptions following from his law ofuniversal gravitation.

Pierre Simon de Laplace (1749-1827) is the one who made thebiggest step forward by describing the whole solar system by asystem of differential equations including attractive power of eachplanet on the other ones. But there is still a problem: the resonancephenomenon of Jupiter and Saturn. This phenomenon occurs in aperiod of 2/5, which means that while Saturn makes two loopsaround the Sun, Jupiter makes five loops. Included into thedifferential equations of celestial bodies, the solution 2/5 is called a«little divisor». According to the above listed concepts, Laplace wasable to explain the location of Saturn following from an observationdated from 228 BC. Therefore, he did solve one question which wasto prove if Newton's law describes the planet's orbits. The answer isaffirmative but there is another question he could not answer, whichis: are the orbits stable?

Later on, Urbain Le Verrier (1811-1877), who did discover theexistance of planet Neptun according to irregularities observed inUranus' movement, tried to answer the second question usingLaplace's equations. Thus after a gigantic amount of calculations (hehad slave calculators to help him) he studied the orbit of eachplanet. On the whole he could not find exact solutions therefore hethought these equations were effective only over a short period oftime. The problem following from the little divisors implied aninitial states subordination phenomenon. Moreover, a mathematicalproblem, which forbids to finding solutions, occurred for somevalues of the little divisors.

Between 1892 and 1899, Henri Poincaré (1854-1912) tried tosolve these equations but could not. Thus he said it was impossibleto solve them therefore impossible to predict the planet's orbits overa long period of time. But everything was not finished. By basingthemeselves on Poincaré's work, Andreï N. Kolmogorov (1903-1987), Vladimir Arnold and Jürgen Moser (K.A.M.) diddemonstrate using a new approach that it was possible to solve theseequations under some conditions. Unfortunately K.A.M. theory cannot be applied to Solar system because the celestial bodies are muchtoo big. It means that we are sentenced to ignorance about the orbitsof Solar system over a long period of time. In other words we oncemore found a chaotic dynamic system which of course does notallow predictions over a long period of time, as well as formeteorology.

3.B. Weak chaos, disasters and self-organization

Remember the earthquake which happened in January 1994 inLos Angeles. Geologists said it was because of the instability of SanAndrea's fault. A big cause was needed for explaining such a bigdisaster. But little causes, like a butterfly's wing's beating, can leadto important consequences as well. Numerous systems, which arecomposed of million of members in interaction and which evolve toa «critical behaviour», can lead to a disaster following from a verylittle cause trigging off a chain reaction.

Let is discuss the dynamic behaviour of a sand heap. When youpour sand grain by grain on a plate, grains are forming a little heapwhich has a weak slope. From that time, when the slope becomeslocally too steep, some grains are sliding down, which forms a littleavalanche. The more the heap swells, the steeper the slope becomes.Therefore, avalanches are becoming more and more important as theheap grows up. But the heap stops growing up when the amount ofgrains which are being poured is equal to the amount of grainssliding down. Thus, the dynamic system reached a «criticalbehaviour». As a result, grains being poured may generate«disastrous» avalanches (a disaster which is proportional to the sandheap's height).

Furthermore, when the sand is being poured at constant speed,the amount of sand sliding down the slope varies in function oftime. Thus a graphic, whose axis are time and amount of sandfalling down, shows an irregular curve, whose variations have verydifferent durations. Such a curve is called «sparkling noise». Thiskind of sparkling noise is typical of a dynamic system which is verystrongly influenced by its past history, or past states. On thecontrary, «white noise» appears in a system which looses thememory of its history, or of its past states. In other words, whitenoise appears in all dissipative dynamic systems.

Other known examples of sparkling noise appear in nature: solaractivity, light given out by galaxies, flow of a river, electric currentgoing through a heating element. Moreover, a lot of «fractal»geometry is showed by nature: mountains, clouds, distribution ofgalaxies, turbulent fluids, and so on. In the case of a fluid, forinstance, the number of whirlpool contained into a sphere of radius ris proportional to rD, where D is the dimension of the dynamicsystem. When D is a fractionary number (fractionary dimensionswill be explained in chapter 4.A.) the distribution of whirlpools is ofa fractal type. Thus fractal structures and sparkling noise seem to bea spatial and temporal signature of all self-organized criticalsystems.

There is a lot of numerical models which show the characteristicsof self-organized critical systems. Let is take the example of seismicactivity. The geologists Beno Gutenberg and Charles Richter (theone who created Richter's scale) found out, in 1956, a link betweenthe number of strong earthquakes with the number of weakearthquakes. Thus the Gutenberg-Richter law says that the numberof annual earthquaques which generate an amount of energy E isconversly proportional to Eb. b is an universal constant which isequal to approximately 1,5. Moreover b does not depend from anyparticular geographic location. The Gutenberg-Richter law thusshows that there are more weak earthquakes than strong ones. So aregion which undergoes every year an earthquake of energy 100 (inarbitrary units) undergoes 1000 earthquakes of energy 1 in the sameyear as well. In spite of a lot of knowledge about earth tremours it isstill very hard, not to say nearly impossible, to predict earthquakes along time in advance because of seismic activity's typical chaoticbehaviour, hence initial states subordination. But here comes thedifference between chaos and weak chaos. The initial states


subordination of a chaotic system follows an exponential lawcontrarily to weak chaos which follows a law of raising numbers toa certain power (think at Gutenberg-Richter's law). This behaviourof «weak chaos» follows from the self-organized critical states,which already is a little piece of information decreasing the part ofuncertainty in previsions. Therefore, a major difference betweenchaos and weak chaos appears: a totally chaotic system is absolutelyunpredictable over a long period of time. This because the part ofuncertainty following from the initial states subordination grows toofast. On the contrary, a weak chaotic system allows previsions overa long period of time but accuracy decreases in a linear way as thetime goes on.

And what about the evolution of a colony of living organisms?Let is take a funny easy model called the «game of life», which wasinvented by the American mathematician John Conway in 1970.This game represents a dynamic system which tends to organizeitself. In other words, applying the rules of the game on a hazardousconfiguration turns it into a well organized structure. We initiallyplace pawns - which represent the organisms - by chance on asquare field. Each square contains at most one pawn and has eightadjoining squares. Then the state of each square is determined bythe following rules:1. When two squares adjoining a given square, which is either freeor occupied, are occupied, the state of the given square does notchange.2. When three squares adjoining a given square are occupied, a neworganism appears in the central square if this one is free, or if thisone is occupied the organism survives.3. In all other cases, the organisms die because of overpopulation orof solitude.Every time that the system reaches a stationary state - or criticalbehaviour - we add a pawn by chance on the square field, count thenumber of births and deaths until the system reaches a newstationary state, then add another pawn and so on (see picture 3.2).

Picture 3.2: The «game of life» as it looks when a stationary state is reached. The firstsquare field represents whatever stationary state whereas the second square field resultsfrom the addition of a pawn in P8. Before obtaining such a result, we first have to place acertain number of organisms randomly on a wide enough square field so that organismsdo not go to the sides of the grid, which bends the development of the game becausethere are not eight adjoining squares anymore. Then we apply the three rules which are:1. When two squares adjoining a given square, which is either free or occupied, areoccupied, the state of the given square does not change. 2. When three squares adjoininga given square are occupied, a new organism appears in the central square if this one isfree, or if this one is occupied the organism survives. 3. In all other cases, the organismsdie because of overpopulation or of solitude. Now that the stationary state is reached weplace a new organism in square P8 what makes the system's state unstable so thatchanges occurs in order to reach a new stationary state. This little change did not lead toan «avalanche», but if we placed the organism in S8 instead of P8, the two squaredfigures around Q6 and T6 would have diseappeared. Remember to make the parallel withreality, which is represented, for instance, by a sand heap or real avalanches. Of course,this game is a lot more interesting using a computer so that we can handle million of

squares over a very long period of time and see very complex underlying fractalstructures or moving objects appear.

