## Welcome to the Journey to Science of Complexity, Chaos Theory & Non Linear System Dynamics:

Apple Computer Advertising 1997

## Follow the links they will take you to the Land of Smart and Ultra Genius

- Valuable Resources on Taleb's Work includes Multimedia
- Our 50 Strong Fcaebook Group on Systems Thinking, Complexity, Chaos and Learning
- Our Group on Fcaebook- The Black Swan Risk Management Stripped
- Societty of Organisational Learning is result of Peter Senge's outstanding work in the fields of Organisational Learning
- Santa Fe Institute- foremost in research of Complex Systems, Non Linear Dynamics Chaos Theory, Complexity and a lot more
- Wikipedia Page on Benoit Mandelbrto, the father of Fractal Geometry. The applications of his research extend far beyond imaging. Zooming into and out of Mandelbrot Set gives an impression as if we were viewing the creation. Our interest here in the work of Mandelbrot is its applications to Risk Management and understanding the dynamics of Financial Markets.
- Nassim Nicholas Taleb's Home Page- My Hero & Guru
- Mandelbrot's Undergraduate Course Resoueces on Fractal Geometry

## Monday, December 22, 2008

## Sunday, November 23, 2008

### Randall On 'Discovery Of Black Swan'

## Thursday, November 6, 2008

## Tuesday, October 28, 2008

### Nassim Taleb Discussion with Benoit Mandelbrot

## Friday, October 17, 2008

### What caused the current Financial Crisis

## Wednesday, August 13, 2008

## Wednesday, July 23, 2008

### Peter Senge's Society for Organisational Learning

Senge defines a learningorganization as the one that can create the results it truly desires.Through learning we are able to do something we never were able to do. Five disciplines of learning organization are:

1) Building shared vision

2) Mental models

3) Team learning

4) Personal mastery

5) Systems thinking- The Fifth Discipline that integrates the other 4.

**The Laws of theFifthDiscipline:**

1: Today's problems come from yesterday's "solutions."

2)

*The harder you push, the harder the system pushes back.*

3) Behavior will grow worse before it grows better.

4) The easy way out usually leads back in.

5) The cure can be worse than the disease.

6) Faster is slower.

**7) Cause and effect are not closely related in time and space.**

8) Small changes can produce big results.but the areas of highest leverage are often the least obvious.8) Small changes can produce big results.but the areas of highest leverage are often the least obvious.

9)You can have your cake and eat it too-but not all at once.

10) Dividing an elephant in half does not produce two small elephants.

11) There is no one to blame but us.

I have created an Orkut community on Senge's Fifth Disciplne. You are invited:

http://www.orkut.com/Community.aspx?cmm=49320492

## Saturday, July 19, 2008

### Chaos Theory every where

## Thursday, July 10, 2008

### Nassim Taleb Video Presentation at LongNow Foundation

## Sunday, June 29, 2008

### An Introduction to Mathematical Chaos Theory and Fractal

Chaos Theory

An Introduction to Mathematical Chaos Theory and Fractal

Geometry

For a printable version of this document, please click here.

The following essay was compiled by me, Manus J. Donahue III (second year Physics and

Philosophy major at Duke University..age 19). It has been unofficially published in four

different countries, has been cited in The New York Times and has been awarded

technology site of the day by TechSightings.com. Please cite this page as a reference if you

use any of the material on this page in essays, documents, or presentaitons. Also, you may

e-mail me at mjd@duke.edu if you have any questions, and I'll try to get back with you as

soon as possible.

Because I compiled this essay for myself and the enjoyment of others, and because I am

presenting it completely free, I am not responsible for any copyright violations or anything

like that. Some of the pictures that are included in this essay (although almost universally

common) were taken from other Web pages. If you are a high school/middle school student

who has to do a report on chaos theory and you print this essay off and turn it in, you will

be violating not only the work of myself, but the various other people who unknowingly

may have contributed to this site. Don't do that - these people deserve credit for their work!

Use this paper merely as a “jumping off point” for your own research, and then write a

paper that is even better – and publish it. I wrote this essay because I was always fascinated

by chaos theory and non-linear math and I could never find explanative essays aimed at the

“average person.” Hopefully this one is.

Finally, I get a lot of e-mails from people asking me to recommend books and other

resources on chaos theory. The books that I used while writing this paper are included

below; they’re the ones that I would recommend:

Chaos: Making a New Science by James Gleick

Fractals: The Patterns of Chaos by John Briggs

In the Wake of Chaos by Stephen H. Kellert

Needless to say, there is also a surfeit of resources on the Web concerning chaos theory and

fractals – so they would be worth checking out too.

"Physicists like to think that all you have to do is say, these are the conditions, now

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what happens next?" -Richard P. Feynman

The world of mathematics 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.

However, the problem arises that we humans do not live in an even remotely linear world;

in fact, our world should indeed be categorized as nonlinear; hence, proportion and linearity

is scarce. How may one go about pursuing and understanding a nonlinear system in a world

that is confined to the easy, logical linearity of everything? This is the question that

scientists and mathematicians became burdened with in the 19th Century; hence, a new

science and mathematics was derived: chaos theory.

The very name "chaos theory" seems to contradict reason, in fact it seems somewhat of an

oxymoron. The name "chaos theory" leads the reader to believe that mathematicians have

discovered some new and definitive knowledge about utterly random and incomprehensible

phenomena; however, this is not entirely the case. The acceptable definition of chaos theory

states, chaos theory is the qualitative study of unstable aperiodic behavior in deterministic

nonlinear dynamical systems. A dynamical system may be defined to be a simplified model

for the time-varying behavior of an actual system, and aperiodic behavior is simply the

behavior that occurs when no variable describing the state of the system undergoes a

regular repetition of values. Aperiodic behavior never repeats and it continues to manifest

the effects of any small perturbation; hence, any prediction of a future state in a given

system that is aperiodic is impossible. Assessing the idea of aperiodic behavior to a relevant

example, one may look at human history. History is indeed aperiodic since broad patterns in

the rise and fall of civilizations may be sketched; however, no events ever repeat exactly.

What is so incredible about chaos theory is that unstable aperiodic behavior can be found in

mathematically simply systems. These very simple mathematical systems display behavior

so complex and unpredictable that it is acceptable to merit their descriptions as random.

An interesting question arises from many skeptics concerning why chaos has just recently

been noticed. If chaotic systems are so mandatory to our every day life, 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. No human is stupid enough to endure the boredom; however, a

computer is always up to the challenge. Computers have always been known for their

excellence at mindless repetition; hence, the computer is our telescope when studying

chaos. For, without a doubt, one cannot really explore chaos without a computer.

Before advancing into the more precocious and advanced areas of chaos, it is necessary to

touch on the basic principle that adequately describes chaos theory, the Butterfly Effect.

The Butterfly Effect was vaguely understood centuries ago and is still satisfactorily

portrayed in folklore:

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"For want of a nail, the shoe was lost;

For want of a shoe, the horse was lost;

For want of a horse, the rider was lost;

For want of a rider, the battle was lost;

For want of a battle, the kingdom was lost!"

