Sardar Patel Institute of Technology

A Report

On

S.E. Information Technology

Group C2

(Roll Numbers: 54-60)

A report submitted in partial fulfillment of the

requirement of Communication and Presentation

Techniques syllabus: Report Writing

October 2012

CERTIFICATE

This to certify that the work on the project titled Chaos Theory has been carried out by the

following students, who are bonafide students of Sardar Patel Institute of Technology,

Mumbai, in partial fulfilment of the syllabus requirement in the subject “Communication and

Presentation Techniques” in the academic year 2012-2013:

1. Gajanan Shewale 2. Nayana Shinde 3. Aditya Shirode 4. Suntej Singh 5. Jayesh Solanki 6. Madhuri Tajane 7. Gaurav Tripathi

Project Guide: ____________________

(Madhavi Gokhale)

Principal: _________________________

(Dr.Prachi Gharpure)

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PREFACE

The trouble with weather forecasting is that it's right too often for us to ignore it and wrong

too often for us to rely on it.

- Patrick Young

The quote perfectly fits. Whenever our meteorological department predicts some kind of

weather, we know that nature is going to do exactly opposite. But why does this happen? If

we can exactly predict at what time any astronomical body is going to enter earth’s

atmosphere and where it is going to strike, why can’t weather? Does God really play dice?

Not quite. The unpredictability of weather can be explained by Chaos Theory.

The word “Chaos” comes from the Greek word “Khaos”, meaning "gaping void". Chaos in

other words means a state of utter confusion or the inherent unpredictability in the behavior

of a complex natural system.

Chaos theory is a field of study in mathematics, with applications in several disciplines

including physics, engineering, economics, biology, and philosophy which primarily states

that small differences in initial conditions (such as those due to rounding errors in numerical

computation) can yield widely diverging outcomes for chaotic systems, rendering long-term

prediction impossible in general.

Acknowledging the fact that chaos theory is widely used in many spheres of life and is

known to very few people inspired us to prepare a report on this topic to make the masses

aware regarding this phenomenon. We hope that this project serves as a useful tool for

anyone who is interested in understanding this topic.

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ACKNOWLEDGEMENTS

We, as a group, have taken efforts in this project. However it would not have been possible

without the kind support and help of many individuals and organization, who helped us in

developing the project and willingly helped out with their abilities. We would like to extend

our sincere thanks to all of them.

We are grateful to Prof. Madhavi Gokhale, Lecturer in ‘Communication and Presentation

Techniques’, for her guidance and support. Her valuable comments and discussions on earlier

versions of this report lead us to improvements and propelled us in right direction. Her apt

suggestions and constant encouragement have enabled us to focus our efforts.

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ABSTRACT

Chaos theory is a mathematical field of study which states that non-linear dynamical systems

that are seemingly random are actually deterministic from much simpler equations. The

phenomenon of Chaos theory was introduced to the modern world by Edward Lorenz in 1972

with conceptualization of ‘Butterfly Effect’. As chaos theory was developed by inputs of

various mathematicians and scientists, it found applications in a large number of scientific

fields.

The purpose of the project is the interpretation of chaos theory which is not as familiar as

other theories. Everything in the universe is in some way or the other under control of Chaos

or product of Chaos. Every motion, behavior or tendency can be explained by Chaos Theory.

The prime objective of it is the illustration of Chaos Theory and Chaotic behavior.

This project includes origin, history, fields of application, real life application and limitations

of Chaos Theory. It explores understanding complexity and dynamics of Chaos.

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TABLE OF CONTENTS

Preface ……………………………………………………………………………… i

Acknowledgement ……………………………………………………………….. ii

Abstract ……………………………………………………………………………… iii

1. Introduction ………………………………………………………………… 1

1.1 What is chaos theory?

1.2 Chaos Theory: Before and After Lorenz

2. Concepts of Chaos Theory ………………………………………………… 5

2.1 Key Terms

2.1.1 Chaotic Systems

2.1.2 Attractors

2.1.3 Fractals

2.2 The Butterfly effect

3. Aspects of Chaos Theory …………………………………………………………. 12

3.1 Predictability

3.2 Control of chaos

3.3 Synchronization of chaos

4. Applications of Chaos theory …………………………………………………. 16

4.1 Stock market

4.2 Population dynamics

4.3 Biology (human being as a chaotic system)

4.3.1 Predicting heart attacks

4.4 Real time applications

4.4.1 Chaos to produce music

4.4.2 Climbing with chaos

4.5 Random Number Generation

5. Limitations of chaos theory …………………………………………………. 23

6. Conclusion …………………………………………………………………. 24

Appendix …………………………………………………………………………. 25

List of References …………………………………………………………………. 31

Bibliography …………………………………………………………………………. 32

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1. INTRODUCTION 1.1 WHAT IS CHAOS THEORY?

Changes in the weather. Cardiac arrhythmias. Traffic flow patterns. Urban development and

decay. Epidemics. The behavior of people in groups. Any idea what holds all of these ideas

together?

Richard Feynman, a well known physicist, quoted that, “Physicists like to think that all you have

to do is say, these are the conditions, now what happens next?”

The world of science has been confined to the linear world for centuries. That is to say,

mathematicians and physicists have overlooked dynamical systems as random and unpredictable.

The only systems that could be understood in the past were those that were believed to be linear,

that is to say, systems that follow predictable patterns and arrangements. Linear equations, linear

functions, linear algebra, linear programming, and linear accelerators are all areas that have been

understood and mastered by the human race. But there were some areas that just could not be

explained, like weather patterns, ocean currents, or the actions of cells. There were too many

things going on to keep track of with linear equations.

Answer to both questions posed above is Chaos, a theory related to which was given by Lorenz,

‘during a coffee break’.

What is Chaos exactly? Mathematicians say it is tough to define chaos, but is easy to “recognize

it when you see it.” Chaos and order are two sides of the same coin. Chaos is interrelated with

order. It keeps stagnation from setting into any one system.

Systems considered to be chaotic are not really chaotic at all – they are just not as predictable as

the cause-and-effect kind of ideas associated with linear dynamics. Mythology and early science

have presented ideas of chaos to explore. In many mythologies the creation of the universe is

symbolized by the gods of order conquering chaos. “While the universe, including the gods, may

originate from chaos, order seems to emerge also. Order banishes chaos but never really destroys

it.”1 Despite its etymological Greek origin ‘����’ (zhaos), the notion of chaos appears in many

different ancient narrations about the origins of the World. Chaos is abundant, but also regime

dependent.

What is Chaos Theory then?

To state as a definition, Chaos theory is the study of complex, nonlinear, dynamic systems.

