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Beginner's Guide to Chaos

"You believe in a God who plays dice, and I in complete law and order."  - AlbertEinstein, Letter to Max Born

Introduction...

This website has been produced for a Bristol University computin course

!"#MM$%&'(%)* +t is, hopefully, a basic and easy to understand overview of "haostheory, its history and applications* hile much of the complex mathematics of chaos is

omitted, such as a complete discussion of topoloy, some specific mathematical

examples are iven, alonside examples of chaos that arises in nature* "omputer-relatedtopics such as fractals, and the use of computers in detectin chaos, are also examined*

lease email any comments.suestions to ill Bolam, address at the bottom of the pae*

Main Menu:

The history of chaos -a brief history, and overview of the idea

"haos in nature -includin a detailed description of The Butterfly Effect  

/ractals -descriptions and pictures of both main sets, Mandlebrot and 0ulia

Mathematical examples of chaos 

-use a poc1et calculator to produce chaos

http://www.monkeylogic.co.uk/chaos/history.htmlhttp://www.monkeylogic.co.uk/chaos/nature.htmlhttp://www.monkeylogic.co.uk/chaos/fractal.htmlhttp://www.monkeylogic.co.uk/chaos/maths.htmlhttp://www.monkeylogic.co.uk/chaos/maths.htmlhttp://www.monkeylogic.co.uk/chaos/fractal.htmlhttp://www.monkeylogic.co.uk/chaos/history.htmlhttp://www.monkeylogic.co.uk/chaos/nature.htmlhttp://www.monkeylogic.co.uk/chaos/fractal.htmlhttp://www.monkeylogic.co.uk/chaos/maths.html

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The History of Chaos Theory

Introduction

This is not intended as an overview of all of chaos theory, it is instead as a loo1 bac1 to

 before chaos theory, and an introduction to the sub2ect* /irst we o bac1 in time***

Before Chaos Theory

3urin the %4th and %5th centuries, 6ewton led a scientific revolution that encompassed

many new theories and ideas* These new ideas brouht about an atmosphere ofexcitement in the scientific community because they wor1ed - phenomenon such as heat,

sound, waves, liht, manetism, electricity and mechanics were all explained* The deree

to which they were explained obviously varied, but the overall feelin was one of

optimism - that deterministic science held all the answers* This led to the view that thefuture of the entire universe was in fact predetermined, and therefore, with enouh

1nowlede, could be predicted* This was the view held by ierre $imon de Laplace, a

leadin %5th century mathematician, in his "Philosophical Essays on Probabilities" 7

"n intellect which at any !iven moment new all the forces that animate #ature and the

mutual positions of the bein!s that comprise it, if this intellect was vast enou!h to submit

its data to analysis, could condense into a sin!le formula the movement of the !reatest

bodies of the universe and that of the li!htest atom$ for such an intellect nothin! could beuncertain% and the future &ust lie the past would be present before its eyes." 

The Advent of Chaos

These views were held until very recently, and by some very important and well

respected people - for example both 6ewton and Einstein, see the 8uote from Einstein on

the first pae* hen examples of chaos were first constructed by mathematicians, theywere inored, on the rounds that the e8uations were obscure, mathematical curiosities*

9owever, examples of chaos in nature were turnin up, and the e8uations that were seen

as obscure constructs turned out to be actually more common than classical deterministice8uations* The maths of chaos bean to be explored* 6ow, thouh the sub2ect is still

youn and much is still to be learnt, the basics of chaos have been discovered, and areed

upon throuhout the world*

The Mathematical Definition of Chaos

chaos !:1eios) n. $tochastic behaviour occurin in a deterministic system*

A deterministic system is a system overned by e8uations which have a definite,

determinable outcome* These were the type of systems which 6ewton and Einsteinwor1ed on* $uch systems were always thouht to have a determinable outcome, ie a

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specific outcome that could be calculated exactly*

'tochastic behaviour  is seeminly unordered, unpredictable, behaviour, ie chaotic

 behaviour*$o, chaos is the mathematics of seeminly normal, predictable systems exhibitin

 behaviour that is anythin but*

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Chaos in ature

Introduction! and on"Chaos Maths in ature

#nly recently have we bean to discover how much mathematics overns nature* There

are many examples of both chaotic and non-chaotic maths in nature* #n the non-chaotic

side, there is, for example, the fibonachi se8uence* This is a simple iterative se8uence,iven by7fn = fn-1 + fn-2 , where f0 = f1 = 0.