Small variations in initial conditions result in huge, dynamic transformations in concluding

events. That is to say that there was no nail, and, therefore, the kingdom was lost. The

graphs of what seem to be identical, dynamic systems appear to diverge as time goes on

until all resemblance disappears.

Perhaps the most identifiable symbol linked with the Butterfly Effect is the famed Lorenz

Attractor. Edward Lorenz, a curious meteorologist, was looking for a way to model the

action of the chaotic behavior of a gaseous system. Hence, he took a few equations from the

physics field of fluid dynamics, simplified them, and got the following three-dimensional

system:

dx/dt=delta*(y-x)

dy/dt=r*x-y-x*z

dz/dt=x*y-b*z

Delta represents the "Prandtl number," the ratio of the fluid viscosity of a substance to its

thermal conductivity; however, one does not have to know the exact value of this constant;

hence, Lorenz simply used 10. The variable "r" represents the difference in temperature

between the top and bottom of the gaseous system. The variable "b" is the width to height

ratio of the box which is being used to hold the gas in the gaseous system. Lorenz used 8/3

for this variable. The resultant x of the equation represents the rate of rotation of the

cylinder, "y" represents the difference in temperature at opposite sides of the cylinder, and

the variable "z" represents the deviation of the system from a linear, vertical graphed line

representing temperature. If one were to plot the three differential equations on a

three-dimensional plane, using the help of a computer of course, no geometric structure or

even complex curve would appear; instead, a weaving object known as the Lorenz Attractor

appears. Because the system never exactly repeats itself, the trajectory never intersects

itself. Instead it loops around forever. I have included a computer animated Lorenz

Attractor which is quite similar to the production of Lorenz himself. The following Lorenz

Attractor was generated by running data through a 4th-order Runge-Kutta fixed-timestep

integrator with a step of .0001, printing every 100th data point. It ran for 100 seconds, and

only took the last 4096 points. The original parameters were a =16, r =45, and b = 4 for the

following equations (similar to the original Lorenz equations):

x'=a(y-x)

y'=rx-y-xz

z'=xy-bz

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The initial position of the projectory was (8,8,14). When the points were generated and

graphed, the Lorenz Attractor was produced in 3-D:

The attractor will continue weaving back and forth between the two wings, its motion

seemingly random, its very action mirroring the chaos which drives the process. Lorenz had

obviously made an immense breakthrough in not only chaos theory, but life. Lorenz had

proved that complex, dynamical systems show order, but they never repeat. Since our world

is classified as a dynamical, complex system, our lives, our weather, and our experiences

will never repeat; however, they should form patterns.

Lorenz, not quite convinced with his results, did a follow-up experiment in order to support

his previous conclusions. Lorenz established an experiment that was quite simple; it is

known today as the Lorenzian Waterwheel. Lorenz took a waterwheel; it had about eight

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buckets spaced evenly around its rim with a small hole at the bottom of each . The buckets

were mounted on swivels, similar to Ferris-wheel seats, so that the buckets would always

point upwards. The entire system was placed under a waterspout. A slow, constant stream

of water was propelled from the waterspout; hence, the waterwheel began to spin at a fairly

constant rate. Lorenz decided to increase the flow of water, and, as predicted in his Lorenz

Attractor, an interesting phenomena 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, there was never and instance where the

waterwheel was in the same position twice. 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.

Now it may be accepted from Lorenz and his comrades that our world is indeed linked with

an eery form of chaos. Chaos and randomness are no longer ideas of a hypothetical world;

they are quite realistic here in the status quo. A basis for chaos is established in the

Butterfly Effect, the Lorenz Attractor, and the Lorenz Waterwheel; therefore, there must be

an immense world of chaos beyond the rudimentary fundamentals. This new form

mentioned is highly complex, repetitive, and replete with intrigue.

"I coined fractal from the Latin adjective fractus. The corresponding Latin verb

frangere means "to break": to create irregular fragments. It is therefore sensible-and

how appropriate for our needs!-that, in addition to "fragmented", fractus should also

mean "irregular," both meanings being preserved in fragment." -Benoit Mandelbrot

The extending and folding of chaotic systems give strange attractors, such as the Lorenz

Attractor, the distinguishing characteristic of a nonintegral dimension. This nonintegral

dimension is most commonly referred to as a fractal dimension. Fractals appear to be more

popular in the status quo for their aesthetic nature than they are for their mathematics.

Everyone who has seen a fractal has admired the beauty of a colorful, fascinating image,

but what is the formula that makes up this glitzy image? The classical Euclidean geometry

that one learns in school is quite different than the fractal geometry mainly because fractal

geometry concerns nonlinear, nonintegral systems while Euclidean geometry is mainly

oriented around linear, integral systems. Hence, Euclidean geometry is a description of

lines, ellipses, circles, etc. However, fractal geometry is a description of algorithms. There

are two basic properties that constitute a fractal. First, is self-similarity, which is to say that

most magnified images of fractals are essentially indistinguishable from the unmagnified

version. A fractal shape will look almost, or even exactly, the same no matter what size it is

viewed at. This repetitive pattern gives fractals their aesthetic nature. Second, as mentioned

earlier, fractals have non-integer dimensions. This means that they are entirely different

from the graphs of lines and conic sections that we have learned about in fundamental

Euclidean geometry classes. By taking the midpoints of each side of an equilateral triangle

and connecting them together, one gets an interesting fractal known as the Sierpenski

Triangle. The iterations are repeated an infinite number or times and eventually a very

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simple fractal arises:

In addition to the famous Sierpenski Triangle, 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. The

end construction of a Koch Snowflake resembles the coastline of a shore.

The two fundamental fractals that I have included provided a basis for much more complex,

elaborate fractals. Two of the leading reasearchers in the field of fractals were Gaston

Maurice Julia and Benoit Mandelbrot. Their discoveries and breakthroughs will be

discussed next.

On February 3rd, 1893, Gaston Maurice Julia was born in Sidi Bel Abbes, Algeria. Julia

was injured while fighting in World War I and was forced to wear a leather strap across his

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face for the rest of his life in order to protect and cover his injury. he spent a large majority

of his life in hospitals; therefore, a lot of his mathematical research took place in the

hospital. At the age of 25, Julia published a 199 page masterpiece entitled "Memoire sur

l'iteration des fonctions." The paper dealt with the iteration of a rational function. With the

publication of this paper came his claim to fame. Julia spent his life studying the iteration of

polynomials and rational functions. If f(x) is a function, various behaviors arise when "f" is

iterated or repeated. If one were to start with a particular value for x, say x=a, then the

following would result:

a, f(a), f(f(a)), f(f(f(a))), etc.

Repeatedly applying "f" to "a" yields arbitrarily large values. Hence, the set of numbers is

partitioned into two parts, and the Julia set associated to "f" is the boundary between the

two sets. The filled Julia set includes those numbers x=a for which the iterates of "f"

applied to "a" remain bounded. The following fractals belong to the Julia set.