Finally, after repeating this loop a great number of times, weobserve that the distribution of «avalanches» (think at the exampleof the sand heap) follows a law of raising a number to a certainpower. Thus the «game of life» gets organized following from thebehaviour of an «self-organized critical dynamic system».Moreover, the distribution of the occupied squares is of a fractaltype: the average number of occupied squares, in a circle of radius raround whatever occupied square, is proportional to rD, where D -the fractal dimension (see chapter 4.A.) - is equal to approximately1,7. Furthermore, the same law r1,7 describes other variants of the«game of life», such as a three-dimensional square field, adding ofpawn during the system's evolution, and so on.

Some studies suggest that the «game of life» could be applied inbiology. In order to explain it, let is consider the «game of life» as aminiature coevolutive system, where every square represents a geneof a very simple animal or vegetal specy. The gene takes either thevalue 1 or 0. Thus stability of each value depends of itsenvironment, in other words of the value of the gene amongneighbouring species. So coevolution process makes the system turnfrom its initial uncertain state into an organized critical state, whichhas very complex static and dynamic configurations. This theorywas developed by the biologist Stuart Kaufmann at the Pennsylvaniauniversity. According to this study, evolutive process would be atthe limit of chaos, in other words, following from a weak chaosbehaviour. Moreover, death of dinosaurians could be nothing morethan a disastrous avalanche within the framework of weak chaos.

Are fluctuations in economy a consequence of avalanches in anself-organized critical system? Oddy enough it behaves that way.But it is true that too many factors have to be studied, thereforeeconomical activity may not yet be simplified that way on. There isa lot of other self-organized critical dynamic systems. For instancethe brain, which is a system composed of a lot of parts ininteraction. Therefore, spreadings of information between neuronescould be a phenomenon of the kind «avalanche», so that suddeninspired ideas could be released by minor events.


4. Fractals

«Clouds are not spheres, mountains are not cones,shores are not arcs of circle, the bark of a tree is notsmooth, and the flash of lightning does not draw astraight line.»

Benoît Mandelbrot

What is the total length of British coast? It depends of the carewe are measuring it. Indeed, there are big shapes as well as verylittle ones such as splits in rocks. When we look at British coast in avery strongly magnificated way we of course see a lot more detailsthan when looking at it on a map. Thus its total length seems tobecome bigger as we measure it in a more precise way. The samephenomenon occurs when trying to measure a human's lung length.Indeed, human's lung is made of a very complex structure in order tostore as much oxygen as possible in a limited volume. But bothabove listed examples have something in common which is the«invariance of scale», which is typical to fractals. It means that suchobjects may be observed at very different scales without loosingtheir apparant structure. A fractal picture may be observed at anyscale - with a magnificient glas or miles away - it shows the samestructure. It is true that these examples are not really invariant ofscale because their structure will obviously change if you pushmagnification forward to atomic level. But the idea is still good andobjects matching invariance of scale in a perfect way may beinvented using geometrical representation and mathematics. So didBenoît Mandelbrot, who invented the name «fractal» in order tocharacterize these geometrical objects. This name comes from theLatin adjective fractus which means broken, and of the idea of afraction. So that the choice of these two ideas, fractus and fraction,becomes more understandable it is useful to say that fractals havefractionary dimensions, which shows the degree of fragmenting of ageometrical picture (see chapter 4.A.). Thus fractals do not have thethree dimensions 1; 2; 3 we are used to, but may have values like1.47; 0.9; 2.83... The concept of fractal is relatively young becauseit first appeared end of 1975 with the parution of BenoîtMandelbrot's famous book «Fractals: form, chance anddimension», then renamed to «The fractal geometry of nature».

So that it does not become too abstract, we are now going toexamine a fractal object come to life from a real situation. For thatlet is take a practical example which Mandelbrot had to deal with.Not much time after he found a job at I.B.M., he discussed withtransmission ingeniors about a problem which occurred ontelephone lines inside the I.B.M. building they worked in. A noiseappeared erratically on telephone lines transmitting informationbetween computers. It occurred under the form of undesirablepackets which sometimes erased transmitted information, thereforebrought on mistakes. Likewise, it was known to appear erratically,so that there were long periods without any disturbing noise as wellas short periods in which it appeared several times. In addition nounderlying structure in time succesion could be found because itoccurred randomly, or in other words, as if it was chaotic. In factthis actually was a chaotic dynamic system. But Mandelbrot, whohad great ability of representing things in a geometrical way, foundout something strange. His way of proceeding to study thisdisturbing noise's behaviour was to divide periods of transmission isalways more little intervals of time in which he counted the numberof errors which occurred. Thus he divided every hour in three partsof twenty minutes. Some of the periods were free of transmissionerrors, some other not. Afterwards he divided all intervals of twentyminutes by three and there once more were periods without

disturbing noise as well as others in which it appeared. In fact,Mandelbrot could not find any period of transmission in whicherrors were distributed within same intervals of time. Thus everysquall of transmission errors, which was either long or short,contained periods with disturbing noise and periods withoutdisturbing noise. Finally, he found a coherent geometricalexplanation which said the relation between periods without errorsto periods with errors stayed constant whatever the scale of time -one hour or one second - was. In other words, this distribution wasinvariant of scale. The way Mandelbrot did study distribution ofsquall errors on phone lines was not new but well-known under thename of Cantor set (see picture 4.1).

Picture 4.1: This is the Cantor set which was invented by the German mathematician

Georg Cantor (1845-1918) in 1899. It is a fractal picture of dimension: D = log 2log 3 = log32

≈ 0,63 (see chapter 4.A). Cantor's set is a topological undelying set E of a segment ofinterval [0; 1] (which means from 0 included to 1 included), which elements E1; E2;... ;

En can be written following the formula ∑n = 1

∞

2xn3n , where: xn ∈ {0; 1} (which means that

every x has either the value 0 or 1). These numbers are written in ternary notation without

using number 1. Thus first three values E1; E2; E3 are: E1 = [0, 1]; E2 = [ ]0 , 13 ∪ [ ] 23 , 1

; E3 = [ ]0, 19 ∪ [ ] 23 , 79 ∪ [ ] 89 , 1 . ∪ is a mathematical synonym for expressing thegathering of elements. In this case, Cantor's set E is equal to intersections shared by allsets E1; E2;... ; En (It means the addition of parts which are shared by all sets, whichrepresents the last part on the right of picture 4.1). This idea expressed in a mathematical

way is written as follows: E = ∩ ∞

n = 1

En. ∩ is a mathematical symbol for «intersection».

Such a formulation leads to the deduction that Cantor's set E is a closed set of lengthzero. In other words, Cantor's set E is composed of parts which are shared by all itsunderlying sets E1; E2;... ; En. But as we see on picture 4.1, underlying sets' lengths tendto zero because they loose 1

3 of their heigth every time that we go one step forward -every time that n gains 1 unit. In a way it is a pathological set because its existance isproved in a mathematical way but it has a heigth equal to zero, which represents a glaringparadox. There are other pathological examples, such as Van Koch's curve, Peano's curveor Weierstrauss' curve. Some of them will be explained in chapter 4.A.