This se8uence of numbers occurs in many natural features, such as the number of leavesor petals on certain plants* There is also the :olden ratio:, which is also observed in many

natural features* The olden ratio can be described as follows7

e define a rectanle whose sides are in this ratio to be a :olden rectanle:* Then if we

ta1e a olden rectanle, and remove the larest s8uare possible from one end !shaded),we are left with another olden rectanle*

9owever, much of the most interestin, and important, maths in nature is chaotic* #neexample is the self repitition that many plants, such as ferns, exhibit* $elf repitition

means that a small part of the structure, enlared, loo1s li1e the whole structure* This is

very closely lin1ed to some chaotic maths, and fractals* e continue, however, with

fluids*

#luids and Tur$ulence

The way that water flows has interested people for many centuries* /or example,

Leonardo da ;inci produced many pictures of the flow of water* 9e was very concernedto try and draw the water exactly as it flowed in reality, thouh this was a diffucult thin

to attempt* The

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But why is water so difficult to draw, and indeed to model= +t is due to the complexities

of fluid motion, and at the heart of the matter is chaos* /irst thouh, a simple experiment,

to show that the behaviour of water is far from deterministic* /or this experiment you willneed nothin but a tap*

The Dri%%ing Ta%Turn your tap on, 2ust enouh for it to drip reularly* The drips come at reular intervals*

 6ow increase the level, slihtly* >ou should be able to et a different, but reular

rhythm, somethin li1e drip-drip***drip-drip***drip-drip etc* 6ow, if you turn the tap a tiny bit more, so that the drips have almost formed a stream of water, you should be able to

et to a point where there is no pattern to the fallin drips* A small increase in the

amount of water bein allowed out, but a bi chane in what:s happenin* This is atypical example, on a very simplistic level, of chaos*

Bac& to #luid dynamics

/irst, a 8uic1 idea of what turbulence is* +t has the same meanin here as in normal use

e* ?turbulent seas?, but we must be slihtly more precise* Turbulence is the state a fluidreaches after smooth laminar flow has bro1en up* Laminar flow is where a fluid slidesover itself in layers* @oin bac1 to the tap, laminar flow is when the water is flowin out

of the tap in a steady stream, maybe with some extra, twisted structure* Turbulence is

when the tap is on full, with spray everywhere, and no obvious structure at all*The classical motion of a viscous fluid was described by "laude 6avier and $ir @eore

$to1es, based on ideas by Euler* The e8uations are deterministic, and therefore

 predictable, but turbulence is very irreular* $o turbulence could not be correctly predicted from these e8uations* Many people since wor1ed on turbulence, and the first

modification of the theory came from Lev Landau and Eberhard 9opf* Their theory was

one of wobbles, accumulatin and causin the transition to turbulence* /or '( years this

was the preferred theory* Then, 3avid

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that these predictions are often wron* Edward Loren discovered the reasonin behind

this, some ( years ao***

The Butterfly (ffect

#ne of the most famous effects of "haos Theory is 1nown as the butterfly effect,

concernin weather* +t was discovered, and named, by Edward Loren, a mathematicianturned meteoroloist, in winter %CD%* The butterfly effect is the theory that a sinle

 butterfly, flappin its wins in one part of the world, could cause a tornado in another

entirely different part of the world* Loren made his discovery entirely by accident*Loren had a number of e8uations, based on wor1 by $altman, that predicted simple

weather systems* "rucially, he had an early computer, a

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#ractals

A member of the mandelbrot set

Introduction

/ractals are beautiful diarams, formed from some more complex chaotic maths* An

inherent feature of fractals is self repitition* $elf repetition is where a small part of thestructure, enlared, loo1s li1e the whole structure* There are many different types of

fractal, and each type yields an infinite number of fractal diarams* This is because

fractals are infinitely complex, with a suitable proram you can oom in to any

manification, and the structure will remain as complex as before the oom* /ractalsutilise complex numbers* A complex number has two parts, one real, one imainary* The

real part is a normal number, whilst the imainary part is a multiple of the s8uare root of

-%, represented by i* The complex plane is the plane with x-axis comprisin the real part,

and the y-axis comprisin the imainary part* +n this way, any complex number can be plotted on the complex plane* All the fractal pictures below were produced with /ractint,

a free fractal producin proram*

The Mandel$rot )et

This set was devised by "the father of fractals" , Benoit Mandelbrot, and is the oriinalfractal set* The fractal is plotted on the complex plane* Then, to plot the fractal, we ta1e

each point on the complex plane in turn* This is called c* Then, startin with ( F (, we

use the iterative formula7zn+1 = zn

2 + c

Accordin to how fast this iteration leads to G &, the point c on the complex plane isshaded a certain colour*

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The *ulia )et

This set is named after @aston 0ulia, a /rench mathematician* The 0ulia set is plotted onthe same axis as the Mandelbrot set, and is closely lin1ed* +t has a very similar formula*

/or each c, ( F c, and then the same iterative formula is used7zn+1 = zn2 + c

Aain, accordin to the behaviour of this iteration, the point c is coloured*

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Maths (+am%les of Chaos

Bifurcation diaram

)im%le Chaos

"haos does not re8uire complex e8uations* +t can be produced by very simple ones* Ta1e

a loo1 at the simple e8uationcx2-1 where c is a real valued constant.