It is a branch of mathematics that deals with systems that appear to be orderly (deterministic)

but, in fact, harbor chaotic behaviors. It also deals with systems that appear to be chaotic, but, in

fact, have underlying order. Chaos theory studies the behavior of dynamical systems that are

highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly

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effect. Small differences in initial conditions (such as those due to rounding errors in numerical

computation) yield widely diverging outcomes for chaotic systems, rendering long-term

prediction impossible in general. This happens even though these systems are deterministic,

meaning that their future behavior is fully determined by their initial conditions, with

no random elements involved. In other words, the deterministic nature of these systems does not

make them predictable. This behavior is known as deterministic chaos, or simply chaos.

Nature is highly complex, and the only prediction you can make is that she is unpredictable. The

amazing unpredictability of nature is what Chaos Theory looks at. Why? Because instead of

being boring and translucent, nature is marvelous and mysterious. And Chaos Theory has

managed to somewhat capture the beauty of the unpredictable and display it in the most

awesome patterns. Nature, when looked upon with the right kind of eyes, presents her as one of

the most fabulous works of art ever.

Chaos Theory holds to the axiom that reality itself subsists in a state of ontological anarchy.

Chaos theory is most commonly attributed to the work of Edward Lorenz. His 1963 paper,

Deterministic Nonperiodic Flow, is credited for laying the foundation for Chaos Theory. Lorenz

was a meteorologist who developed a mathematical model used to model the way the air moves

in the atmosphere. He discovered by chance that when he entered a starting value at three

decimal .506 instead of entering the full .506127. It caused vast differences in the outcome of the

model. In this way he discovered the principle of Sensitive Dependence on Initial Conditions

(SDIC), which is now viewed as a key component in any chaotic system. This idea was then

immortalized when Lorenz gave a talk at the 139th meeting of the American Association for the

Advancement of Science in 1972 entitled “Predictability: Does the Flap of a Butterfly’s Wings in

Brazil Set Off a Tornado in Texas”. With this speech the idea of the Butterfly Effect was born

and has been used when talking about chaos theory ever since. The basic principle is that even in

an entirely deterministic system the slightest change in the initial data can cause abrupt and

seemingly random changes in the outcome.

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1.2 CHAOS THEORY: BEFORE AND AFTER LORENZ

While Lorenz’s discovery has achieved widespread attention that it did not initially receive upon

its initial publication is well-deserved outcome, largely being read initially by meteorologists and

climatologists who did not fully appreciate its broader mathematical implications. Nevertheless,

the possibility of SDIC had been realized by others much earlier, including Maxwell (1876),

Hadamard (1898), and Poincaré (1890). The idea of fractality also had a long history preceding

him going back at least to Cantor (1883), with the idea of endogenously erratic dynamics that

come close to following periodicity arguably going back as far as the pre-Socratic Greek

philosopher, Anaxagoras [according to Rössler (1998)], although first clearly shown by Cayley

(1879).

Playing a foundational role, Anaxagoras created the qualitative mathematical notions used so

successfully later by the Poincaré school: deterministic flow, cross-section through a flow, i.e.

and – most important – the notions of mixing and unmixing.

Yet another figure who has been seen to foreshadow modern chaos theory is the Renaissance

polymath, Leonardo da Vinci. Some of his drawings involving wind depict spectacular

turbulence, which in fluid dynamics has long been a central area of study associated with chaos

theory. Leibniz (1695) posited the possibility of fractional derivatives.

“Big whorls have little whorls, Which feed on their velocity;

And little whorls have lesser whorls, And so on to viscosity”

Another mathematician, Richardson, stated the above, in the molecular sense, in 1922.

Professor of Political Economy in Britain, Thomas Robert Malthus, in the first edition of his

famous Essay on the Principle of Population (1798), mentioned that history would tend greatly

to elucidate the manner in which the constant check upon population acts.

Alan Turing conducted work on the process of Morphogenesis in the early 1950s that argued for

a mathematical explanation of the process. Morphogenesis is the biological process that causes

an organism to develop its shape.

Now seeing the work done by his predecessors, one might infer that Lorenz did not do anything

of any particularly great importance. But Lorenz can be claimed to have “discovered chaos” both

because he put all the elements together, sensitive dependence on initial conditions, with strange

(fractal) attractors, and the resulting erratic dynamics, and because he saw them all together and

in the context of a computer simulation situation that could be replicated. He was the one who

derived the mathematical understanding.2

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What Lorentz did in weather, Robert May did in ecology. His work, in the early 1970s, helped

pin down the concepts of bifurcation (a split or division by two) and period doubling (one way in

which order breaks down into chaos). A period is the time required for any cyclic system to

return to its original state. In the early 1970s, May was working on a model that addressed how

insect birthrate varied with food supply. He found that at certain critical values, his equation

required twice the time to return to its original state--the period having doubled in value. After

several period-doubling cycles, his model became unpredictable, rather like actual insect

populations tend to be unpredictable. Since May’s discovery with insects, mathematicians have

found that this period-doubling is a natural route to chaos for many different systems.

Even though Lorenz coined the term Butterfly Effect, the name chaos was coined by Jim Yorke

in 1974, an applied mathematician at the University of Maryland.

In 1971, David Ruelle and Floris Takens described a phenomena they called a strange

attractor (a special type of attractor today called achaotic attractor). This strange phenomenon

was said to reside in what they called phase space (a geometric depiction of the state space of a

system). Strange attractor is recognized as the general pattern followed by chaotic systems.

Another pioneer of the new science was Mitchell Feigenbaum. His work, in the late 1970s, was

so revolutionary that several of his first manuscripts were rejected for publication because they

were so novel they were considered irreverent. He discovered order in disorder. He looked

deeply into turbulence, the home of strange attractors, and saw universality. Feigenbaum showed

that period doubling is the normal way that order breaks down into chaos.

How come mathematicians have not studied chaos theory earlier? The answer can be given in

one word: computers. The calculations involved in studying chaos are repetitive, boring and

number in the millions. A multidisciplinary interest in chaos, complexity and self-organizing

systems started in 1970’s with the invention of computers.

Benoît Mandlebrot found the piece of the chaos puzzle that put all things together. Mandelbrot

published a book, The Fractal Geometry of Nature (1982), which looked into a mathematical

basis of pattern formation in nature, much like the earlier work of Turing. In this he outlined his

principle of self�similarity, which describes anything in which the same shape is repeated over

and over again at smaller and smaller scales. His fractals (the geometry of fractional dimensions)

helped describe or picture the actions of chaos, rather than explain it. Chaos and its workings

could now be seen in color on a computer. As a graphical representation of the mathematical rule

behind self similarity Mandelbrot created an image that has come to be known as the Mandelbrot

Set, or the Thumbprint of God.3 The Mandelbrot Set’s properties of being similar at all scales

mirrors a fundamental ordering principle, repeated in nature.