Cantor's set has a fractal dimension, which is equal to D ≈ 0,63(see chapter 4.B.). In order to build Cantor's set consider an intervalwhich goes from zero to one. Then take its central thirdth away in


order to keep the two thirds which are on the sides of the initialinterval. Repeat this operation on each of the two obtained segmentsin order to obtain four segments of a length of one thirdmultiplicated by one third, which makes one ninth. Then repeat thesame loop an infinite number of times. At this point, the total lengthtaken away is equal to the initial interval however there subsists annon demonstrable infinity of points. Therefore, such a paradoxalbehaviour is called pathological set.

Thus Mandelbrot did not solve the problem of transmitted errorsbut was able to tell about its origin. It was not due, as the ingeniorsthought, to human activity disturbing the telephone net, but to thewhole net which had hardware problems. Thus they could not solvethe problem and adopted a strategy of information redundance inorder not to get disturbed by errors anymore. But the most importantlesson from this experience was not having helped these ingeniorsbut having discovered a link between one of these pathological setswith reality. Oddy enough Cantor's set described a real situation.This led Benoît Mandelbrot to study fractionary dimensions,invariance of scale, geometry of nature, distribution of galaxies, andfinally to create a new theory following from theory of chaos.

Another example of a fractal which explains a naturalphenomenon appears in molecular motion, which was studied by thefamous physicist Jean Perrin (1870-1942). Little particles insuspension in a fluid are animated by quick and irregularmovements which are brought about by thermic roughness of fluid'smolecules. If we trace a segment linking the locations occupied bythe same particle at very close instants we observe that thesesegments' directions vary in an absolutely irregular way, or in achaotic way. The suites of segments forms a pathological fractalcurve which has no derivables. If we observe the particles' locationin intervals of time which are two times smaller, each segment willbe replaced by two segments of a higher total length. Therefore, thisfractal curve has no finite perimeter. Moreover, magnifyingmolecular motion's graphic by a factor two and dividing theintervals of time by two would show exactly the same structure.Therefore, it shows invariance of scale, which prooves this curve isof a fractal type. Furthermore, molecular motion's fractal curve is ofdimension D = 2 (see chapter 4.A) because the curve fills the planein which it is contained in an homogeneous way.

Now we are going to dive far into abstraction of fractionarydimensions (chapter 4.A.) and iterative process (chapter 4.B.) tofinally emerge under a new concrete light diffused by fractalpictures (chapter 4.C.).

4.A. Heard about dimension 2,5?Humans are used to deal with only two dimensions. Granted that

they do not have the ability of flying, they usually move and think intwo dimensions, however technological improvements sometimesallow the use of the third dimension. Some other animals are used tothink in three dimensions because they move themeselves using allthree axis. It is more particulary the case of fish and birds. The mostinteresting thing is to study differences of formation between thebrain of an animal who travels in three dimensions with the brain ofan animal who travels in two dimensions. According to the laws ofDarwinism's evolutive process, living beings have to adaptthemeselves to the environment they are living in. Thus animals whotravel in three dimensions have a lot bigger cerebellum than otherswho make use of only two dimensions. For instance roaches or hensdo have a bigger cerebellum than frogs or snakes. It is known thatcerebellum is the part of the brain which controls coordination andequilibrium. Thanks to intensive use of his brain, human being wasable to find existance of one more dimension which attaches itself to

the three basic ones. Humans understood they lived in a four-dimensional world. Time, which is the fourth dimension, exertedinfinite fascination on human being and once humans understoodthey had a short lifetime it called forth culture, poetry, art,philosophy and science. The development of science suddenly led toa very uncomfortable certified report: we are not living in a four-dimensional world but in a world of fractionary dimensions.Britain's coast has a dimension of 1.25 whereas Portugal's frontierhas a dimension of 1.15. Such an overthrown of the mosttrustworthy values is enough to turn some brains to crazyness. I amchallenging anyone of being able to represent himself how adimension of 2.43 looks according to our traditional way ofmeasuring dimensions.

But there is no need to get shocked. This way of measuringdimensions has its reasons for being. The big advantage of fractaldimensions is that they describe an object without according anyimportance to the scale from which the observer looks at it, which isnot the case of the old way of measuring dimensions. Thus we saythat fractal dimensions show invariance of scale. So that it becomesmore understandable, let is take the example of an ant and a fox. Wenow consider the bark of a tree and the behaviour of these twoanimals in front of it. The fox will not be able to use the bark in anyway, or perhaps he could rub himself onto it. On the contrary, theant will be able to find shelter in bark's hollows because she is sosmall that she makes use of the bark's geometry, what the fox isunable to do. Ants and foxes are two animals which have such adifference of size that they see the world from an absolutelydifferent scale. A fractal dimension could show, because of itsinvariance of scale, the ability of the tree to give shelter. Anotherexample is the one of a wool ball. If looked at it from a certaindistance, it looks just like an one dimension point. Getting closer toit will show us a circle, which has two dimensions. Getting evenmore closer will show that this wool ball is made of string which isrolled up itself and therefore is a three dimension object. Describinga wool ball using fractal dimension spares us such a confusionbecause its dimension is invariant of scale, therefore the same fromany point we look at the wool ball. All fractal objects havefractionary dimensions, which makes a lot of objects in nature. Forinstance trees, clouds, water, whirlpools, distribution of galaxies,ferns, all have fractal structures, therefore a fractionary dimension.Moreover, researches are being made to find out fractal dimensionsto all objects of the universe. The French physicist Laurent Nottaleis even trying to find out a new theory which unifies generalrelativity and quantum physics. This new theory is based upon theidea of a fractal space-time continuum. I now will explain how tofind out the fractal dimension of objects.

Let is take a piece of string which we extend by another piece ofthe same length. The result is of course a new piece of string whichis twice as long as the initial one. Let is keep on with the sameexperience but now with a square. In order to obtain a square whichis twice as big as the initial one we this time need four squares.Using the same reasoning we need eight cubes to obtain a new cubetwice as big as the original one. Moreover, if we could manipulateobjects in four dimensions, we would need sixteen of these four-dimensional objects. Altogether, in order to double an hypercube ofdimension d we need c = 2d examplaries. Therefore, d, whichrepresents the number of dimensions, is determined making a simplealgebrical conspicuouseness which is:

log c = d ⋅ log 2

d = log c log 2 or: d = log2 c

In a more general way, it is more accurate to replace 2 by a variable.Thus we say that to multiply by a the measures of an object ofdimension d, we need c = ad copies of the initial object. As a resultthe formula then becomes:


d = log c log a or: d = loga c

Remember that c represents the number of needed exemplaries anda the number of times we want the initial figure to be multiplied by.Let is take a concrete example, which is Van Koch's curve, to makethis formula more understandable (see picture 4.2).

Picture 4.2: Van Koch's curve. In order to show how complex this curve becomes, wewould have to scale it down a lot of times because its length on Ox axis increases veryfast. This is a pathologicat curve as well as Cantor's set. What makes it pathological isthe fact that it has no finite perimeter. As the number of steps increases, Van Koch'scurve's perimeter increases as well so that its total perimeter is in a mathematical wayinfinite. When observing the first step, it becomes obvious that the picture's length hasbeen multiplicated by three, therefore a takes the value a = 3. Moreover, the initialpicture appears four times after one step, sixteen times after the second one. It is verylikely it would appear sixty-four times after the third step, after the fourth one twohundred and fifty-six times, and so on. Each step multiplies the previous number by four,hence a value for c which is c = 4. We are now ready to find out this picture's fractal

dimension: d = log c log a = loga c = log3 4 ≈ 1,26.