 6ow, by iteratin this e8uation on x, varyin results can be obtained, some displayinchaos*

By :iteratin on x: + mean ta1e some startin value of x F x(, then 1eep puttin the value

of the e8uation bac1 into itself, such that7xn+1 = cxn

2-1

hat we et is an infinite se8uence of values, x(, x%, x&,*** which can then be analysised*

• The se8uence miht eventually settle down to some value, the se8uence is said toconvere to this value*

• The se8uence miht settle into a cycle of a finite number of values, in which case

the se8uence is said to be a cycle*• The se8uence may 2ust head off towards H.- infinity* This is called diverence*

• 9owever the se8uence may not display any sort of pattern, and in fact will

continue not to exhibit any sort of pattern infinitely* +t is these se8uences which

are said to exhibit chaos*

ou may

already be familiar with the concept, and the word mappin here has the same meanin as

http://www.monkeylogic.co.uk/chaos/maths.html#bifurcation

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it always does in maths* A mappin is another word for a function, that is a rule to et

from an input x to an output f!x)* /or example, above the mappin iven is7x -> cx2-1

The loistic mappin is similar in form, and is iven by7x -> kx(1 - x) where k is a real valued constant etween 0 and !, and x 0

has initial value etween 0 and 1

+teratin, as above, we et7xt+1 = kxt(1 - xt)

@eometrically, the Loistic mappin represents stretchin.compressin a line sement ina non-uniform way, and then foldin it in half* This stretchin.foldin combination is a

classic recipe for chaos* e shall now examine the loistic mappin, and the way it

 behaves*

)teady )tate -egimes

/or values of 1 between ( and ' iteratin on any x( between ( and % will produce a

se8uence that converes* /or example, ta1e 1 F &, and x( F (*C* The values for the

se8uence IxtJ are as follows7

value of t xt

( (*C

% (*%5

& (*&C&

' (*%D%%'C&

(*5C&D&%%D

(*CCD('5C%54

D (*CCCCCD5D%

4 (*5 (*

C (*

As can be seen, the se8uence has convered to (** (* is called a point attractor for thesystem, a steady stable state* +n the lon term, such a system remains at rest, it does

nothin* 6ow this is not the most interestin bit of maths, but it leads onwards***

eriod"dou$ling Cascades

/or 1 F ', we have a 1ind of intermediate stae* The se8uence will convere at 1 F ', but

it converes infinitely slowly* As 1 is increased, the se8uence becomes unstable, but then

settles down into a period & cycle - there are & point attractors* But at about 1 F '* this &-cycle becomes unstable, and as 1 is increased we et a period cycle* The rate at which

this doublin occurs increases, and by 1 F '*D there is a period 5 cycle, by 1 F '*D4

there is a %D-cycle* This continues infinitely, but happens so 8uic1ly that by 1 F '*5 ithas finished* At this point the mappin becomes chaotic* There are more period doublin

cascades to come thouh* At 1 F '*5' there is a period ' cycle, doublin to D, %&, & etc

as 1 is increased very slihtly* There is a period attractor at 1 F '*4'C, which aain

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forms a period doublin cascade !, %(, &(, (,***)* The simple loistic mappin produces

this extremely complex mixture of chaos and order*

Bifurcation Diagrams

The behaviour of the loistic mappin can be described raphically by a bifurcation

diaram* This is a raph which plots the value of 1 across, aainst the attractors of these8uence vertically7

Bifurcation diaram - clic1 to enlare

The sinle branch at the beinnin represents the steady state, which branches into two*

This is the period & cycle* 6ow we can raphically see the period doublin cascade, asthe branches each divide into two, until there is chaos* Amidst the chaos, new branches

sprin up, and then double themselves - these are the new period doublin cascades* !Tosee the proper detail, you will need to examine the larer version of the diaram*) + wrote

a simple 8basic proram to produce the bifurcation diaram, clic1 here to see the code*

http://www.monkeylogic.co.uk/chaos/bif1.txthttp://www.monkeylogic.co.uk/chaos/bifur.jpghttp://www.monkeylogic.co.uk/chaos/bif1.txt