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2. CONCEPTS

2.1 KEY TERMS

The typical features of chaos include:

• Nonlinearity. If it is linear, it cannot be chaotic.

• Determinism. It has deterministic (rather than probabilistic) underlying rules every future state

of the system must follow.

• Sensitivity to initial conditions. Small changes in its initial state can lead to radically different

behavior in its final state. This “butterfly effect” allows the possibility that even the slight

perturbation of a butterfly flapping its wings can dramatically affect whether sunny or cloudy

skies will predominate days later.

• Sustained irregularity in the behavior of the system. Hidden order including a large or infinite

number of unstable periodic patterns (or motions). This hidden order forms the infrastructure of

irregular chaotic systems---order in disorder for short.

• Long-term prediction (but not control!) is mostly impossible due to sensitivity to initial

conditions, which can be known only to a finite degree of precision.4

2.1.1 CHAOTIC SYSTEMS

Chaos is qualitative in that it seeks to know the general character of a system's long-term

behavior, rather than seeking numerical predictions about a future state.

Chaotic systems are unstable since they tend not to resist any outside disturbances but instead

react in significant ways. In other words, they do not shrug off external influences but are partly

navigated by them. These systems are deterministic because they are made up of few, simple

differential equations, and make no references to implicit chance mechanisms. A dynamic

system is a simplified model for the time-varying behavior of an actual system. These systems

are described using differential equations specifying the rates of change for each variable.

A deterministic system is a system in which no randomness is involved in the development of

future states of the system. 5It is said to be chaotic whenever its evolution sensitively depends on

the initial conditions. This property implies that two trajectories emerging from two different

close-by initial conditions separate exponentially in the course of time.

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Lorenz's experiment: Difference between starting values of these curves is only .000127

Source: http://www.imho.com/sites/default/files/images/fig1_0.gif

The fact that some dynamical model systems showing the above necessary conditions possess

such a critical dependence on the initial conditions was known since the end of the nineteenth

century. However, only in the last thirty years of twentieth century, experimental observations

have pointed out that, in fact, chaotic systems are common in nature. They can be found, for

example, in Chemistry (Belouzov-Zhabotinski reaction), in Nonlinear Optics (lasers), in

Electronics (Chua-Matsumoto circuit), in Fluid Dynamics (Rayleig-BeHnard convection), etc.

Many natural phenomena can also be characterized as being chaotic. They can be found in

meteorology, solar system, heart and brain of living organisms and so on.

The peculiarities of a chaotic system can be listed as follows:6

1. Strong dependence of the behavior on initial conditions

2. The sensitivity to the changes of system parameters

3. Presence of strong harmonics in the signals

4. Fractional dimension of space state trajectories

5. Presence of a stretch direction, represented by a positive Lyapunov exponent

The last can be considered as an “index” that quantifies a chaotic behavior.

Solutions of chaotic systems can be complex and typically they cannot be easily extrapolated

from current trends. The game of Roulette is an interesting example that might illustrate the

distinction between random and chaotic systems: If we study the statistics of the outcome of

repeated games, then we can see that the sequence of numbers is completely random. That led

Einstein to remark: “The only way to win money in Roulette is to steal it from the bank.'' On the

other hand we know the mechanics of the ball and the wheel very well and if we could somehow

measure the initial conditions for the ball/wheel system, we might be able to make a short term

prediction of the outcome.7

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A double pendulum is a classic chaotic system. It consists of one pendulum attached to another.

Double Pendulum

Source: http://web.mat.bham.ac.uk/C.J.Sangwin/Teaching/pendulum/dp1.jpg

2.1.2 ATTRACTORS

An attractor is a set of states (points in the phase space), invariant under the dynamics, towards

which neighboring states in a given basin of attraction asymptotically approach in the course of

dynamic evolution. An attractor is defined as the smallest unit which cannot itself be

decomposed into two or more attractors with distinct basins of attraction

In chaos theory, systems evolve towards states called attractors. The evolution towards a

specific state is governed by a set of initial conditions. An attractor is generated within the

system itself.8

There are several types of attractors9. The first is a point attractor, where there is only one

outcome for the system. Death is a point attractor for human beings. No matter who we are, how

we lived our life or whatever, we die at the end of our life.

The second type of attractor is called a limit cycle or periodic attractor. Instead of moving to a

single state as in a point attractor, the system settles into a cycle. While we can then not predict

the exact state of the system at any time, we know it will be somewhere in the cycle.

The third type of attractor is called a strange attractor or a chaotic attractor. It is a double spiral

which never repeats itself (or it would be periodic attractor), but the values always move towards

a certain range of values. There are certain states in which the system can exist and others it

cannot. If the system were to somehow move out from the acceptable range of states it would be

“attracted” back into the attractor. It is a geometrical shape which represents the limited region of

phase space ultimately occupied by all trajectories of a dynamical system.

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The giant red spot on Jupiter, also called ‘Soliton’, is a good example of a strange attractor. We

know it has been stable since first viewed around 1660. The surface of Jupiter consists of belts of

highly volatile gasses rotating around the planet at extremely high speeds. The friction between

the layers and the turbulence created resulted in the formation of the spot. Lorenzian waterwheel

is also experimental example of a strange attractor.

A.Point attractor B.Limit Cycle C.Limit Tori D. Strange attractor

Source: http://psycnet.apa.org/journals/amp/49/1/images/amp_49_1_5_fig2a.gif

Try this:

• Pick a number, any number

• Plug it in for x in .8x + 1

• Take the result and plug that back in for x again in .8x + 1

• Repeat this process

After some number of iterations, you should notice the list of numbers (the "orbit") converging

on a particular number

Strange attractors are shapes with fractional dimension; they are fractals.

2.1.3 FRACTALS

Fractals are objects that have fractional dimension.

A fractal is a mathematical object that is self-similar and chaotic. Fractals are infinitely complex:

the closer you look the more detail you see. Fractals are pictures that result from iterations of

nonlinear equations, usually in a feedback loop. Using the output value for the next input value, a

set of points is produced. Graphing these points produces images.

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Again, by creating a vast number of points using computers to generate those points,

mathematicians discovered these wonderfully complex images which were called fractals, a term

coined by Benoit Mandelbrot, one of the first to discover and examine these images.

Two important properties of fractals are: self-similarity and fractional dimensions.

Self-similarity means that at every level, the fractal image repeats itself. Fractals are shapes or

behaviors that have similar properties at all levels of magnification. Just as the sphere is a

concept that unites raindrops, basketballs and Mars, so fractal is a concept that unites clouds,

coastlines, plants and strange attractors. (Look Appendix I for images of various fractals.)