Each step multiplies the picture's length by three, therefore a takesthe value a = 3. Moreover, the first step shows very well that fourinitial pictures are needed to multiply its dimensions by three,therefore c takes the value c = 4. Now we are able to find the fractaldimension of this picture. A synonym for fractal dimension isHausdorff-Besicovitch dimension, which means approximately thesame. Thus its dimension is:

d = log c log a =

log 4 log 3 ≈ 1,26.

This is a pathologicat curve as well as Cantor's set. If we scale itdown by three after each step, this picture's length will stay in thesame interval but its perimeter will get always more uneven. Scalingit down and observing Van Koch's curve behaviour as the number ofsteps increases reveal this curve as a pathological one. Thus as thenumber of steps increases Van Koch's curve's perimeter increases aswell. Therefore, the perimeter is in a mathematical way infinite.Furthermore, another characteristic of this kind of curve is the lackof derivatives. As reminder derivatives are the inclination of thecurve's tangent at one point. But it is impossible to find anyderivative to Van Koch's curve because every point is composed ofan infinity of underlying starting curves so that it becomesimpossible to know the curve's declivity at one point, therefore itstangent. In order to illustrate fractal dimensions there are other verysimple geometrical objects on which fractionary dimension mayeasily be found (see picture 4.3).

Picture 4.3: Applying the same process as the one shown in picture 4.2, these curves allhave fractionary dimensions. Moreover they all show pathological behaviour. Thus theydo not have derivatives and have an infinite perimeter. It is very easy to calculate theirfractal dimension. Curve A needs five examplaries for being multiplicated by fourtherefore c gets the value c = 5 and a gets the value a = 4. Object B becomes four timesbigger when using its picture seven times. Objects C and D have something specialwhich makes fractal dimension a bit more suble to find out. Indeed, they both havesegments of different lengths. In other words they both have a segment twice as big asthe others ones. In such cases we take the most little segment as common denominatorthen count how many times it appears. Thus the bigger segments are counted two timesinstead of one time for all other segments, so that curve D needs six examplaries in orderto multiply its length by four. The same reasoning has to be applied to figure D.

Another interesting characteristic of fractal pictures is invarianceof scale. Hence their structure is the same at any scale from whichthe observer is watching at the object. It means that a fractal picturemagnificated to an infinite factor looks the same as if it was notmagnificated. In spite of the fact that invariance of scale seems notto call forth imagination, applying this characteristic on fractals givethem their fascinating power. Imagine yourself watching a nicecomplex coloured abstract picture, for instance in a museum.Traditional pictures we are used to are just a composition of verylittle units of colour, which is not the case of fractal pictures.Applying invariance of scale concept means that you technically areable to «dive into» this picture, which then shows you always morestrange structures. Many people got totally hooked by fractalpictures' beauty, what is more particulary the case of Arthur C.Clarke - the author of «2001: a space odyssey» - who givesnumerous conferences on fractals. Thanks to always increasingcomputer calculation power it is now possible to admire realtimemagnifications of fractal pictures, which is very impressive. But wefirst will have to prove invariance of scale with the help of somepictures. Michael Barnsley and his co-workers of Georgia Instituteof Technology did push forward the development of a special kindof fractals, which are good samples for showing invariance of scale.Therefore, we first will look at a Barnsley fractal tree (see picture4.4).


Picture 4.4: This fractal tree shows branches which repeat the same structure an infinitenumber of times. It seems branches have an end but magnification would show they aremade of always more little structures. The fractal tree is obtained using relative easymathematics. In this case, this tree is defined by four functions which are obtained on thebasis of a list of numbers. A mathematical definition of Barnsley's tree can be found inappendix.

However Michael Barnsley did call this picture a tree it looks in mypoint of view a bit more like a dandelion. I will now describe thistree. We climb up the trunk until first split is reached. At this pointthe tree shows a vertical axis of symmetry which divides the picturein two equal parts. Next we follw the right branch until next split isreached. There is another axis of symmetry, which has an incline offorty-five degrees. We once more follow the right branch until wereach next split. At this stade we of course see another axis ofsymmetry which this time has an incline of zero degree. Altogethereach split reveal an axis of symmetry which incline either losesforty-five degrees if turning to the right, or gains fourty-five degreesif turning to the left. Furthermore, all branches' length areproportional among themeselves. It means, for instance, that if wedivide the branch's length which is between split number three andtwo by the branch's length which is between split number two andone, we will find the same value as if we divide, for instance, thelength between split number twenty-four and twenty-two by thelength between split number twenty-two and twenty-one. In otherwords, Barnley's tree is invariant of scale since it is a fractal picture.This way of proceeding is similar to how Michael Feigenbaum didfind out Feigenbaum's constant (see chapter 2.B.). So thatinvariance of scale becomes obvious, I will now magnify the firstsplit of the tree's trunk (see picture 4.5).

Picture 4.5: Magnification has been achieved on the first split of the trunk. It reveals asplit made of the same previously observed structure. Enlarging the first split an infinitenumber of times would show the same structure to appear an infinite number of times aswell. Therefore, splits of this picture are pathological because they are made of an infinitenumber of underlying structures, so that they have an infinite perimeter. Moreover thesesplits seldom have derivatives.

Invariance of scale becomes obvious. The same structure appearsand it is likely it appears an infinite number of times as we magnifythe first tree's split. The example of the tree was used for itspedagogic value but this tree does not look very natural. Barnsleyand his co-workers wanted to put into equation natural objects inorder to obtain fractal pictures which are the exact copy of reality.Some of these pictures were not very accurate, but others lookedsurprisingly the same as in reality. It is more particulary the case ofan obtained fractal picture of a fern (see picture 4.6).

Picture 4.6: A fractal fern. However this picture is composed of a lot of points coming atrandom places, the set of all these points draws a fern. In fact, the fractal fern, which israther complex, is obtained following the same relative simple mathematical process asfor the fractal tree. A mathematical definition of Barnley's fern can be found in appendix.

Any biologist who is asked to give a name to picture 4.6 will nothesitate to give the partially right answer which is a fern. Very fewpeople will say it is Barnley's fractal fern, however it is the rightanswer. This example shows how small the difference between areal fern and a computer generated fern is. Oddy enough thiscomplex looking picture is obtained following from relative simpleequations (see appendix). We now are going to magnify the fern'stip so that invariance of scale is shown (see picture 4.7).

Picture 4.7: Magnification shows once more the same structure, which stays the same atany scale from which we look at this picture. Therefore, it shows invariance of scale.


We establish that exactly the same structure appears. Pushingforward magnification on the fern's tip would show that it is rolledupon itself. In fact, the tip of Barnley's fern does not have anyending because it is rolled upon itself an infinite number of times. Ifwe could keep on enlarging its tip we could see that it goes rounditself in order to form a spiral. It is true that it does not exactlycorrespond to reality. But real ferns are under influence of otherexternal factors which avoid such a geometrical construction.However, it is surprising to see nature which tends to get organizedfollowing very basic mathematical fractal shemes such as this one.

This chapter showed how to find fractal dimensions as well asinvariance of scale. These two themes are important characteristicsof fractals which will help us go beyond simple appearance whenwatching fractal pictures. But we now will get closer to technicalbirth of fractals by iterating complex functions.

