A remarkable comilation on Chaos Theory and Fractals Geometry by Manus J. Donahue. There are certain images in the document which I am unable to upload here. Please email me
motasim.ahmad@gmail.com or Mr.Donahue
mjd@duke.edu for full PDF File and other remarkable compilations on Chaos Theory and Fractals Geometry.
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
Chaos Theory
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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.
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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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Chaos Theory: Printable Version
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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