Fractals are quite real and incredible; however, what do these have to do with real life? Is there a

purpose behind these fascinating images? Fractals make up a large part of the biological world.

Clouds, arteries, veins, nerves, parotid gland ducts, and the bronchial tree all show some type of

fractal organization. In addition, fractals can be found in regional distribution of pulmonary

blood flow, pulmonary alveolar structure, regional myocardial blood flow heterogeneity,

surfaces of proteins, mammographic parenchymal pattern as a risk for breast cancer, and in the

distribution of arthropod body lengths.

There is a strong link between chaos and fractals. Fractal geometry is the geometry that describes

the chaotic systems we find in nature. Fractals are a language, a way to describe this geometry.

Euclidean geometry is a description of lines, circles, triangles, and so on. Fractal geometry is

described in algorithms- a set of instructions on how to create the fractal. Computers translate the

instructions into the magnificent patterns, we see as fractal images.

Somewhere after Fractals:

A scientist by the name of Feigenbaum was looking at the bifurcation diagram again. He was

looking at how fast the bifurcations come. He discovered that they come at a constant rate. He

calculated it as 4.669. In other words, he discovered the exact scale at which it was self-similar.

Make the diagram 4.669 times smaller, and it looks like the next region of bifurcations. He

decided to look at other equations to see if it was possible to determine a scaling factor for them

as well. Much to his surprise, the scaling factor was exactly the same. Not only was this

complicated equation displaying regularity, the regularity was exactly the same as a much

simpler equation. He tried many other functions, and they all produced the same scaling factor,

4.669. This was a revolutionary discovery. He had found that a whole class of mathematical

functions behaved in the same, predictable way. This universality would help other scientists

easily analyze chaotic equations. Universality gave scientists the first tools to analyze a chaotic

system. Now they could use a simple equation to predict the outcome of a more complex

equation.

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2.2 THE BUTTERFLY EFFECT

Butterfly effect is a way of describing how, unless all factors can be accounted for, large systems

like the weather remain impossible to predict with total accuracy because there are too many

unknown variables to track. It is also called as sensitive dependence on initial condition.

Lorenz coined the term and put forward the idea of ‘Butterfly Effect’. His paper was titled as-

‘Predictability: Does the Flap of a Butterfly’s wings in Brazil Set Off a Tornado in Texas?’10

Intrigued by results he obtained as a mistake in inputs during his computer simulation, Lorenz

began creating a mathematical explanation that would show the sensitive dependence of large,

complex systems like the weather. To simplify his findings, Lorenz coined

the butterfly explanation.

What he meant broadly was the butterfly effect is a way of describing how, unless all factors can

be accounted for, large systems like the weather remain impossible to predict with total accuracy

because there are too many unknown variables to track.

The butterfly does not cause the tornado. The flap of the wings is a part of the initial conditions;

one set of conditions leads to a typhoon while the other set of conditions does not. The flapping

wing represents a small change in the initial condition of the system, which causes a chain of

events leading to large-scale alterations of events.

A better analogy for butterfly effect is an avalanche. It can be provoked with a small input (a

loud noise, some burst of wind), it's mostly unpredictable, and the resulting energy is huge.

As a literary device used to entertain, the concept of Butterfly Effect suggests that our present

conditions can be dramatically altered by the most insignificant change in the past.

Chaos theory and butterfly effect are not the same. The Butterfly effect is a symbol of chaos. It is

a simple and entertaining way of describing one component of the greater Chaos Theory, namely

"Sensitive dependence on initial conditions." Essentially, the Butterfly Effect describes how

small changes at one point of a nonlinear system can result in larger differences to a later state.

Chaos Theory itself is a much larger system of theorems and formulas for predicting and

understanding the behaviors of complex, nonlinear systems. Saying that the Butterfly Effect is

the same as Chaos Theory is a bit like saying that the cheese and the cheese burger are the same

thing.

Because of the "Butterfly Effect", it is now accepted that weather forecasts can be accurate only

in the short-term, and that long-term forecasts, even made with the most sophisticated computer

methods imaginable, will always be no better than guesses.

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The presence of chaotic systems in nature seems to place a limit on our ability to apply

deterministic physical laws to predict motions with any degree of certainty. The discovery of

chaos seems to imply that randomness lurks at the core of any deterministic model of the

universe.11

Because of this fact, some scientists have begun to question whether or not it is meaningful at all

to say that the universe is deterministic in its behavior. This is an open question which may be

partially answered as science learns more about how chaotic systems operate.

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3. ASPECTS OF CHAOS THOERY

3.1 PREDICTABILITY OF CHAOS

Notion of chaos can be made more precise by seeing its sign, mapping them and observing the

conditions under which a system drifts into a chaotic state.

Characteristics of a Chaotic System:

• No periodic behavior

• Sensitivity to initial conditions

• Chaotic motion is difficult or impossible to forecast

• The motion 'looks' random

• Non-linear

Because of the various factors involved in chaotic systems, they are hard to predict. A lot of

complicated computations and mathematical equations are involved. But still the prediction may

vary from observed output.

Chaos in the atmosphere:

The atmosphere is a chaotic system, and as a result, small errors in our estimate of the current

state can grow to have a major impact on the subsequent forecast. Because of the limited number

of observations available and the uneven spread of these around the globe, there is always some

uncertainty in our estimate of the current state of the atmosphere. In practice this limits detailed

weather prediction to about a week or so ahead.

Accepting the findings from chaos theory about the sensitivity of the prediction to uncertainties

in the initial conditions, it is becoming common now to run in parallel a set, or ensemble, of

predictions from different but similar initial conditions. The Ensemble Prediction System (EPS)

provides a practical tool for estimating how these small differences could affect the forecast.

This weather prediction model is run 51 times from slightly different initial conditions. To take

into account the effect of uncertainties in the model formulation, each forecast is made using

slightly different model equations. The 51 scenarios can be combined into an average forecast

(the ensemble-mean) or into a small number of alternative forecasts (the clusters), or they can be

used to compute probabilities of possible future weather events.

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3.2 CONTROL OF CHAOS 12

The idea of controlling chaos is that when a trajectory approaches ergodically a desired

periodic orbit embedded in the attractor, one applies small perturbations to stabilize such an

orbit. If one switches on the stabilizing perturbations, the trajectory moves to the neighbourhood

of the desired periodic orbit that can now be stabilized. This fact has suggested the idea that the

critical sensitivity of a chaotic system to changes (perturbations) in its initial conditions may be,

in fact, very desirable in practical experimental situations. (This is known as Ott, Grebogi, and

Yorke (OGY) approach of controlling chaos.)