4.B. Iteration of complex functionsIterative process is already known to us because it was described

in chapter 2.B. However, it is useful to once more repeat the way ofproceeding to iterate functions or equations. Thus an iteration iswhen the function's solution is used as new entry value in order tofind a new solution, which is once more used as new entry value,and so on. After a great number of iterations when the last obtainedvalues tend to be infinite or tend to zero, it is said that the iteratedfunction diverges. On the contrary, if the function's successivevalues tend to a number this function converges to this number,which is called an attractor. Chapter 2.B. described iterative processapplied to the logistical function, which is a function made of realnumbers. But there are imaginary numbers as well, what is thischapter's only technical topic. Instead of iterating a functioncomposed of real numbers we will this time iterate a complexfunction containing imaginary numbers. Imaginary numbers' maincharacteristic is to have negative squares. As reminder it isimpossible to find a solution for the square root of negativenumbers. But it becomes possible if you allow the use of complexnumbers. Moreover complex numbers always have a part which isreal - traditional numbers - and the other made of imaginarynumbers. So that it loses of its abstraction it is useful to practisecomplex's numbers manipulation by finding some of these numbers'location on a graphic.

The first example is complex number: Z = 4 + 2i. At first it iswise to describe this number. Thus, complex number Z is composedof a real number, which value is 4, and an imaginary number, whichvalue is 2. Hence two informations are known which leads, in theframe of a graphical representation, to the deduction that to eachvalue corresponds an axis. Therefore, real numbers are assigned thehorizontal Ox axis and imaginary numbers get the vertical Oi axis.Such a graphic is called complex plane (see picture 4.8).

Picture 4.8: The complex plane where imaginary numbers are represented on vertical Oiaxis. As reminder there are five sets of numbers which classify all known numbers. Theyare N (entire natural numbers), Z (entire rational numbers), Q (rational numbers), R (realnumbers), C (complex numbers), where C includes all other sets, R as well excepted C,and so on. This picture shows a representative plane of set C in which every pointrepresents a complex number of the form: Z = x-coordinate + i-coordinate. So that acomplex function may be represented onto a graphic it is wise to take away one of itsvariables. If not taken away, this complex function's graphic looks like a big stainbecause every point of the complex plane is then a solution of the function. Thus we setarbitrary imaginary value 1i to complex number Z so that function: ƒ(Z) = Z2 thenbecomes: ƒ(x + 1i) = (x + 1i)2.

Now that geometrical representation is explained efforts have to bedone to understand algebrical characteristics of complex numbers.For that the example of complex number: Z = (1 + 2i)2 will beuseful. As reminder, complex numbers raised to a pair power givenegative values. Consequently the value of Z becomes: Z = 1 + 4i +(2i)2 = 1 + 4i - 4 = -3 + 4i. Representing a complex function insteadof a simple equation is a bit more subtle because all complexnumbers have two variables, one which is real, the other which isimaginary. The problem following from two variables is that allpoints of the complex plane are a solution of the complex function,therefore the graphic will look like a big stain. Thus a way ofshuning this problem is to take away one variable by chosing a valuefor it so that a well looking graphic is obtained (see picture 4.8).

The required theory has been explained so that it now will bepossible to find out Benoît Mandelbrot's famous set. Mandelbrot didstudy the iterative behaviour of a particular complex function whichis: ƒ(Z) = Z2 + C. Because this function is going to be iterated it ismore accurate to write it in a recurrent form so that it shows by itselfit has to be iterated. Therefore, it becomes: Zn+1 =Zn 2 + C. FollowingMandelbrot's reasoning, divergent behaviours - when the function'svalue tend to be infinite or tend to zero - are not part of the setwhereas convergent behaviours - when the function's value tend to aparticular value - are part of the set wich is linked with Mandelbrot'sname. Moreover, explanations about Mandelbrot's recurrentcomplex equation have to be given. The difference between a lot ofvery basic fractals sets, pictures - for instance between Julia's set andMandelbrot's set - is the value of Z0. In the case of Mandelbrot's set,complex number Z0 has the value zero. Changing this value can leadto absolutely different results, hence pictures. The other point to bediscussed is the value of complex number C. C represents the pointfor which we want to know if the recurrent complex equationconverges or not. If it converges, point C is part of Mandelbrot's set.Benoît Mandelbrot's idea was to iterate all points of a complex planethen give a black colour to all points which are part of the set, sothat he eventually could see an underlying structure appear. Thisway of proceeding is very slow, indeed, fractal pictures are to beseen only since computers with a certain calculation power


appeared. A human iterating by himself 64000 different points -which corresponds to a very low computer screen resolution - at arate of twelve hours a day would need two and a half year tocomplete this task. Nowadays an average personal computer is ableto perform this task within one second.

In spite of slowness of human calculation power, I think it is agood idea to iterate some of the complex plane's points so that theabove listed concepts become a matter of course. At first, let is takepoint C = 1 + 1i. Iterative process then looks as follows:

Z1 = Z0 2 + C = (0 + 0i)2 + 1 + 1i = 1 + 1iZ2 = Z1 2 + C = (1 + 1i)2 + 1 + 1i = 1 + 3iZ3 = Z2 2 + C = (1 + 3i)2 + 1 + 1i = -7 + 7iZ4 = Z3 2 + C = (-7 + 7i)2 + 1 + 1i = 1 -97iZ5 = Z4 2 + C = (1 + -97i)2 + 1 + 1i = -9407 - 193i

Obviously the function diverges, therefore the point which waschosen belongs not to Mandelbrot's set. At present let is try withanother point, which is: C = -1 + 0i.

Z1 = Z0 2 + C = (0 + 0i)2 - 1 + 0i = -1Z2 = Z1 2 + C = (-1)2 - 1 + 0i = 0Z3 = Z2 2 + C = (0)2 - 1 + 0i = -1

Oddy enough the function's value oscillates between 0 and -1. Butaccording to chapter 2.B., oscillation in iterative process shows theexistance of two attractors, therefore the function converges, whichmeans that the point represented by the value of C belongs toMandelbrot's set. Following Mandelbrot's reasoning this pointbecomes black. Thanks to computer's calculation power we do nothave to iterate by ourselves all points of the complex plane forobtaining the fractal picture which represents Mandelbrot's set (seepicture 4.9).

Picture 4.9: A fractal picture of a complex set which is Mandelbrot's set. Some guess itlooks like a snowman, others like a gingerbread. In fact it is neither the first guess nor thesecond one but a simple geometrical representation of the solutions of iterative processapplied on complex function: ƒ(Z) = Z2 + C. So that a link with reality is established inorder not to get lost in abstraction, it is useful to say that some chaotic dynamic systemsmay be represented by complex functions such as this one. Thus when iterating a realchaotic dynamic system expressed under the form of a complex function in order to findout an «image of chaos» we speak of the «linearisation of a dynamic system». It isfascinating to think at these fractal pictures as signatures of chaos. A Turbo Pascalprogram which shows how to find out fractal pictures can be found in appendix.

Of course, the very basic Mandelbrot set is not representative of thefractals' beauty, which makes them so popular. For instance, fractalshave colours and sometimes are in three dimensions. I will nowexplain the technical path to colours and later on to three-dimensional representation of fractals, which is linked with thecolours' values.

