Chaos
Theory: A Brief
Introduction
by Greg Rae
What exactly is
chaos? The name
"chaos theory"
comes from the
fact that the
systems that the
theory describes
are apparently
disordered, but
chaos theory is
really about
finding the
underlying order
in apparently
random data.
When was
chaos first
discovered? The
first true
experimenter in
chaos was a
meteorologist,
named Edward
Lorenz. In 1960,
he was working
on the problem
of weather
prediction. He
had a computer
set up, with a
set of twelve
equations to
model the
weather. It
didn't predict
the weather
itself. However
this computer
program did
theoretically
predict what the
weather might
be.
One day in
1961, he wanted
to see a
particular
sequence again.
To save time, he
started in the
middle of the
sequence,
instead of the
beginning. He
entered the
number off his
printout and
left to let it
run.
When he came
back an hour
later, the
sequence had
evolved
differently.
Instead of the
same pattern as
before, it
diverged from
the pattern,
ending up wildly
different from
the original.
(See figure 1.)
Eventually he
figured out what
happened. The
computer stored
the numbers to
six decimal
places in its
memory. To save
paper, he only
had it print out
three decimal
places. In the
original
sequence, the
number was
.506127, and he
had only typed
the first three
digits, .506.
By all
conventional
ideas of the
time, it should
have worked. He
should have
gotten a
sequence very
close to the
original
sequence. A
scientist
considers
himself lucky if
he can get
measurements
with accuracy to
three decimal
places. Surely
the fourth and
fifth,
impossible to
measure using
reasonable
methods, can't
have a huge
effect on the
outcome of the
experiment.
Lorenz proved
this idea wrong.
This effect
came to be known
as the butterfly
effect. The
amount of
difference in
the starting
points of the
two curves is so
small that it is
comparable to a
butterfly
flapping its
wings.
The
flapping of
a single
butterfly's
wing today
produces a
tiny change
in the state
of the
atmosphere.
Over a
period of
time, what
the
atmosphere
actually
does
diverges
from what it
would have
done. So, in
a month's
time, a
tornado that
would have
devastated
the
Indonesian
coast
doesn't
happen. Or
maybe one
that wasn't
going to
happen,
does. (Ian
Stewart,
Does
God Play
Dice? The
Mathematics
of Chaos,
pg. 141)
This
phenomenon,
common to chaos
theory, is also
known as
sensitive
dependence on
initial
conditions. Just
a small change
in the initial
conditions can
drastically
change the
long-term
behavior of a
system. Such a
small amount of
difference in a
measurement
might be
considered
experimental
noise,
background
noise, or an
inaccuracy of
the equipment.
Such things are
impossible to
avoid in even
the most
isolated lab.
With a starting
number of 2, the
final result can
be entirely
different from
the same system
with a starting
value of
2.000001. It is
simply
impossible to
achieve this
level of
accuracy - just
try and measure
something to the
nearest
millionth of an
inch!
From this
idea, Lorenz
stated that it
is impossible to
predict the
weather
accurately.
However, this
discovery led
Lorenz on to
other aspects of
what eventually
came to be known
as chaos theory.
Lorenz
started to look
for a simpler
system that had
sensitive
dependence on
initial
conditions. His
first discovery
had twelve
equations, and
he wanted a much
more simple
version that
still had this
attribute. He
took the
equations for
convection, and
stripped them
down, making
them
unrealistically
simple. The
system no longer
had anything to
do with
convection, but
it did have
sensitive
dependence on
its initial
conditions, and
there were only
three equations
this time.
Later, it was
discovered that
his equations
precisely
described a
water wheel.
At the
top, water
drips
steadily
into
containers
hanging on
the wheel's
rim. Each
container
drips
steadily
from a small
hole. If the
stream of
water is
slow, the
top
containers
never fill
fast enough
to overcome
friction,
but if the
stream is
faster, the
weight
starts to
turn the
wheel. The
rotation
might become
continuous.
Or if the
stream is so
fast that
the heavy
containers
swing all
the way
around the
bottom and
up the other
side, the
wheel might
then slow,
stop, and
reverse its
rotation,
turning
first one
way and then
the other.
(James
Gleick,
Chaos
- Making a
New Science,
pg. 29)
The
equations for
this system also
seemed to give
rise to entirely
random behavior.
However, when he
graphed it, a
surprising thing
happened. The
output always
stayed on a
curve, a double
spiral. There
were only two
kinds of order
previously
known: a steady
state, in which
the variables
never change,
and periodic
behavior, in
which the system
goes into a
loop, repeating
itself
indefinitely.
Lorenz's
equations were
definitely
ordered - they
always followed
a spiral. They
never settled
down to a single
point, but since
they never
repeated the
same thing, they
weren't periodic
either. He
called the image
he got when he
graphed the
equations the
Lorenz
attractor. (See
figure 2)
In 1963,
Lorenz published
a paper
describing what
he had
discovered. He
included the
unpredictability
of the weather,
and discussed
the types of
equations that
caused this type
of behavior.