Julia became famous around the 1920's; however, upon his demise, he was essentially

forgotten. It was not until 1970 that the work of Gaston Maurice Julia was revived and

popularized by Polish born Benoit Mandelbrot.

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark

is not smooth, nor does lightning travel in a straight line." -Benoit Mandelbrot

Benoit Mandelbrot was born in Poland in 1924. When he was 12 his family emigrated to

France and his uncle, Szolem Mandelbrot, took responsibility for his education. It is said

that Mandelbrot was not very successful in his schooling; in fact, he may have never

learned his multiplication tables. When Benoit was 21, his uncle showed him Julia's

important 1918 paper concerning fractals. Benoit was not overly impressed with Julia's

work, and it was not until 1977 that Benoit became interested in Julia's discoveries.

Eventually, with the aid of computer graphics, Mandelbrot was able to show how Julia's

work was a source of some of the most beautiful fractals known today. The Mandelbrot set

is made up of connected points in the complex plane. The simple equation that is the basis

of the Mandelbrot set is included below.

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changing number + fixed number = Result

In order to calculate points for a Mandelbrot fractal, start with one of the numbers on the

complex plane and put its value in the "Fixed Number" slot of the equation. In the

"Changing number" slot, start with zero. Next, calculate the equation. Take the number

obtained as the result and plug it into the "Changing number" slot. Now, repeat (iterate) this

operation an infinite number or times. When iterative equations are applied to points in a

certain region of the complex plane, a fractal from the Mandelbrot set results. A few fractals

from the Mandelbrot set are included below.

Benoit Mandelbrot currently works at IBM's Watson Research Center. In addition, he is a

Professor of the Practice of Mathematics at Harvard University. He has been awarded the

Barnard Medal for Meritorious Service to Science, the Franklin Medal, the Alexander von

Humboldt Prize, the Nevada Medal, and the Steinmetz Medal. His work with fractals has

truly influenced our world immensely.

It is now established that fractals are quite real and incredible; however, what do these

newly discovered objects have to do with real life? Is there a purpose behind these

fascinating images? The answer is a somewhat surprising yes. Homer Smith, a computer

engineer of Art Matrix, once said, "If you like fractals, it is because you are made of them.

If you can't stand fractals, it's because you can't stand yourself." 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.

Understanding and mastering the concepts that govern fractals will undoubtedly lead to

breakthroughs in the area of biological understanding. Fractals are one of the most

interesting branches of chaos theory, and they are beginning to become ever more key in

the world of biology and medicine.

George Cantor, a nineteenth century mathematician, became fascinated by the infinite

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number of points on a line segment. Cantor began to wonder what would happen when and

infinite number of line segments were removed from an initial line interval. Cantor devised

an example which portrayed classical fractals made by iteratively taking away something.

His operation created a "dust" of points; hence, the name Cantor Dust. In order to

understand Cantor Dust, start with a line; remove the middle third; then remove the middle

third of the remaining segments; and so on. The operation is shown below.

The Cantor set is simply the dust of points that remain. The number of these points are

infinite, but their total length is zero. Mandelbrot saw the Cantor set as a model for the

occurerence of errors in an electronic transmission line. Engineers saw periods of errorless

transmission, mixed with periods when errors would come in gusts. When these gusts of

errors were analyzed, it was determined that they contained error-free periods within them.

As the transmissions were analyzed to smaller and smaller degrees, it was determined that

such dusts, as in the Cantor Dust, were indispensable in modeling intermittency.

"It's an experience like no other experience I can describe, the best thing that can

happen to a scientist, realizing that something that's happened in his or her mind

exactly corresponds to something that happens in nature. It's startling every time it

occurs. One is surprised that a construct of one's own mind can actually be realized in

the honest-to-goodness world out there. A great shock, and a great, great joy." -Leo

Kadanoff

The fractals and iterations are fun to look at; the Cantor Dust and Koch Snowflakes are fun

to think about, but what breakthroughs can be made in terms of discovery? Is chaos theory

anything more than a new way of thinking? The future of chaos theory is unpredictable, but

if a breakthrough is made, it will be huge. However, miniature discoveries have been made

in the field of chaos within the past century or so, and, as expected, they are mind boggling.

The first consumer product to exploit chaos theory was produced in 1993 by Goldstar Co.

in the form of a revolutionary washing machine. A chaotic washing machine? The washing

machine is based on the principle that there are identifiable and predictable movements in

nonlinear systems. The new washing machine was designed to produce cleaner and less

tangled clothes. The key to the chaotic cleaning process can be found in a small pulsator

that rises and falls randomly as the main pulsator rotates. The new machine was

surprisingly successful. However, Daewoo, a competitor of Goldstar claims that they first

started commercializing chaos theory in their "bubble machine" which was released in

1990. The "bubble machine" was the first to use the revolutionary "fuzzy logic circuits."

These circuits are capable of making choices between zero and one, and between true and

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false. Hence, the "fuzzy logic circuits" are responsible for controlling the amount of

bubbles, the turbulence of the machine, and even the wobble of the machine. Indeed, chaos

theory is very much a factor in today's consumer world market.

The stock markets are said to be nonlinear, dynamic systems. Chaos theory is the

mathematics of studying such nonlinear, dynamic systems. Does this mean that chaoticians

can predict when stocks will rise and fall? Not quite; however, chaoticians have determined

that the market prices are highly random, but with a trend. The stock market is accepted as a

self-similar system in the sense that the individual parts are related to the whole. Another

self-similar system in the area of mathematics are fractals. Could the stock market be

associated with a fractal? Why not? In the market price action, if one looks at the market

monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance.

However, just like a fractal, the stock market has sensitive dependence on initial conditions.

This factor is what makes dynamic market systems so difficult to predict. Because we

cannot accurately describe the current situation with the detail necessary, we cannot

accurately predict the state of the system at a future time. Stock market success can be

predicted by chaoticians. Short-term investing, such as intra day exchanges are a waste of

time. Short-term traders will fail over time due to nothing more than the cost of trading.

However, over time, long-term price action is not random. Traders can succeed trading

from daily or weekly charts if they follow the trends. A system can be random in the

short-term and deterministic in the long term.

Perhaps even more important than stock market chaos and predictability is 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. It is easy to understand how two bodies will revolve

around a common center of gravity. However, what happens when a third, fourth, fifth or

infinite number of gravitational attractions are introduced? The vectors become infinite and

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

of motion. Even with the advanced computers that we have today, the long term

calculations are far too lengthy. Stephen Hawking once said, "If we find the answer to that

(the universe), it would be the ultimate triumph of human reason-for then we would know

the mind of God.

The applications of chaos theory are infinite; seemingly random systems produce patterns

of spooky understandable irregularity. From the Mandelbrot set to turbulence to feedback

and strange attractors; chaos appears to be everywhere. Breakthroughs have been made in

the past in the area chaos theory, and, in order to achieve any more colossal

accomplishments in the future, they must continue to be made. Understanding chaos is

understanding life as we know it.