Iteration achieved on the complex plan's points shows eitherconvergent or divergent behaviour. It was said above that whenconvergence occurs the point C gets a colour, which was black but

could as well be blue, yellow, cyan, or any other colour. But inorder to give colours to points which are not parts of Mandelbrot'sset or any fractal set, it is important to study divergence's behaviour.In fact it is the basis on which colours are given to points. Thus thespeed of divergence has to be studied, which means for instance thatafter ten iterations the function's value for a point Ca is a lot biggerthan the function's value for Cb, therefore Ca will get a brightercolour than Cb. Consequently we have to decide of a whole scale ofdivergence speed, which indicates for instance that the fastestdiverging points get a bright colour like cyan or yellow hence theslowest diverging points a dark colour like magenta or purple. Sothat a realistic scale of divergence may be found, a good idea is todivide the intervals of divergence speed by the number of colourswe want to use. In other words, let is consider the value of thefastest diverging point after one hundred iterations. Now let isobserve the value of the slowest diverging point after the samenumber of iterations. Then we compare these two values and noticea big difference of value, which represents an interval of, forinstance, one million. Further on we want to use one hundredcolours. Thus for finding out a realistic scale of divergence speed inorder to give colours to the complex plan's points, the interval has tobe divided by the number of used colours. In the case of myexample, one million, which represents the interval, has to bedivided by one hundred, which represents the number of colours.The obtained result, under these circumstances of value tenthousand, represents the rungs' value which decide of the point'scolour. For example a point which has value fourty-four after onehundred iterations does not cover first rung, therefore gets thelowest colour value. Indeed, such a point, for instance inMandelbrot's set's case: C = -1 + 0,4 (see its location on picture 4.9to get a better understanding of this concept), does diverge veryslowly because it is at the fractal set's frontier. Another example is apoint which has value fifty-four thousand after one hundrediterations. In this case five rungs are covered therefore point C getscolour value number five. In fact this is one of the easiest methodsfor assigning colours to divergent points but there are other morecomplex ones, which may give better good-looking results. Thuslogarithmic method creates rungs which are very little for slowdivergence but which grow in a logarithmic proportional way as thespeed of divergence increases. Such a way of assigning coloursallows to give a better differentiation of all points which are veryclose to the fractal set. Thus, according to the studied example,when taking picture 4.9 as basis, the point C of value C = 1 + 1i wedid iterate above would get a relative bright colour because of itshigh divergence speed.

Three-dimensional representation is even easier to understand. Itis meanwhile important to have understood the way colours aregiven to all complex plane's points for reaching a really goodassimilation of the use of three dimensions. Thus let is take theexample of a basic fractal coloured picture for instance. As it wasexplained above, colours are not given by chance but according todivergence speed. Consequently we now will add a third axis, whichis perpendicular to both other axis. This new axis, called Oz, doesnot yet change anything at the way the picture looks because wewatch the fractal in a parallel direction to Oz. Now the point'scoulour will define these points' location on Oz axis. For instancebright colours mean high values on Oz axis whereas dark colourslow values. Thus we now have three coordinates for each point ofthe complex plane but the picture still looks two-dimensional. Sothat three-dimensional representation is achieved the picture has tobe observed from another direction which is not parallel to Oz.Therefore, rotating the picture reveal it as being in three dimensions.To summarize, all points are given a certain colour whichcorresponds to their speed of divergence. Then these colours define


a certain value on the third axis. Finally rotating the picture showsthree-dimensional representations.

In conclusion, chapter 4.B. taught in an exhaustive way how tofind coloured three-dimensional fractals. At first sight it does notseem as if great information leading to very interesting results weregiven however the key to fantastic pictures was given. Indeed,zooming on a three-dimensional fractal allows to dive into animaginary world which has absolutely no limits because of itsinvariance of scale characteristic which guarantees an infiniteamount of infinitely little details. Moreover these pictures may begiven certain colours and structures so that they match reality in anastonishingly realistic way. For instance flight simulators designedfor commercial or army pilots have fractal landscapes, imaginaryclouds, imaginary mountains, rivers or trees which all look soincredibly realistic however these landscapes directly follow fromabstract complex numbers, which are made of imaginary numbers.Oddy enough imaginary numbers gave birth to imaginary landscapeswhich look so realistic. Or was it the opposite which was imaginary?Further some nice little philosophical play on words may be createdpushing forward this concept. I let the reader juggle a bit with hisintellect in order to find out what they are and their funnyimplications.

We are going in next chapter to study some representative fractalpictures and discuss them according to the above explained theory.

4.C. Fractal picturesAfter discussing so many abstract concepts, which required a

more or less exigent use of our brain's power to be understood, wefinally emerge under a new concrete light diffused by fractalpictures. Thus it will be possible to discuss these pictures in a moreexhaustive way which goes forder than simple appearance. Theshowed pictures were chosen for their representative value of fractalcharacteristics. Moreover, some of them look really very attractive,which is more particulary the case of three-dimensional pictures.Thus first part is composed of two-dimensional pictures whereassecond part shows astonishingly realistic landscapes. More powerfulcomputers than usual PCs which can be bought on the market maygenerate landscapes using a lot higher resolution than 1024 pixelswidth multiplicated by 768 pixels height, which corresponds to theresolution of all coloured fractals shown in this dossier. As a resultthese very high resolution fractal pictures look absolutely incrediblyrealistic.

We first are going to describe relative abstract pictures beforecoming up to dessert. However, it should not be forgotten whenwatching landscapes that these pictures after all only are the resultof complex functions' iterations. The following commentscorrespond to the order in which the pictures appear.

Picture 1: This is Mandelbrot's set with a logarithmic palettewhich assigns colours to pixels. Mandelbrot's set is represented bythe middle white parcel. Contrarily to picture 4.9, the interestingpoints are the ones which do not belong to the set because theirdifferent divergence speed, which is showed by the points' colourstherefore by divergence speed rungs, brings a new structures toappear. The structures following from different colours' parcels arecoloured fractals' most attractive characteristic, as it is obvious onpicture 2.

Picture 2: This is a fractal picture made by myself. In fact, thereis not a single point which belongs to the set. In other words none ofthese pixels converges, therefore such a black and white picturewould look completely white according to example of picture 4.9.

Thus observed parcels of colour represent intervals of divergencespeed. I obtained this picture thanks to invariance of scalecharacteristic which allowed me to magnify a very little detail of afractal picture which did not look as well as this one. Pushingforward magnification on the center of the «propeller» would revealalways more new fascinating structures.

Picture 3: «Lambdafn» by Michael Coddington. Any fractallover already did admire this fantastic picture. In fact it is a Julia setwhich has been a bit modified in order to look as nice as possible. Ithink the result is conclusive.

Picture 4: «Atomic glow» by Pieter Branderhorst. This one lookslike atoms, whence its name. Moreover, one of its remarkablecharacteristics is to give a feeling of three dimensions however it isa perfectly two-dimensional fractal picture. Watching this picturemakes us want to dive between atoms and look what is behind them.Unfortunately since this picture is two-dimensional it is impossibleto see anything behind these pseudo atoms, which would not be thecase of a three-dimensional fractal. Anyway, next picture shows azoom on this fractal's upper third.

Picture 5: A magnification achieved on Pieter Branderhorst's«Atomic glow». Thanks to invariance of scale this magnificationshows new details which were not visible on previous picture. Theatoms in background seem as if they are dissembled by a blueshadow, which contributes to give an impression of depth. It is hardto keep in mind that this picture in fact is made of only two axis.

Picture 6: «Ribbon and bows» by Pieter Branderhorst. This onewas chosen for its unusual aspect as well as for its representativeability of invariance of scale. Thus infinite spirals are easily visible.

Picture 7: «The Matterhorn». This is the first three-dimensionalfractal which is shown in this dossier. Furthermore, this picture isnot made of only one fractal but is a composition of two ones. Thefirst one is the sky, which is two-dimensional but seen from a pointon a third axis. The second one is the mountains, which are a bitspecial because they even show shadow effect as well as snow ontheir heigths. Giving the aspect of snow is not very difficult. Thusthe pixel's colour depends of its position on vertical axis as well asof the slope of the mountain at this point. Rendering shadow is a bitmore subtle and requires more complex mathematics.