There are three ways to control chaos13

:

1. Alter organizational parameters so that the range of fluctuations is limited.

2. Apply small perturbations to the chaotic system to try and cause it to organize.

3. Change the relationship between the organization and the environment.

Due to the critical dependence on the initial conditions of chaotic systems and due to the fact

that, in general, experimental initial conditions are never known perfectly, these systems are

intrinsically unpredictable. Indeed, the prediction trajectory emerging from a bonafide initial

condition and the real trajectory emerging from the real initial condition diverge exponentially

in course of time, so that the error in the prediction (the distance between prediction and real

trajectories) grows exponentially in time, until making the system's real trajectory completely

different from the predicted one at long times.

For many years, this feature made chaos undesirable, and most experimentalists considered such

characteristic as something to be strongly avoided. Besides their critical sensitivity to initial

conditions, chaotic systems exhibit two other important properties:

• Firstly, there are an infinite number of unstable periodic orbits embedded in the

underlying chaotic set. In other words, the skeleton of a chaotic attractor is a collection of

an infinite number of periodic orbits, each one being unstable.

• Secondly, the dynamics in the chaotic attractor is ergodic, which implies that during its

temporal evolution the system ergodically visits small neighbourhood of every point in

each one of the unstable periodic orbits embedded within the chaotic attractor.

A relevant consequence of these properties is that a chaotic dynamics can be seen as shadowing

some periodic behavior at a given time, and erratically jumping from one to another periodic

orbit. Indeed, if it is true that a small perturbation can give rise to a very large response in the

course of time, it is also true that a judicious choice of such a perturbation can direct the

trajectory to wherever one wants in the attractor, and to produce a series of desired dynamical

states. This is the idea of control of chaos.

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The important point here is that, because of chaos, one is able to produce an infinite number of

desired dynamical behaviors (either periodic or not periodic) using the same chaotic system, with

the only help of tiny perturbations chosen properly. We stress that this is not the case for a non-

chaotic dynamics, wherein the perturbations to be done for producing a desired behavior must, in

general, be of the same order of magnitude as the unperturbed evolution of the dynamical

variables.

There are two methods used in Control of Chaos, namely:

- Ott-Grebogi-Yorke Method

Calculated swift, tiny perturbations are applied to the system once every cycle.

- Pyragas Method

A continuous controlling signal is injected into the system which approaches zero as the

system reaches the desired orbit

The applications of controlling chaos are enormous, ranging from the control of turbulent flows,

to the parallel signal transmission and computation to the parallel coding-decoding procedure, to

the control of cardiac fibrillation, and so forth.

3.3 SYNCHRONIZATION OF CHAOS

Chaos has long-term unpredictable behavior. This is usually couched mathematically as

sensitivity to initial conditions—where the system’s dynamics takes it is hard to predict from the

starting point. Although a chaotic system can have a pattern (an attractor) in state space,

determining where on the attractor the system is at a distant, future time given its position in the

past is a problem that becomes exponentially harder as time passes.

Can we force two chaotic systems to follow the same path on the attractor? Perhaps we could

‘lock’ one to the other and thereby cause their synchronization? The answer is, yes.

Why would we want to do this? The noise-like behavior of chaotic systems suggested early on

that such behavior might be useful in some type of private communications. One glance at the

Fourier spectrum from a chaotic system will suggest the same. There are typically no dominant

peaks, no special frequencies. The spectrum is broadband. There have been suggestions to use

chaos in robotics or biological implants. If we have several parts that we would like to act

together, although chaotically, we are again led to the synchronization of chaos.

There is identical synchronization in any system, chaotic or not, if the motion is continually

confined to a hyperplane in phase space. The most general and minimal condition for stability, is

to have the Lyapunov exponents associated with be negative for the transverse subsystem. Two

Lorenz systems can be synchronized together using CR (Complete Replacement) technique.14

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Systems with more than one positive Lyapunov exponent, called hyperchaotic systems, can be

synchronized using one drive signal.

Pecora & Carroll method of synchronization:

15Synchronization of chaos can be achieved by introducing a coupling term between two chaotic

systems and providing a controlling feedback in one that will eventually cause its trajectory to

converge to that of the other and then remain synchronized with it.

The method of Pecora and Carroll is to split an autonomous dynamical system

u = f(u) into a drive system v = g(v; w) and a response subsystem � = h(v; w). One then makes a

copy of the response subsystem �’= h(v; w) and asks under which conditions we can expect � �

= �’ – � � 0 as t � �

Lasers that behave chaotically could be synchronized. Two solid state lasers can couple through

overlapping electromagnetic lasing fields. The coupling is similar to mutual coupling except it is

negative. This causes the lasers to actually be in oppositely signed states. Such laser

synchronization opens the way for potential uses in fiber optics.16

17Synchronization of chaos may be used to build a means of secure communication over a public

channel.

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4. APPLICATIONS OF CHAOS THEORY

4.1 CHAOS THEORY IN STOCK MARKET

The science of chaos supplies us with a new and provocative paradigm to view the markets and

provides a more accurate and predictable way to analyze the current and future action of a

commodity or stock. It gives us a better map with which to trade. It does not depend on

constructing a template from the past and applying it to the future. But it concentrates on the

current market behavior, which is simply a composite of (and is quite similar to) the individual

Fractal behavior of the mass of traders. Chaos analysis has determined that market prices are

highly random, but with a trend. The amount of the trend varies from market to market and from

time frame to time frame.

The price movements that take place over the period of several minutes will resemble price

movements that take place over the period of several years. In theory, big market crashes should

never happen. But Mandelbrot predicts that a market crash should occur about once a decade.

Given the fact that we've had major crashes in 1987, 1998 and 2008 - roughly once a decade - it's

clear that Mandelbrot made a pretty good prediction.

The new Fractal Market Hypothesis, based on Chaos Theory explains the phenomena in financial

branch, which the Efficient Market Hypothesis could not deal with. In the hypothesis, Hurst

exponent determines the rate of chaos and distinguished fractal from random time series.

Lyapunov exponent determines the rate of predictability. A positive Lyapunov exponent

indicates chaos and it sets the time scale which makes the state of prediction possible. Plotting

stock market variations and matching them with chaotic analyses of above exponents, one might

predict future behavior of market.

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4.2 POPULATION DYNAMICS

Another concept where chaos theory can be applied is the prediction of biological populations.

The simplest equation that takes this into account is the following:

Next year's population = R * this year's population * (1 - this year's population)

In this equation, the population is a number between 0 and 1, where 1 represents the maximum

possible population and 0 represents extinction. R is the growth rate determined by two factors:

the rate of reproduction, and the rate of death from old age. The question is, how does this

parameter affect the equation?

One biologist, Robert May, decided to see what would happen to the equation as the growth rate

value changes18

. At low values of the growth rate, the population would settle down to a single

number. For instance, if the growth rate value is 2.7, the population will settle down to .6292. As

the growth rate increases, the final population would increase as well. Then, something weird

happens.