However, if we do discover a complete theory, it should in time be understandable in

broad principle by everyone, not just a few scientists. Then we shall all, philosophers,

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scientists, and just ordinary people, be able to take part in the discussion of the

question of why it is that we and the universe exist.

-Stephen Hawking

TC

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Chaos Theory: Printable Version

An Introduction to Chaos Theory and Fractal

Geometry

Author: Manus J. Donahue III

Copyright Fall 1997, all rights reserved.

"Physicists like to think that all you have to do is say, these are the conditions, now what

happens next?" -Richard P. Feynman

The world of mathematics 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. However, the problem arises that we humans do not live in an even

remotely linear world; in fact, our world should indeed be categorized as nonlinear; hence, proportion

and linearity is scarce. How may one go about pursuing and understanding a nonlinear system in a

world that is confined to the easy, logical linearity of everything? This is the question that scientists

and mathematicians became burdened with in the 19th Century; hence, a new science and

mathematics was derived: chaos theory.

The very name "chaos theory" seems to contradict reason, in fact it seems somewhat of an oxymoron.

The name "chaos theory" leads the reader to believe that mathematicians have discovered some new

and definitive knowledge about utterly random and incomprehensible phenomena; however, this is

not entirely the case. The acceptable definition of chaos theory states, chaos theory is the qualitative

study of unstable aperiodic behavior in deterministic nonlinear dynamical systems. A dynamical

system may be defined to be a simplified model for the time-varying behavior of an actual system,

and aperiodic behavior is simply the behavior that occurs when no variable describing the state of the

system undergoes a regular repetition of values. Aperiodic behavior never repeats and it continues to

manifest the effects of any small perturbation; hence, any prediction of a future state in a given

system that is aperiodic is impossible. Assessing the idea of aperiodic behavior to a relevant example,

one may look at human history. History is indeed aperiodic since broad patterns in the rise and fall of

civilizations may be sketched; however, no events ever repeat exactly. What is so incredible about

chaos theory is that unstable aperiodic behavior can be found in mathematically simply systems.

These very simple mathematical systems display behavior so complex and unpredictable that it is

acceptable to merit their descriptions as random.

An interesting question arises from many skeptics concerning why chaos has just recently been

noticed. If chaotic systems are so mandatory to our every day life, 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. No human is stupid

enough to endure the boredom; however, a computer is always up to the challenge. Computers have

always been known for their excellence at mindless repetition; hence, the computer is our telescope

when studying chaos. For, without a doubt, one cannot really explore chaos without a computer.

http://www.duke.edu/~mjd/chaos/chaosp.html (1 sur 9)2004-01-31 19:36:44

Chaos Theory: Printable Version

Before advancing into the more precocious and advanced areas of chaos, it is necessary to touch on

the basic principle that adequately describes chaos theory, the Butterfly Effect. The Butterfly Effect

was vaguely understood centuries ago and is still satisfactorily portrayed in folklore:

"For want of a nail, the shoe was lost;

For want of a shoe, the horse was lost;

For want of a horse, the rider was lost;

For want of a rider, the battle was lost;

For want of a battle, the kingdom was lost!"

Small variations in initial conditions result in huge, dynamic transformations in concluding events.

That is to say that there was no nail, and, therefore, the kingdom was lost. The graphs of what seem

to be identical, dynamic systems appear to diverge as time goes on until all resemblance disappears.

Perhaps the most identifiable symbol linked with the Butterfly Effect is the famed Lorenz Attractor.

Edward Lorenz, a curious meteorologist, was looking for a way to model the action of the chaotic

behavior of a gaseous system. Hence, he took a few equations from the physics field of fluid

dynamics, simplified them, and got the following three-dimensional system:

dx/dt=delta*(y-x)

dy/dt=r*x-y-x*z

dz/dt=x*y-b*z

Delta represents the "Prandtl number," the ratio of the fluid viscosity of a substance to its thermal

conductivity; however, one does not have to know the exact value of this constant; hence, Lorenz

simply used 10. The variable "r" represents the difference in temperature between the top and bottom

of the gaseous system. The variable "b" is the width to height ratio of the box which is being used to

hold the gas in the gaseous system. Lorenz used 8/3 for this variable. The resultant x of the equation

represents the rate of rotation of the cylinder, "y" represents the difference in temperature at opposite

sides of the cylinder, and the variable "z" represents the deviation of the system from a linear, vertical

graphed line representing temperature. If one were to plot the three differential equations on a threedimensional

plane, using the help of a computer of course, no geometric structure or even complex

curve would appear; instead, a weaving object known as the Lorenz Attractor appears. Because the

system never exactly repeats itself, the trajectory never intersects itself. Instead it loops around

forever. I have included a computer animated Lorenz Attractor which is quite similar to the

production of Lorenz himself. The following Lorenz Attractor was generated by running data through

a 4th-order Runge-Kutta fixed-timestep integrator with a step of .0001, printing every 100th data

point. It ran for 100 seconds, and only took the last 4096 points. The original parameters were a =16,

r =45, and b = 4 for the following equations (similar to the original Lorenz equations):

x'=a(y-x)

y'=rx-y-xz

z'=xy-bz

The initial position of the projectory was (8,8,14). When the points were generated and graphed, the

Lorenz Attractor was produced in 3-D:

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The attractor will continue weaving back and forth between the two wings, its motion seemingly

random, its very action mirroring the chaos which drives the process. Lorenz had obviously made an

immense breakthrough in not only chaos theory, but life. Lorenz had proved that complex, dynamical

systems show order, but they never repeat. Since our world is classified as a dynamical, complex

system, our lives, our weather, and our experiences will never repeat; however, they should form

patterns.

Lorenz, not quite convinced with his results, did a follow-up experiment in order to support his

previous conclusions. Lorenz established an experiment that was quite simple; it is known today as

the Lorenzian Waterwheel. Lorenz took a waterwheel; it had about eight buckets spaced evenly

around its rim with a small hole at the bottom of each . The buckets were mounted on swivels, similar

to Ferris-wheel seats, so that the buckets would always point upwards. The entire system was placed

under a waterspout. A slow, constant stream of water was propelled from the waterspout; hence, the

waterwheel began to spin at a fairly constant rate. Lorenz decided to increase the flow of water, and,

as predicted in his Lorenz Attractor, an interesting phenomena arose. The increased velocity of the

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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, there was never and instance where the waterwheel was in the same position twice.

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.

Now it may be accepted from Lorenz and his comrades that our world is indeed linked with an eery

form of chaos. Chaos and randomness are no longer ideas of a hypothetical world; they are quite

realistic here in the status quo. A basis for chaos is established in the Butterfly Effect, the Lorenz

Attractor, and the Lorenz Waterwheel; therefore, there must be an immense world of chaos beyond

the rudimentary fundamentals. This new form mentioned is highly complex, repetitive, and replete

with intrigue.