Picture 8: «El Capitan, Yosemite Valley, CA». This fractalpicture, which looks surprisingly realistic, is a bit more complexthan the previous one. Thus this picture is composed of fourdifferent fractals which are the sky, mountains, trees, and river.Furthermore, there are many types of trees which are generatedfollowing approximately the same process as for Barnsley's tree.These trees «grow» according to the same criterions as for setting acolour to pixels. Thus height and slope of the mountain determine ifa specy of tree either may grow or not, as it is the case in reality.

Picture 9: «Lyapounov» fractal. This one represents a desolatedplace. Anyway, it is interesting to observe the left stone's pike'sshadow which is put out of shape by a wave. In fact, a Lyapounovfractal is made of a set ,which represents a concrete physical chaoticdynamic system, discovered by A. Lyapounov. It is thereforeawesome to realize that this picture is a real dynamic system whichis translated with the help of iterations into a wonderful landscape.

Picture 10: «Druids». A more appropriate title would be in mypoint of view «Stonehenge». This is a very interesting fractalbecause it obviously does not represent a natural landscape but an


edifice erected by human, or by any intelligent being. In factmountains and the stone ring are two different fractals which are puttogether to render the remarkable impression of a non naturalarchitecture.

Picture 11. «Grand Canyon of the Tuolumne, C». This picture ismy favourite one because it nearly would be possible asserting it is aphotograph taken during one's holidays. An interesting fact is toobserve the remote mountains' colours which become pale, as inreality. In fact this optical illusion is easy to render: the computercalcluates the distance between the observer and all points of thepicture then fades the colour proportionaly to the obtained removal.Soon people will be able to create fractal pictures in order to maketheir neighbours beleive they spent marvelous holidays...


Appendix

Barnsley's treeWe saw in chapter 4.A., pictures 4.4 and 4.5, that Michael

Barnsley's tree is made of a very strict structure. In fact it is notsurprising because this tree is obtained using relative easymathematics. Thus Barnsley's tree is defined by four functions basedon a list of numbers. Each line of the list of numbers, whichrepresents the mathematical definition of the tree, contains one ofthe generating function. The list looks as follows:

a b c d e f p0 0 0 0.15 0 0 0.05

0.42 -0.42 0.42 0.42 0 0.2 0.40.42 0.42 -0.42 0.42 0 0.2 0.40.1 0 0 0.1 0 0.2 0.15

The values of each line define one function which is composed of amatrix, a vector and a probability. Finally, these functions areiterated and their solutions are represented on a graphic. Thesefunctions' values thus are defined as follows:

matrix vector probabilitya bc d

ef

p

Barnsley's FernThe well looking fern of Michael Barnsley (pictures 4.6 and 4.7)

is obtained following exactly the same process as for his tree. This ismore surprising because the fern looks really the same as a real one.However, the only difference between the fern and the tree are thevalues of the numbers' list. Therefore, the four functions describingthis fern are defined according to the following list of numbers:

a b c d e f p0 0 0 0.16 0 0 0.01

0.85 0.04 -0.04 0.85 0 0.16 0.850.2 -0.26 0.23 0.22 0 1.6 0.07

-0.15 0.28 0.26 0.24 0 0.44 0.07The values of each line define one function which is composed of amatrix, a vector and a probability. Finally, these functions areiterated and their solutions are represented on a graphic. Thesefunction's values thus are defined as follows:

matrix vector probabilitya bc d

ef

p

Turbo Pascal fractal representationprogram

Iterative process achieved on complex functions indeed is veryeasy and does not require heavy mathematical tools. Such arepetitive mathematical process is what computers do at besttherefore showing how to find out Mandelbrot's set's picture wasimperative. Notwithstanding great computer calculation power, thisTurbo Pascal program is a lot slower than the one second limitmentioned in chapter 4.B. In fact this slowness is due to the wayTurbo Pascal translates its information to the computer. Thisprograming language has become very popular thanks to the fact it

is very easy to handle, but on the other hand it slows down a lotcalculation speed. A less popular language because of its complexitybut which is at least ten times faster is Assembler language. Indeed,a well programed assembler routine of Mandelbrot's set mightyeldingly give a picture of this set within less than one second. Onthe other hand only very few people master Assembler languagegood enough to produce such a program however it is a relativebasic one. Altogether Turbo Pascal's biggest advantage is to offervery easily understandable listings which allow even people notknowing this programing language to manage understanding theselistings' guidelines. This program was written by H. Cattin.

1. uses crt,graph;2. var gd,gm,m,n,nbiter:integer;3. a,b:real;4. procedure dessinepoint(x,y:real;couleur:word);5. begin setcolor(couleur);6. putpixel(320+round(150*x),240-round(150*y),couleur)end;7. procedure iter(a,b:real;var nbiter:integer);8. var x,y,u:real;9. begin10. nbiter:=0;x:=0;y:=0;11. repeat u:=x*x-y*y+a;y:=2*x*y+b;x:=u;nbiter:=nbiter+112. until ((x*x)+(y*y) > 1000) or (nbiter = 100) or keypressed13. end;14. begin15. gd:=detect;initgraph(gd,gm,'c:\utils\tp');clrscr;setgraphmode(gm);16. for n:=-320 to 320 do for m:= -240 to 240 do begin a:=n/150;b:=m/150;17. iter(a,b,nbiter);18. If nbiter = 100 then dessinepoint(a,b,black) else dessinepoint(a,b,white);19. end;20. readln;21. closegraph;22. end.

Some explanations have to be given so that even non programersunderstand this routine's guidelines. If anyone wants to createanother set than Mandelbrot's one, a very easy method is to changethe value of Z0 in complex function ƒ(Z) = Z0 2 + C. If Z0 is not equalto zero the obtained picture corresponds to one of Julia's sets.Changing the value of Z0 is very easy because Z0 's value is definedin line 10. Therefore, if someone wants to set value Z0 = 2 - 3i thisline will look as follows: nbiter:=0;x:=2;y:=-3; This little changeleads to an absolutely different picture. Next line defines the form ofthe function, which is in this case Z0 2 + C . u represents the real partafter iterating the function whereas y the imaginary part. But thisline needs too many explanations which certainly would bore thereader, therefore I will skip them. Next line shows under whatcircumstances the computer has to stop iterations for a given valueof C. There are three circumstances which are following ones: 1. Ifthe square of addition of both real and imaginary parts is bigger thanone thousand. 2. If the number of iterations exceeds one hundred. 3.If a key is pressed. Of course, values 1000 and 100 may either belowered in order to gain speed - however the picture then loosesaccuracy - or raised in order that the picture gains accuracy -however the process then looses speed. If changing the number ofiterations do not forget to enter the same value in line 18. Line 15 isimportant because it specifies the path where Turbo Pascal iswritten. If not entering the right path this program will not be able torun. Do not forget as well that if you want this program to run youwill need the required Turbo Pascal libraries which are crt.lib andgraphic.lib.