As soon as the growth rate passes 3, the line breaks in two. Instead of settling down to a single

population, it would jump between two different populations. It would be one value for one year,

go to another value the next year, then repeat the cycle forever. Raising the growth rate a little

more would cause it to jump between four different values. As the parameter rose further, the

line bifurcates (doubles) again. The bifurcations came faster and faster until suddenly, chaos

appears. Past a certain growth rate, it becomes impossible to predict the behavior of the equation.

However, upon closer inspection, it is possible to see white strips. Looking closer at these strips

reveals little windows of order, where the equation goes through the bifurcations again before

returning to chaos.

Population biology illustrates the deep structure that underlies the apparent confusion in the

surface behavior of chaotic systems. Some animal populations exhibit a boom-and-bust pattern

in their numbers over a period of years. This boom-and-bust pattern has been seen elsewhere,

including disease epidemics.

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Bifurcation graph showing chaotic behavior of population

Source: http://mathewpeet.org/science/chaos/chaos.jpg

4.3 BIOLOGY

Chaos theory can also be applied to human biological rhythms. The human body is governed by

the rhythmical movements of many dynamical systems: the beating heart, the regular cycle of

inhaling and exhaling air that makes up breathing, the circadian rhythm of waking and sleeping,

the saccadic (jumping) movements of the eye that allow us to focus and process images in the

visual field, the regularities and irregularities in the brain waves of mentally healthy and

mentally impaired people as represented on electroencephalograms. None of these dynamic

systems are perfect all the time, and when a period of chaotic behavior occurs, it is not

necessarily bad. Healthy hearts often exhibit brief chaotic fluctuations, and sick hearts can have

regular rhythms. Applying chaos theory to these human dynamic systems provides information

about how to reduce sleep disorders, heart disease, and mental disease.

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4.3.1 PREDICTING HEART ATTACKS

19

Science Daily, in its July 23, 2010 article reported that Chaos models may someday help model

cardiac arrhythmias -- abnormal electrical rhythms of the heart, according to researchers in the

journal Chaos, published by the American Institute of Physics.

A space-time plot of the alternans along a cardiac fiber is a solution to the Echebarria-Karma equation.

Source: http://images.sciencedaily.com/2010/07/100721145105-large.jpg

In recent years, medical research has drawn more attention to chaos in cardiac dynamics.

Although chaos marks the disorder of a dynamical system, locating the origin of chaos and

watching it develop might allow researchers to predict, and maybe even counteract, certain

outcomes.

An important example is the chaotic behavior of ventricular fibrillation, a severely abnormal

heart rhythm that is often life-threatening. One study found chaos in two and three dimensions in

the breakup of spiral and scroll waves, thought to be precursors of cardiac fibrillation. Another

study found that one type of heartbeat irregularity, a sudden response of the heart to rapid

beating called "spatially discordant alternans," leads to chaotic behavior and thus is a possible

predictor of a fatal heart attack.

Assigning extreme parameter values to the model, the team was able to find chaotic behavior in

space over time. The resulting chaos may have a unique origin, which has not yet been

identified.20

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4.4 REAL TIME APPLICATIONS

4.4.1 CHAOS TO PRODUCE MUSIC

21

In 1995, Diana S. Dabby, An Associate Professor of Electrical Engineering and Music in Olin

College, applied Chaotic Mapping to generate musical variations. The goal was to inspire

composers from the generated ideas.

A chaotic mapping provides a technique for generating musical variations of an original work.

This technique, based on the sensitivity of chaotic trajectories to initial conditions, produces

changes in the pitch sequence of a piece. A sequence of musical pitches {pi, is paired with the

x-components {xi} of a Lorenz chaotic trajectory. In this way, the x axis becomes a pitch axis

configured according to the notes of the original composition. Then, a second chaotic trajectory,

whose initial condition differs from the first, is launched. Its x-components trigger pitches on the

pitch axis (via the mapping) that vary in sequence from the original work, thus creating a

variation. There are virtually an unlimited number of variations possible, many appealing to

experts and others alike.

The technique’s success with a highly context-dependent application such as music22

, indicates it

may prove applicable to other sequences of context dependent symbols, e.g., DNA or protein

sequences, pixel sequences from scanned art work, word sequences from prose or poetry,

textural sequences requiring some intrinsic variation, and so on.

4.4.2 CLIMBIMG WITH CHAOS

23In 2009, Caleb Phillips, a Colorado Graduate and a rock climbing enthusiast, thought that

mathematics could help him use computers to design new climbing routes. Because in indoor

rock climbing, the real challenge is to scale the wall using only some of the hand and foot holds,

following a specific route.

He made a computer program, which he named Strange Beta, which starts with an existing

climbing route, alters some climbing moves, like a left sidepull or a right match, and gives a

resulting varied route plan, using a chaotic variation generator and a fourth order Runge-Kutta

numerical integrator. It was discovered that computer aided route setting can produce routes

which climbers prefer to those set traditionally.

This proved that a chaotic systems like this can be described mathematically and computers can

produce some seemingly random results, which can be used to approximate human creativity.

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4.5 RANDOM NUMBER GENERATION

24The programming community is majorly interested in design and maintenance of standard

pseudo-random number generators that are portable and reusable. The pseudo-random number

generators, familiar to all programmers, are derived from deterministic chaotic dynamical

systems. As we shall see, when pseudo-random number generators are designed properly, the

sequences produced are completely predictable.

Random number generation has been of great interest from the beginnings of computing.

Philosophically and mathematically, the concept of randomness poses many problems.

Intuitively, we equate randomness with unpredictability.

In the past 25 years, largely due to the separate efforts of Chaitin and Kolmogorov the concept of

randomness has been made definitive. They accomplished this by developing the concept of

algorithmic complexity. The complexity of an n-digit sequence is the length in bits of the

shortest computer program that can produce the sequence. For a very regular sequence,

1111111111, for example, a very small program is needed. As the sequence becomes highly

irregular and as the length of the sequence grows beyond bounds, it can be shown that the

shortest program needed to produce the sequence is slightly larger than the sequence itself.

Clearly, any algorithmic implementation of the theoretical ideal of randomness on a computer

will be imperfect. From all view points above, a random sequence is a non-computable infinite

sequence. A measure of the "goodness" of a pseudo-random number generator is aperiodicity.

However, any finite computer algorithm implementation yields only periodic sequences--

although of very long period. Thus, random number generators found in computer languages are

referred to as "pseudo-random" number generators.