"I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere

means "to break": to create irregular fragments. It is therefore sensible-and how appropriate

for our needs!-that, in addition to "fragmented", fractus should also mean "irregular," both

meanings being preserved in fragment." -Benoit Mandelbrot

The extending and folding of chaotic systems give strange attractors, such as the Lorenz Attractor,

the distinguishing characteristic of a nonintegral dimension. This nonintegral dimension is most

commonly referred to as a fractal dimension. Fractals appear to be more popular in the status quo for

their aesthetic nature than they are for their mathematics. Everyone who has seen a fractal has

admired the beauty of a colorful, fascinating image, but what is the formula that makes up this glitzy

image? The classical Euclidean geometry that one learns in school is quite different than the fractal

geometry mainly because fractal geometry concerns nonlinear, nonintegral systems while Euclidean

geometry is mainly oriented around linear, integral systems. Hence, Euclidean geometry is a

description of lines, ellipses, circles, etc. However, fractal geometry is a description of algorithms.

There are two basic properties that constitute a fractal. First, is self-similarity, which is to say that

most magnified images of fractals are essentially indistinguishable from the unmagnified version. A

fractal shape will look almost, or even exactly, the same no matter what size it is viewed at. This

repetitive pattern gives fractals their aesthetic nature. Second, as mentioned earlier, fractals have noninteger

dimensions. This means that they are entirely different from the graphs of lines and conic

sections that we have learned about in fundamental Euclidean geometry classes. By taking the

midpoints of each side of an equilateral triangle and connecting them together, one gets an interesting

fractal known as the Sierpenski Triangle. The iterations are repeated an infinite number or times and

eventually a very simple fractal arises:

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In addition to the famous Sierpenski Triangle, 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. The end construction of a Koch Snowflake resembles the coastline

of a shore.

The two fundamental fractals that I have included provided a basis for much more complex, elaborate

fractals. Two of the leading reasearchers in the field of fractals were Gaston Maurice Julia and Benoit

Mandelbrot. Their discoveries and breakthroughs will be discussed next.

On February 3rd, 1893, Gaston Maurice Julia was born in Sidi Bel Abbes, Algeria. Julia was injured

while fighting in World War I and was forced to wear a leather strap across his face for the rest of his

life in order to protect and cover his injury. he spent a large majority of his life in hospitals; therefore,

a lot of his mathematical research took place in the hospital. At the age of 25, Julia published a 199

page masterpiece entitled "Memoire sur l'iteration des fonctions." The paper dealt with the iteration

of a rational function. With the publication of this paper came his claim to fame. Julia spent his life

studying the iteration of polynomials and rational functions. If f(x) is a function, various behaviors

arise when "f" is iterated or repeated. If one were to start with a particular value for x, say x=a, then

the following would result:

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a, f(a), f(f(a)), f(f(f(a))), etc.

Repeatedly applying "f" to "a" yields arbitrarily large values. Hence, the set of numbers is partitioned

into two parts, and the Julia set associated to "f" is the boundary between the two sets. The filled Julia

set includes those numbers x=a for which the iterates of "f" applied to "a" remain bounded. The

following fractals belong to the Julia set.

Julia became famous around the 1920's; however, upon his demise, he was essentially forgotten. It

was not until 1970 that the work of Gaston Maurice Julia was revived and popularized by Polish born

Benoit Mandelbrot.

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not

smooth, nor does lightning travel in a straight line." -Benoit Mandelbrot

Benoit Mandelbrot was born in Poland in 1924. When he was 12 his family emigrated to France and

his uncle, Szolem Mandelbrot, took responsibility for his education. It is said that Mandelbrot was

not very successful in his schooling; in fact, he may have never learned his multiplication tables.

When Benoit was 21, his uncle showed him Julia's important 1918 paper concerning fractals. Benoit

was not overly impressed with Julia's work, and it was not until 1977 that Benoit became interested in

Julia's discoveries. Eventually, with the aid of computer graphics, Mandelbrot was able to show how

Julia's work was a source of some of the most beautiful fractals known today. The Mandelbrot set is

made up of connected points in the complex plane. The simple equation that is the basis of the

Mandelbrot set is included below.

changing number + fixed number = Result

In order to calculate points for a Mandelbrot fractal, start with one of the numbers on the complex

plane and put its value in the "Fixed Number" slot of the equation. In the "Changing number" slot,

start with zero. Next, calculate the equation. Take the number obtained as the result and plug it into

the "Changing number" slot. Now, repeat (iterate) this operation an infinite number or times. When

iterative equations are applied to points in a certain region of the complex plane, a fractal from the

Mandelbrot set results. A few fractals from the Mandelbrot set are included below.

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Benoit Mandelbrot currently works at IBM's Watson Research Center. In addition, he is a Professor

of the Practice of Mathematics at Harvard University. He has been awarded the Barnard Medal for

Meritorious Service to Science, the Franklin Medal, the Alexander von Humboldt Prize, the Nevada

Medal, and the Steinmetz Medal. His work with fractals has truly influenced our world immensely.

It is now established that fractals are quite real and incredible; however, what do these newly

discovered objects have to do with real life? Is there a purpose behind these fascinating images? The

answer is a somewhat surprising yes. Homer Smith, a computer engineer of Art Matrix, once said, "If

you like fractals, it is because you are made of them. If you can't stand fractals, it's because you can't

stand yourself." 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. Understanding

and mastering the concepts that govern fractals will undoubtedly lead to breakthroughs in the area of

biological understanding. Fractals are one of the most interesting branches of chaos theory, and they

are beginning to become ever more key in the world of biology and medicine.

George Cantor, a nineteenth century mathematician, became fascinated by the infinite number of

points on a line segment. Cantor began to wonder what would happen when and infinite number of

line segments were removed from an initial line interval. Cantor devised an example which portrayed

classical fractals made by iteratively taking away something. His operation created a "dust" of points;

hence, the name Cantor Dust. In order to understand Cantor Dust, start with a line; remove the middle

third; then remove the middle third of the remaining segments; and so on. The operation is shown

below.

The Cantor set is simply the dust of points that remain. The number of these points are infinite, but

their total length is zero. Mandelbrot saw the Cantor set as a model for the occurerence of errors in an

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electronic transmission line. Engineers saw periods of errorless transmission, mixed with periods

when errors would come in gusts. When these gusts of errors were analyzed, it was determined that

they contained error-free periods within them. As the transmissions were analyzed to smaller and

smaller degrees, it was determined that such dusts, as in the Cantor Dust, were indispensable in

modeling intermittency.

"It's an experience like no other experience I can describe, the best thing that can happen to a

scientist, realizing that something that's happened in his or her mind exactly corresponds to

something that happens in nature. It's startling every time it occurs. One is surprised that a

construct of one's own mind can actually be realized in the honest-to-goodness world out there.

A great shock, and a great, great joy." -Leo Kadanoff

The fractals and iterations are fun to look at; the Cantor Dust and Koch Snowflakes are fun to think

about, but what breakthroughs can be made in terms of discovery? Is chaos theory anything more

than a new way of thinking? The future of chaos theory is unpredictable, but if a breakthrough is

made, it will be huge. However, miniature discoveries have been made in the field of chaos within

the past century or so, and, as expected, they are mind boggling.