IndexA

algorithmic complexity...................................................... 4Andreï N. Kolmogorov.................................................... 12Arnold Vladimir .......................................................... 4; 12Assembler ........................................................................ 22astronomy ................................................................ 3; 4; 11astrophysics ....................................................................... 2atmosphere................................................................. 3; 4; 6avalanche ................................................................... 12; 13

B

Barnsley Michael........................................... 16; 17; 20; 22Belousov-Zhabotinsky reaction (B.Z.) .............................. 7bifurcation ......................................................................... 9billiard ............................................................................... 3biology..................................................................... 3; 4; 13brain..................................................................... 13; 15; 20British coast..................................................................... 14bromate.............................................................................. 7butterfly effect ............................................................... 4; 7

C

Cantor Georg ....................................................... 14; 15; 16cerebellum ....................................................................... 15cerebral chaos .................................................................. 11cerium................................................................................ 7chance................................................2; 3; 5; 11; 13; 14; 19chemistry ........................................................................... 3Clarke Arthur C. .............................................................. 16complex function ................................................. 18; 19; 22complex plane............................................................ 18; 19computer................................4; 6; 9; 13; 16; 17; 19; 21; 22computer screen resolution........................................ 19; 20convective wave ................................................................ 6convergence................................................. 7; 8; 18; 19; 20Conway John ................................................................... 13Copernicus....................................................................... 12critical behaviour ....................................................... 12; 13curve

Peano's curve ............................................................... 14Van Koch's curve................................................... 14; 16Weierstrauss' curve ...................................................... 14

D

Darwinism ................................................................. 11; 15derivative ................................................................... 16; 17determinism ............................................................... 2; 3; 7developed turbulence......................................................... 4dinosaurians............................................................... 11; 13disaster............................................................................. 12divergence ....................................................... 8; 18; 19; 20doubling of period ................................................... 7; 9; 11

E

Earth...........................................................................11; 12earthquake ........................................................................12economics.......................................................................3; 4Einstein Albert ...................................................................5electrons .............................................................................4equations

differential equations................................... 3; 4; 6; 7; 12logistical equations...............................................8; 9; 10recurrent equations ...............................................7; 8; 18

F

Feigenbaum Mitchell............................... 7; 8; 9; 10; 11; 17Feynman Richard P. ...........................................................5Fields medail ......................................................................4fluid mixing......................................................................11fractal landscape...............................................................20fractal picture .......................................... 14; 16; 17; 19; 20fractal space-time continuum ...........................................15fractionary dimension...........................................13; 15; 16

G

game of life ......................................................................13gene ..................................................................................13geodesics ............................................................................4Georgia Institute of Technology.......................................16Gutenberg Beno .........................................................12; 13

H

Hausdorff-Besicovitch dimension ....................................16Heisenberg Werner ............................................................5Hénon Michel.....................................................................7Hipparque.........................................................................12hydrodynamics .................................................................10

I

I.B.M................................................................................14initial states subordination............................. 3; 4; 7; 11; 12ion ......................................................................................7iteration ............................................................. 7; 8; 10; 18

J

Jupiter.........................................................................11; 12

K

Kadanoff Leo ...................................................................11Kaufmann Stuart ..............................................................13Kepler Johannes ...............................................................12Kolmogorov Andreï N. ......................................................4

L

Laplace Pierre Simon ...................................................5; 12lasers ..................................................................................4Laskar Jacques .................................................................11


Le Verrier Urbain ............................................................ 12Lorenz Edward .................................................... 4; 6; 7; 11Los Alamos.................................................................. 7; 11Los Angeles ..................................................................... 12

M

magnetic field .............................................................. 7; 11Mandelbrot Benoît......................................... 11; 14; 18; 25Mars................................................................................. 11Massachusetts Institute of Technology M.I.T.................... 6mathematics ...........................................3; 4; 14; 17; 20; 22matrix............................................................................... 22Mercury ........................................................................... 11meteorology................................................................... 3; 4Microsoft Word ................................................................. 2molecular motion............................................................. 15molecules..................................................................... 4; 15Moser Jürgen ............................................................... 4; 12

N

Neptun ....................................................................... 11; 12neurones........................................................................... 13Newton Isaac ......................................................... 6; 11; 12Nice observatory................................................................ 7Nobel price .................................................................... 4; 5non euclidian geometry...................................................... 4Nottale Laurent................................................................ 15number

complex number .................................................... 18; 20entire natural number................................................... 18entire rational number.................................................. 18imaginary number .................................................. 18; 20rational number............................................................ 18real number.............................................................. 8; 18

O

orbitalmost periodic orbit...................................................... 6periodic orbit ................................................................. 6

P

Pennsylvania university ................................................... 13Perrin Jean ....................................................................... 15philosophy ................................................................... 3; 15photosynthesis ................................................................... 4physics

physics of particles ........................................................ 3quantum physics .............................................. 2; 4; 5; 15

Pluto ................................................................................ 11Poincaré Henri....................................................... 3; 11; 12Portugal's frontier ............................................................ 15Prandtl, number of ............................................................. 6principle of uncertainty...................................................... 5probability ............................................................... 2; 5; 22Ptolémée .......................................................................... 12

Q

quantum optics........................................................... 4; 7; 8

R

Rayleigh-Bénard ................................................................6redox pair ...........................................................................7relativity .......................................................................4; 15research ......................................................................3; 4; 7resonance................................................................4; 11; 12Richter Charles...........................................................12; 13

S

San Andrea's fault ............................................................12Saturn .........................................................................11; 12seismic activity...........................................................11; 12self-organization.......................................................1; 4; 12set

Cantor's set ...................................................................14fractal set ......................................................................19Julia's set ..........................................................18; 20; 22Mandelbrot's set ........................................ 18; 19; 20; 22pathological set ......................................................14; 15

sociology ............................................................................4Solar system .................................................. 1; 4; 5; 11; 12solar wind...........................................................................4space of phases...................................................................6sparkling noise .................................................................12stationary state..................................................................13strange attractor................................................... 4; 6; 7; 11system

chaotic system ....................................... 3; 4; 6; 7; 12; 13coevolutive system .......................................................13dissipative dynamic system ....................................11; 12dynamic system ......................... 4; 5; 6; 7; 11; 12; 13; 19linearisation of a dynamic system.............................4; 19self-organized critical system.................................12; 13

T

theoryergodic theory ................................................................4K.A.M. theory ....................................................4; 11; 12theory of chaos ...................................... 2; 3; 4; 5; 11; 15

thermic convection .............................................................6topology .............................................................................4trigonometric function......................................................10Turbo Pascal ..............................................................19; 22

U

unpredictability ..................................................................3Uranus ........................................................................11; 12

V

vector ...........................................................................6; 22Venus ...............................................................................11

W

weak chaos ...........................................................11; 12; 13white noise .......................................................................12

Y

Yoccoz Jean-Christophe...............................................4; 25


References

Books:- Chaos et déterminisme, A. Dahan Dalmedico & J.-L. Chabert & J.-C. Yoccoz, Seuil Ed.1992

- Le hasard et la nécessité, Jacques Monod, Seuil Ed.1970

- Patience dans l'azur, Hubert Reeves, Seuil Ed.1988

- La théorie du chaos, James Gleick, Flammarion Ed.1989

- Une brève histoire du temps, Stephen Hawking, Flammarion Ed.1989

- Entre le temps et l'éternité, Ilya Prigogine & Isabelle Stengers, Flammarion Ed.1992

- Les objets fractals, Benoît Mandelbrot, Flammarion Ed.1995

- L'optique quantique, Peter Knight, Flammarion Ed.1993

- Physique des systèmes loin de l'équilibre et auto-organisation, Grégoire Nicolis, Flammarion Ed.1993

- Qu'est-ce que le chaos pour que nous l'ayons à l'esprit?, Joseph Ford, Flammarion Ed.1993

- Hasard et Chaos, David Ruelle, Oldie Jacob Ed.1991

- Les mathématiques, Ian Stewart, Pour la science Ed.1989

- Cours de mathématiques supérieures: 4-Équations différentielles, J. Quinet, Bordas Ed.1977

Dictionaries:- Grand Dictionnaire Encyclopédique Larousse, Larousse Ed.1982-1985

Periodicals:- Le chaos, Pour la science Ed.1995

- Science & Vie no938, novembre 1995

Software (fractal pictures):- Two dimension pictures and strange attractors: Fractint V.19.1

- Three dimension pictures: Vistapro V.3.13

Documents

The theory of Chaos