In 1951, Lehmer proposed an algorithm that has become the de facto standard pseudo-random

number generator. As it is usually implemented, the algorithm is known as a Prime Modulus

Multiplicative Linear Congruential Generator. It is better known as a Lehmer generator. The

form of the Lehmer generator is:

f(xn) = xn+1 = a xn mod m.

Park and Miller proposed a portable minimal standard Lehmer generator. Their choice of m, the

prime modulus, is 231 - 1, and their choice of the multiplier is a = 75. Their minimal standard

Lehmer generator is given by:

f(x) = 16807 x mod 2147483647.

Qualitatively the behavior of chaotic systems is non-periodic, greatly disordered, deterministic--

yet apparently unpredictable and random. In current usage, a system is chaotic if it has sensitive

dependence on initial conditions. Associated with this sensitive dependence is a geometric

growth in small differences with time.

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The simplest chaotic dynamical system is the Bernoulli shift described by Palmore:

Xn+1 = D xn mod 1,

where D is an integer larger than 1.

The Bernoulli shift is a simple algorithm that exhibits complex chaotic behavior. It is

deterministic in that it is a specific algorithm implemented by a definite set of instructions in a

computer language on a computer with finite precision. It is characterized by exponential growth

and disorder.

By inspection, it is similar in form to the Lehmer generator. With the example of the Bernoulli

shift above, we observe that prime modulus multiplicative linear congruential generators are

implementations of deterministic chaotic processes. They are Bernoulli shifts on integers. The

multiplier a and the modulus m are chosen with great care to ensure that a full period is

produced. A specific implementation of a full period generator is a single cycle system. Seeding

starts the generator at a given point on the cycle--the entire sequence will be produced if the

generator is called m times. This algorithm depends sensitively on the choice of multiplier and

modulus.

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5. LIMITATIONS OF CHAOS THEORY

The major and most significant limitation of chaos theory is the feature that defines it: sensitive

dependence on initial conditions.

The limitations of applying chaos theory are in due mostly from choosing the input parameters.

The methods chosen to compute these parameters depend on the dynamics underlying the data

and on the kind of analysis intended, which is in most cases highly complex and not always

accurate. Theoretical chaos results are seriously constrained by the need for large amounts of

preliminary data.

Chaos theory in its current form is limited. At its present stage of development, it can be used to

ask if experimental data were generated by a random or deterministic process, but it is a difficult

and frustrating analytical approach to use. It is not clear how much data are required in order to

construct the phase space set and determine its fractal dimension; the amounts of data are likely

to be extremely large, and biological systems may not remain in a single state long enough to

gather the required amounts of data. Present technology too, while managing calculations and

data, also falls short at some level.

Because this theory is still developing, many ideas continue to emerge, and hence concepts are

redefined or complemented continuously. Scientists are trying to link chaos theory with other

scientific disciplines to establish a more general theory.

Some systems do not seem to benefit from the results of chaos theory in general. Chaos will not

appear in slow systems, i.e., where events are infrequent or where a great deal of friction

dissipates energy and damps out disturbances.

One may encounter scenarios and systems with erratic behavior where a source of chaos is not

immediately evident. In this event, it may be necessary to examine different scales of behavior.

For example, chaos theory does not help study the flight of a single bird, free to choose where

and when to fly. However, there is evidence of chaos in how groups of birds flock and travel

together.

Even though chaos theory helps us in taking better decisions and designing new strategies for

future, we cannot completely rely on its outcomes, else this habit of short-term observe-and-

manage would put organization’s long term future in danger. Chaos theory is not as simplistic to

find an immediate and direct application in the business environment, but mapping of the

business environment using the knowledge of chaos theory definitely is worthwhile.

A new statistical hypothesis testing must be designed in near future to analyze the ever growing

chaotic patterns and analyze subtle changes in them.

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6. CONCLUSION

It’s well known that the heart has to be largely regular or you die. But the brain has to be largely

irregular; if not, you have epilepsy. This shows that irregularity, chaos, leads to complex

systems. It’s not all disorder. On the contrary, I would say chaos is what makes life and

intelligence possible.

- Ilya Prigogine

The natural world has always had a chaotic way about it. We can find chaos theory everywhere

around us: in simple pendulum, stock market, solar system, weather forecasting, image

processing, biological systems, human body and so on. Chaotic systems are deterministic in

nature but they may appear to be random. Chaotic systems are very sensitive to the initial

conditions which means that a slight change in the starting point can lead to enormously different

outcomes. This makes the system fairly unpredictable. Chaos systems never repeat but they

always have some order. Most of the systems we find in the world predicted by classical physics

are the exceptions but in this world of order, chaos rules. Chaos theory is a new way of thinking

about what we have. It gives us a new concept of measurements and scales. It looks at the

universe in an entirely different way. Understanding chaos is understanding life as we know it.

Because of chaos, it is realized that even simple systems may give rise to and, hence, be used as

models for complex behavior.

Chaos forms a bridge between different fields. Chaos offers a fresh way to proceed with

observational data, especially those data which may be ignored because they proved too erratic.

Just as relativity eliminated the Newtonian illusion of absolute space and time, and as quantum

theory eliminated the Newtonian and Einsteinian dream of a controllable measurement

process, chaos eliminates the Laplacian fantasy of long-time deterministic predictability. That

is the reason why chaos theory has been seen as potentially “one of the three greatest triumphs of

the 21st century.” In 1991, James Marti speculated that ‘Chaos might be the new world order.’

And it definitely might.

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APPENDIX I – Other Applications of Chaos Theory

APPLICATION OF CHAOS THEORY IN NEGOTIATIONS

Richard Halpern1, in an article published in 2008, summarizes impact of Chaos Theory and

Heisenberg Uncertainty Principle on case negotiations in law, as follows-

(1) Never rely on someone else's measurement to formulate a key component of strategy.

� A small mistake can cause huge repercussions (SDIC). Hence better do it yourself.

(2) Keep trying something new, unexpected; sweep the defence of its feet.

� Make the system chaotic.

(3) If the process is going the way you wanted, simplify it as much as possible.

� Predictability would increase and chance of blunders is minimized.

(4) If the tide is running against you, add new elements: complicate.

� Nothing to lose, and with a little help from Chaos, everything to gain. You might turn

a hopeless case into a winner.

CHAOS THEORY IN PHILOSOPHY

You can stay at home and be happily introspective or you can make a choice, step out, and be the

Butterfly that begins the tempest that changes the world.

- John Sanford

In this infinite world, amongst billions of people, we, alone, are just one person. One might think

nobody would care what we think. But our intentions and actions matter, because of the Butterfly

Effect. We are profoundly affected by the world around us, but sometimes the reverse is true as

well. If we keep the intention to save all beings from suffering, that influences our actions.