The first consumer product to exploit chaos theory was produced in 1993 by Goldstar Co. in the form

of a revolutionary washing machine. A chaotic washing machine? The washing machine is based on

the principle that there are identifiable and predictable movements in nonlinear systems. The new

washing machine was designed to produce cleaner and less tangled clothes. The key to the chaotic

cleaning process can be found in a small pulsator that rises and falls randomly as the main pulsator

rotates. The new machine was surprisingly successful. However, Daewoo, a competitor of Goldstar

claims that they first started commercializing chaos theory in their "bubble machine" which was

released in 1990. The "bubble machine" was the first to use the revolutionary "fuzzy logic circuits."

These circuits are capable of making choices between zero and one, and between true and false.

Hence, the "fuzzy logic circuits" are responsible for controlling the amount of bubbles, the turbulence

of the machine, and even the wobble of the machine. Indeed, chaos theory is very much a factor in

today's consumer world market.

The stock markets are said to be nonlinear, dynamic systems. Chaos theory is the mathematics of

studying such nonlinear, dynamic systems. Does this mean that chaoticians can predict when stocks

will rise and fall? Not quite; however, chaoticians have determined that the market prices are highly

random, but with a trend. The stock market is accepted as a self-similar system in the sense that the

individual parts are related to the whole. Another self-similar system in the area of mathematics are

fractals. Could the stock market be associated with a fractal? Why not? In the market price action, if

one looks at the market monthly, weekly, daily, and intra day bar charts, the structure has a similar

appearance. However, just like a fractal, the stock market has sensitive dependence on initial

conditions. This factor is what makes dynamic market systems so difficult to predict. Because we

cannot accurately describe the current situation with the detail necessary, we cannot accurately

predict the state of the system at a future time. Stock market success can be predicted by chaoticians.

Short-term investing, such as intra day exchanges are a waste of time. Short-term traders will fail

over time due to nothing more than the cost of trading. However, over time, long-term price action is

not random. Traders can succeed trading from daily or weekly charts if they follow the trends. A

system can be random in the short-term and deterministic in the long term.

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Perhaps even more important than stock market chaos and predictability is 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. It is easy to

understand how two bodies will revolve around a common center of gravity. However, what happens

when a third, fourth, fifth or infinite number of gravitational attractions are introduced? The vectors

become infinite and the system becomes chaotic. This prevents a definitive analytical solution to the

equations of motion. Even with the advanced computers that we have today, the long term

calculations are far too lengthy. Stephen Hawking once said, "If we find the answer to that (the

universe), it would be the ultimate triumph of human reason-for then we would know the mind of

God.

The applications of chaos theory are infinite; seemingly random systems produce patterns of spooky

understandable irregularity. From the Mandelbrot set to turbulence to feedback and strange attractors;

chaos appears to be everywhere. Breakthroughs have been made in the past in the area chaos theory,

and, in order to achieve any more colossal accomplishments in the future, they must continue to be

made. Understanding chaos is understanding life as we know it.

However, if we do discover a complete theory, it should in time be understandable in broad

principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and

just ordinary people, be able to take part in the discussion of the question of why it is that we

and the universe exist.

-Stephen Hawking

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## Saturday, June 21, 2008

### Taleb Takes Alan Greenspan To Task

Sitting 17 weeks on the New York Times best-seller list, “Black Swan” outsold former U.S. Federal Reserve Governor Alan Greenspan's “The Age of Turbulence” months ago.

So, how does it feel?

“Greenspan is an empty suit,” he told the Turkish Daily News in Istanbul, one of the latest stops on his lecture tour. “He does not understand economic life and he does not know that he doesn't know. And his book is boring. I despise the man.”

Taleb says the turmoil vindicates him once again. “Greenspan is a man who plays with economic life without understanding its basic structure. In today's world, links between action and consequences are not as visible as they were in the past.”

A major mistake of Greenspan was letting the banking system cluster, he said. “Thus, you end up with a gigantic bank and lose the natural ecology. If a restaurant does not give decent food, the owner goes bust. But banks get clustered. So you end up with one single source of risk and that is JP Morgan!”

“In the U.S., you trade with any bank, you are trading with JP Morgan. I barked about this for years, but then Bear Stearns went bust and JP Morgan ended up taking it,” he said. For Taleb, a system that banks do not go bust means a system that risk is highly concentrated.

He cited an example from another realm. “Which one has more political volatility? Italy or Saudi Arabia? Of course Italy, because they had 62 governments since World War II. But Saudi Arabia has had the same family in power since you guys left them,” he said, referring to the Ottoman Empire. “But Italy has much less risk than Saudi Arabia.”

So, some entities like Bear Stearns do not have volatility but are very risky, while some that are risky do not have volatility. “Greenspan and others do not understand this,” he said. “They never let the banks fail. I want them to fail, because I love the banking system. Finance is too important to be left to U.S. central bankers.”

In his trading days, Taleb was a legend due to a few incredible “hits.” The most legendary of these was in 1987, when he was working for First Boston. At 28 years of age, he made the right bet on Eurodollar futures when nobody else did. On Oct. 19, the Dow Jones Industrial Average declined 22.6 percent, the biggest one-day drop in the United States ever. Eurodollar futures surged after the Fed pumped liquidity into the banking system in a rush, lowering interbank borrowing rates.

Investment choices:

The majority of his personal fortune today is still based on that lucky day. His choice of investing that fortune tells something about Taleb's philosophy. “I like things that are volatile. Instead of investing in medium-risk securities, I invested 90 percent in no-risk government bonds. But my 10 percent is in extremely risky choices.”

“Some businesses, such as biotech, or emerging markets, can benefit from the black swan,” he said. “The problem is, some businesses, like banks in the U.S., have a lot of downside exposure, but no upside exposure.”

The basic rule for Taleb is simple: “If you need a mathematician to understand what you have in your books, you're a blowup.”

“I trained lots of these people,” he continued. “And I tell you, my students were incompetent. I would not give them my car to drive, or even to wash. Mathematics does not work in real life.”

Does “Extremistan” mean the old saying that history repeats itself is not valid anymore? “People tend to learn first order from history. The best example would be the Maginot Line. When Germans came, the French built a wall. What did the Germans do? They went around it,” he continued. “First order thinking is like, ‘Let's make sure we are prepared for a second 22 percent stock market crash.' Because it had never been that worse. But then, the 22 percent crash did not have a predecessor, so history would not have taught you that.”

Taleb has told “the guys at Morgan Stanley” that they are “morons” precisely because of that. “They were doing historical stress testing on their subprime portfolio. But how can you do that when history does not have a predecessor?”

Then he explains his “second order thinking” so rapidly, one might think he cannot repeat these words again: “There is a past, the past's past and the past's future. Then there is today, today's past and today's future. You should work with today's future in relation to today's past the way the past's future worked with the past's past.”

“Simple peasants understand this thinking. But bring in someone with a PhD who works at a bank on risk management, he does not. It's like autism. Thus, the more mathematicians you have in a bank, the more likely it is to blow up.”

From Lebanon to war on terror:

A political “black swan” from Taleb's childhood was the Lebanese civil war. “Nobody saw it coming,” he said. “My father was telling me that it would be over in a week. It went on 17 years. But today, the black swan for Lebanon is peace.”

The black swan takes on another quality if it is spotted. “Anytime you identify a source of randomness, you overestimate its probability and commit mistakes,” Taleb explained. “Today we overestimate terrorism. Give a retard like [George W.] Bush an army and he starts inventing sources of risk.”

The biggest source of risk for humankind is not terrorism but diabetes, which kills 80 million people every year, he argued. “Our reaction to terror causes more people to die than terrorism itself. Nearly 3,000 people died on Sept. 11, 2001. But in the aftermath, many more died due to traffic accidents because they were afraid of flying.” Nearly 600 extra deaths on U.S. and European roads per month after 9/11, he said.

If diabetes is the biggest source of risk today, economists come after it. “We have too many economists,” he said. “The Federal Reserve is dangerous. So is Davos. All pseudo-experts.”

Overoptimization:

Now, this reporter was warned before the interview that Taleb was a “hard one to crack,” and a couple of previous interviews went astray due to colleagues' insistence on asking his prediction on oil prices, the U.S. dollar or the Turkish economy.

This time, Taleb answers without receiving the question in some sort of verbal preemptive strike. “Why did the price of food and oil rise so much?” he said himself. “Because the system is too optimized. A small imbalance of 1 percent in the demand for wheat causes prices to double. But if you look at the facts, demand for wheat is up 2 percent while supply is up 5 percent.”

Such vast price swings tell us that “forecastability in that domain is worse.” So, nobody can guarantee that a barrel of oil will not cost $40 the next day, instead of continuing its rise toward $140. And that is why Taleb is reluctant to predict.

Then, is there an alternative to be paranoid and expect the unexpected? Maybe one has to look at what Karl Marx had said decades ago, a suggestion surprisingly made by prominent businessman İshak Alaton in April.

Taleb strongly disagrees. “According to Marx, the idea is how to turn knowledge into action, and that is pure enlightenment arrogance,” he said. “My point is how to turn absence of knowledge and understanding into action.”

For that, the world has to wait for “Tinkering,” the next book of the trader-turned-philosopher. Until then, ranks of Taleb fans are sure to get more crowded. The world is hungry for new ideas and perspectives, a common phenomenon for times of such deep crises. And that is exactly what Taleb delivers.

## Wednesday, June 11, 2008

### Mandelbrot on Modern Portfolio Theory

stock and currency

traders know better than ever

that prices quoted in any financial market often

change with heart-stopping swiftness. Fortunes are

made and lost in sudden bursts of activity when the market

seems to speed up and the volatility soars. Last September,

for instance, the stock for Alcatel, a French telecommunications

equipment manufacturer, dropped about 40 percent

one day and fell another 6 percent over the next few days. In a

reversal, the stock shot up 10 percent on the fourth day.

The classical financial models used for most of this century

predict that such precipitous events should never happen. A

cornerstone of finance is modern portfolio theory, which tries

to maximize returns for a given level of risk. The mathematics

underlying portfolio theory handles extreme situations with

benign neglect: it regards large market shifts as too unlikely to

matter or as impossible to take into account. It is true that

portfolio theory may account for what occurs 95 percent of

the time in the market. But the picture it presents does not

reflect reality, if one agrees that major events are part of the

remaining 5 percent. An inescapable analogy is that of a sailor

at sea. If the weather is moderate 95 percent of the time, can

the mariner afford to ignore the possibility of a typhoon?

The risk-reducing formulas behind portfolio theory rely on

a number of demanding and ultimately unfounded premises.

First, they suggest that price changes are statistically independent

of one another: for example, that today’s price has no

influence on the changes between the current price and tomorrow’s.

As a result, predictions of future market movements

become impossible. The second presumption is that all

price changes are distributed in a pattern that conforms to

the standard bell curve. The width of the bell shape (as measured

by its sigma, or standard deviation)

depicts how far price changes

diverge from the mean; events at the extremes

are considered extremely rare. Typhoons

are, in effect, defined out of existence.

Do financial data neatly conform to such assumptions?

Of course, they never do. Charts of stock or currency changes

over time do reveal a constant background of small up and

down price movements—but not as uniform as one would

expect if price changes fit the bell curve. These patterns, however,

constitute only one aspect of the graph. A substantial

number of sudden large changes—spikes on the chart that

shoot up and down as with the Alcatel stock—stand out

from the background of more moderate perturbations.

Moreover, the magnitude of price movements (both large

and small) may remain roughly constant for a year, and then

suddenly the variability may increase for an extended period.

Big price jumps become more common as the turbulence of

the market grows—clusters of them appear on the chart.

According to portfolio theory, the probability of these large

fluctuations would be a few millionths of a millionth of a millionth

of a millionth. (The fluctuations are greater than 10

standard deviations.) But in fact, one observes spikes on a regular

basis—as often as every month—and their probability

amounts to a few hundredths. Granted, the bell curve is often

described as normal—or, more precisely, as the normal distribution.

But should financial markets then be described as abnormal?

Of course not—they are what they are, and it is portfolio

theory that is flawed.

Modern portfolio theory poses a danger to those who believe

in it too strongly and is a powerful challenge for the theoretician.

Though sometimes acknowledging faults in the

present body of thinking, its adherents suggest that no other

premises can be handled through mathematical

modeling. This contention leads to the

question of whether a rigorous quantitative description

of at least some features of major financial upheavals can be

developed. The bearish answer is that large market swings

are anomalies, individual “acts of God” that present no conceivable

regularity. Revisionists correct the questionable

premises of modern portfolio theory through small fixes that

lack any guiding principle and do not improve matters

sufficiently. My own work—carried out over many years—

takes a very different and decidedly bullish position.

I claim that variations in financial prices can be accounted

for by a model derived from my work in fractal geometry.

Fractals—or their later elaboration, called multifractals—do

not purport to predict the future with certainty. But they do

create a more realistic picture of market risks. Given the recent

troubles confronting the large investment pools called

hedge funds, it would be foolhardy not to investigate models

providing more accurate estimates of risk.

## Thursday, May 29, 2008

### Benoit Mandelbrot

## Friday, April 25, 2008

### The Black Swan Ideas Gaining Momentum

"WHAT'S WRONG WITH MARKET ECONOMICS AND GDP?" by HAZEL HENDERSON:24/04/2008

(MaximsNews Network)

**UNITED NATIONS - / MaximsNews Network / 24 April 2008 -- ** The credibility of the economics profession and its macroeconomic and risk models has been shattered by the Wall Street-led financial meltdown. Many analysts see this worst crisis since World War II as the beginning of the end of market fundamentalism as the driver of globalization. Coming into focus is also the fact that the USA is no longer the world’s lone super power. Military force is giving way to the new weapons of choice in today’s geopolitics: currency and cyber-attacks.