Because this universe we find ourselves in is most certainly a complex system, there's no telling

what effect those intentions and actions might have. They will, however, most certainly have an

effect on the system... small or large. Everything we think, everything we do, is significant. Only

by looking backward, can we identify those events that would later prove to have

“revolutionary” consequences.2

When we realize that every action matters, then there are no wasted moments, no insignificant

gestures. Each moment is filled with portent, and possibilities. Maybe the next thing you do will

tip the scales, and the world will go one way rather than another. We just don't know.

What a responsibility!

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CHAOS THEORY IN DISASTER MANAGEMENT

Disaster response organizations are dynamic systems. A dynamical system consists of two parts:

a rule or 'dynamic,' which specifies how a system evolves, and an initial condition or 'state' from

which the system starts. Some dynamical systems evolve in exceedingly complex ways, being

irregular and initially appearing to defy any rule. Chaos theory gives a way to analyze such

systems and develop disaster management techniques.

The interaction of the workplace rules of motion with the field of action determines the direction

and result of motion in the workplace. By applying the logistic equation (Xn+1 = kXn(1-Xn)) to the

appropriate disaster response data, it is possible to determine if a disaster organization or

response system traces the universal route to chaos. Recent work in evolutionary theory and

simulation studies supports the view that organizations at the edge of chaos tend to be highly

adaptive and hence succeed.

The application of chaos theory to disaster management has some preliminary empirical support.

The terms make metaphorical sense and seem to be consistent with disaster manager's

experiences, hence leading to more effective models.3

CHAOS THEORY IN ART

Complexity and self-organization emerge from disorder as the result of a simple process, giving

rise to exquisite patterns.

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a) Pattern formed by the vibration of sand on a metal plate

(Source: http://www.mi.sanu.ac.rs/vismath/jaynew/fig01a.jpg)

b) Vibration of a thin film of glycerine, From Cymatics by Hans Jenny

(Source: http://www.mi.sanu.ac.rs/vismath/jaynew/fig01b.jpg)

Van Gogh's painting, "Starry Night", Fractals in art

Source: http://www.mi.sanu.ac.rs/vismath/jaynew/fig12.jpg

Mandelbrot and Julia sets, Lyapunov diagrams also produce beautiful visuals. Computer

graphics generated with equations obtained through Chaos Theory also breed brilliant pieces of

machine-made art.

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CHAOS THEORY IN POPULAR CULTURE

Chaos theory has been mentioned in numerous movies and works of literature. For instance, it

was mentioned extensively in Michael Crichton’s novel Jurassic Park and more briefly in its

sequel, The Lost World. Other examples include the film Chaos, The Science of Sleep, The

Butterfly Effect, the sitcom The Big Bang Theory and Community, Tom Stoppard's play Arcadia,

Lawrence Sterne’s novel Tristram Shandy, Pfitz by Andrew Crumey, Ray Bradbury’s short story

A Sound Of Thunder and the video games Tom Clancy's Splinter Cell: Chaos Theory and

Assassin's Creed.

The influence of chaos theory in shaping the popular understanding of the world we live in was

the subject of the BBC documentary High Anxieties — The Mathematics of Chaos directed by

David Malone. Chaos theory was also the subject of discussion in the BBC documentary "The

Secret Life of Chaos" presented by the physicist Jim Al-Khalil. Indian movie “Dasavathaaram”

also used idea of butterfly effect where the 2004 Tsunami in the Indian Ocean is portrayed as

being caused by the sinking of Lord Vishnu’s idol along with Kamal Hassan in the 12th

century.4

SOLAR SYSTEM CHAOS

Astronomers and cosmologists have known for quite some time that the solar system does not

run with the precision of a Swiss watch. Inabilities occur in the motions of Saturn's moon

Hyperion, gaps in the asteroid belt between Mars and Jupiter, and in the orbit of the planets

themselves. For centuries astronomers tried to compare the solar system to a gigantic clock

around the sun; however, they found that their equations never actually predicted the real planets'

movement.5

Solar system represents a typical 3-body system. The vectors become infinite and the system

becomes chaotic. This prevents a definitive analytical solution to the equations of motion.

Extreme sensitivity to initial conditions is quantified by the exponential divergence of nearby

orbits.

The recent advances are the beginning of a quest to tease out the critical properties of our solar

system (and its subsystems) that give it the curious character of being only marginally chaotic or

marginally stable on time spans comparable with its current age. It is but a part of the quest to

understand what processes of formation (and perhaps initial conditions) led to this remarkable

system in nature and how common such systems are in our galaxy and the universe.6

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APPENDIX II – EXAMPLES

Examples of Fractals:

Sierpinski's Triangle demonstrates this quite well: a triangle within smaller triangles within

smaller triangles within ever smaller triangles, on and on. Many shapes in nature display this

same quality of self-similarity. Clouds, ferns, coastlines, mountains, etc. all possess this feature.

Sierpenski Triangle and its fractal dimensions

Source: http://eldar.mathstat.uoguelph.ca/dashlock/ftax/Gallery/Siepinski1D960.gif

The Koch Snowflake is also a well noted, simple fractal image. To construct a Koch Snowflake,

begin with a triangle with sides of length 1. At the middle of each side, add a new triangle one-

third the size; and repeat this process for an infinite amount of iterations. The length of the

boundary is 3 X 4/3 X 4/3 X 4/3...-infinity. However, the area remains less than the area of a

circle drawn around the original triangle. What this means is that an infinitely long line

surrounds a finite area.

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Koch Snowflake

Source: http://www.zeuscat.com/andrew/chaos/vonkoch.html

LORENZIAN WATER WHEEL

To check the equations that he had derived from the weather model, Lorenz created a thought

experiment. He used a waterwheel, with a set number of buckets, usually more than seven,

spaced equally around its rim. The buckets are mounted on swivels, much like Ferris-wheel

seats, so that the buckets will always open upwards. At the bottom of each bucket is a small hole.

The entire waterwheel system is than mounted under a waterspout.

A slow, constant stream of water is propelled from the waterspout. The waterwheel would begin

to spin at a fairly constant rate. Lorenz decided to increase the flow of water, and, as predicted in

his Lorenz Attractor, an interesting phenomenon arose. The increased velocity of the water

resulted in a chaotic motion for the waterwheel. The waterwheel would revolve in one direction

as before, but then it would suddenly jerk about and revolve in the opposite direction. The filling

and emptying of the buckets was no longer synchronized; the system was now chaotic. Lorenz

observed his mysterious waterwheel for hours, and, no matter how long he recorded the positions

and contents of the buckets, he could not find a specific relation binding the movements. The

waterwheel would continue on in chaotic behavior without ever repeating any of its previous

conditions. A graph of the waterwheel would resemble the Lorenz Attractor.7

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LIST OF REFERENCES

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