Even US Treasury Secretary Henry Paulson (former head of one of the over-leveraged Wall Street investment banks – Goldman Sachs) now calls for regulation of these reckless, risk-taking, private banks. Former options-trader/mathematician Nassim Nicholas Taleb predicted their downfall in __The Black Swan__ (2007), as did former hedge fund "quant" Richard Bookstaber in __A Demon of Our Own Design__ (2007). Ivory tower mathematicians lured to Wall Street’s big bucks simply didn't understand the real behavior of markets – as was demonstrated back in 1998 when their faulty models led to the collapse of hedge fund Long-Term Capital Management and its bail-out orchestrated by the US Federal Reserve.

The Nobel Prize Committee shares some blame by its recognition of the faulty options pricing model, Black-Scholes Merton, with its Bank of Sweden Prize in 1993. In recent editorials, Taleb has called on the Nobel Committee to withdraw this prize while Peter Nobel himself says that the Bank of Sweden should de-link its prize in economics from the Nobels. As I have noted in my previous editorials for IPS, many other scientists agree, since economics is not a science but a profession.

Meanwhile, the long-simmering critiques of money-based GDP/GNP national accounts are coming to a head. These popular critiques, including my own, were summarized by the late Senator Robert F. Kennedy in 1968 in a speech delivered to the University of Kansas. Even GDP's creator, Simon Kuznets, worried about using GDP as an overall indicator of national progress and well-being, saying that “the welfare of a nation can scarcely be informed from a measure of national income.”

The cracks in GDP as a scorecard of national progress began appearing at the UN Earth Summit in Rio de Janeiro in 1992, followed by the European Parliament's conference in 1995 on "Taking Nature Into Account." In November 2007, the European Parliament again took up the issue at the urging of the European Commission (www.beyond-gdp.eu). Its “Beyond GDP” debate was keynoted by EU President José Manuel Barroso of Portugal before almost 700 parliamentarians and statisticians of sustainability and quality of life. Statisticians themselves also emphasized the need for better measures of national progress, with over 13,000 attending their conference in Istanbul, convened by the OECD (Organization for Economic Co-operation and Development) in June 2007. And, EU Commissioner of Economic Policy Joaquín Almunia noted that GDP “cannot distinguish between economic activities that have a negative or positive impact on wellbeing. In fact, war and natural disasters may register as an increase in GDP.”

By March 2008, the US Senate picked up these critical debates and the plethora of new, broader indicators of health, education and environment. The Senate's Committee on Commerce held its own hearing on "Rethinking GDP as a Measure of National Strength" – a low-key academic exploration on how all of these new measures of overall quality could be used to correct all the now-recognized errors in GDP that economic textbooks perpetuate.

In its March 13, 2008 issue, even __The Economist__ weighed in with "Grossly Distorted Picture," criticizing the widespread focus on GDP-growth. This "growth fetish" has long been the subject of countless critiques by environmentalists and even a few economists. To see this journal of economic and free-trade orthodoxy now also criticizing GDP-growth signals a tipping point in this long debate. Echoing so many earlier critiques, __The Economist__ pointed out that a better measure than rates of GDP-growth would be to compare GDP per head – a much more tangible sign of progress that takes into account the growth of population. For example, Japan's GDP growth has been about 2.1% over the past five years, while GDP in the USA has grown 2.9%.

Yet, comparing the average growth of income per capita between the two countries, a different story emerges: the USA saw only a 1.9% increase while Japanese citizens’ income grew by 2.1%. This was among the reasons I have urged Japan to shift from GDP growth to quality-of-life indicators (__Nikkei Ecology__, August 2000). I pointed out that Japan had matured beyond the need for more material growth and could now concentrate on higher-level services and improving quality of life. Japan’s average income-per-head also was greater because Japan's population is shrinking while the US population is rising. India has enjoyed rapid GDP-growth, but its population has grown much faster, leaving more people to share that income.

__The Economist__ is correct that the growth of average income per capita is the more realistic indicator. But, they omit another problem with these GDP measures: averaging per capita of growth in incomes masks how that income is distributed. Averaging incomes across the whole population could mean that a country might have a few billionaires while most of its citizens live in poverty.

Let’s agree that GDP has outlived its usefulness (started as a World War II measure of war production). There are now many new, better indicators, from the Canadian Index of Wellbeing (CIW), the UN's Human Development Index (HDI), the World Bank’s Wealth Index to Genuine Progress Index (GPI), Bhutan's Gross National Happiness (GNH) to the Calvert-Henderson Quality of Life Indicators I created with the Calvert Group of socially responsible mutual funds (the only private-sector effort so far, updated regularly at www.calvert-henderson.com).

Once again, the public is ahead of the experts and politicians on this issue. A GlobeScan survey in 10 countries in November 2007, in conjunction with the Beyond GDP Conference in the European Parliament, found large majorities in India, Russia, Germany, France, Italy, Britain as well as Australia, Brazil and Kenya favored broader scorecards of progress beyond money-based GDP, including indicators of health, education and environment. Real wealth and progress can never be quantified only in money. The economics textbooks are overdue for revision.

Hazel Henderson is author of __Ethical Markets: Growing the Green Economy__ (2007) and other books. She co-organized the Beyond GDP Conference in Brussels, representing the Club of Rome. www.hazelhenderson.com

## Thursday, April 24, 2008

### Sub Prime mess- A nail in the coffin of Traditional Risk Management

Here is the first indication:

Death of VaR Evoked as Risk-Taking Vim Meets Taleb

Jan. 28 (Bloomberg) -- The risk-taking model that emboldened Wall Street to trade with impunity is broken and everyone from Merrill Lynch & Co. Chief Executive Officer John Thain to Morgan Stanley Chief Financial Officer Colm Kelleher is coming to the realization that no algorithm or triple-A rating can substitute for old-fashioned due diligence.Value at risk, the measure banks use to calculate the maximum their trades can lose each day, failed to detect the scope of the U.S. subprime mortgage market's collapse as it triggered more than $130 billion of losses since June for the biggest securities firms led by Citigroup Inc., Merrill, Morgan Stanley and UBS AG.The past six months have exposed the flaws of a financial measure based on historical prices that securities firms use idiosyncratically and that doesn't anticipate every potential disaster, such as the mistaken credit ratings on defaulted subprime debt.``Finance is an area that's dominated by rare events,'' said Nassim Taleb, a research professor at London Business School and former options trader. ``The tools we have in quantitative finance do not work in what I call the `Black Swan' domain.''Taleb's book ``The Black Swan,'' published last year by Random House, describes how people underestimate the impact of infrequent occurrences. Just as it was assumed that all swans were white until the first black species was spotted in Australia during the 17th century, historical analysis is an inadequate way to judge risk, he said. for full articlehttp://www.bloomberg.com/apps/news?pid=20601109&sid=axo1oswvqx4s&refer=home