Unfortunately,
the only journal
he was able to
publish in was a
meteorological
journal, because
he was a
meteorologist,
not a
mathematician or
a physicist. As
a result,
Lorenz's
discoveries
weren't
acknowledged
until years
later, when they
were
rediscovered by
others. Lorenz
had discovered
something
revolutionary;
now he had to
wait for someone
to discover him.
Another
system in which
sensitive
dependence on
initial
conditions is
evident is the
flip of a coin.
There are two
variables in a
flipping coin:
how soon it hits
the ground, and
how fast it is
flipping.
Theoretically,
it should be
possible to
control these
variables
entirely and
control how the
coin will end
up. In practice,
it is impossible
to control
exactly how fast
the coin flips
and how high it
flips. It is
possible to put
the variables
into a certain
range, but it is
impossible to
control it
enough to know
the final
results of the
coin toss.
A similar
problem occurs
in ecology, and
the prediction
of biological
populations. The
equation would
be simple if
population just
rises
indefinitely,
but the effect
of predators and
a limited food
supply make this
equation
incorrect. The
simplest
equation that
takes this into
account is the
following:
next year's
population = r *
this year's
population * (1
- this year's
population)
In this
equation, the
population is a
number between 0
and 1, where 1
represents the
maximum possible
population and 0
represents
extinction. R is
the growth rate.
The question
was, how does
this parameter
affect the
equation? The
obvious answer
is that a high
growth rate
means that the
population will
settle down at a
high population,
while a low
growth rate
means that the
population will
settle down to a
low number. This
trend is true
for some growth
rates, but not
for every one.
One
biologist,
Robert May,
decided to see
what would
happen to the
equation as the
growth rate
value changes.
At low values of
the growth rate,
the population
would settle
down to a single
number. For
instance, if the
growth rate
value is 2.7,
the population
will settle down
to .6292. As the
growth rate
increased, the
final population
would increase
as well. Then,
something weird
happened.
As soon as the
growth rate
passed 3, the
line broke in
two. Instead of
settling down to
a single
population, it
would jump
between two
different
populations. It
would be one
value for one
year, go to
another value
the next year,
then repeat the
cycle forever.
Raising the
growth rate a
little more
caused it to
jump between
four different
values. As the
parameter rose
further, the
line bifurcated
(doubled) again.
The bifurcations
came faster and
faster until
suddenly, chaos
appeared. Past a
certain growth
rate, it becomes
impossible to
predict the
behavior of the
equation.
However, upon
closer
inspection, it
is possible to
see white
strips. Looking
closer at these
strips reveals
little windows
of order, where
the equation
goes through the
bifurcations
again before
returning to
chaos. This
self-similarity,
the fact that
the graph has an
exact copy of
itself hidden
deep inside,
came to be an
important aspect
of chaos.
An employee
of IBM, Benoit
Mandelbrot was a
mathematician
studying this
self-similarity.
One of the areas
he was studying
was cotton price
fluctuations. No
matter how the
data on cotton
prices was
analyzed, the
results did not
fit the normal
distribution.
Mandelbrot
eventually
obtained all of
the available
data on cotton
prices, dating
back to 1900.
When he analyzed
the data with
IBM's computers,
he noticed an
astonishing
fact:
The
numbers that
produced
aberrations
from the
point of
view of
normal
distribution
produced
symmetry
from the
point of
view of
scaling.
Each
particular
price change
was random
and
unpredictable.
But the
sequence of
changes was
independent
on scale:
curves for
daily price
changes and
monthly
price
changes
matched
perfectly.
Incredibly,
analyzed
Mandelbrot's
way, the
degree of
variation
had remained
constant
over a
tumultuous
sixty-year
period that
saw two
World Wars
and a
depression.
(James
Gleick,
Chaos
- Making a
New Science,
pg. 86)
Mandelbrot
analyzed not
only cotton
prices, but many
other phenomena
as well. At one
point, he was
wondering about
the length of a
coastline. A map
of a coastline
will show many
bays. However,
measuring the
length of a
coastline off a
map will miss
minor bays that
were too small
to show on the
map. Likewise,
walking along
the coastline
misses
microscopic bays
in between
grains of sand.
No matter how
much a coastline
is magnified,
there will be
more bays
visible if it is
magnified more.
One
mathematician,
Helge von Koch,
captured this
idea in a
mathematical
construction
called the Koch
curve. To create
a Koch curve,
imagine an
equilateral
triangle. To the
middle third of
each side, add
another
equilateral
triangle.
Keep on adding
new triangles to
the middle part
of each side,
and the result
is a Koch curve.
(See figure 4) A
magnification of
the Koch curve
looks exactly
the same as the
original. It is
another
self-similar
figure.
The Koch
curve brings up
an interesting
paradox. Each
time new
triangles are
added to the
figure, the
length of the
line gets
longer. However,
the inner area
of the Koch
curve remains
less than the
area of a circle
drawn around the
original
triangle.
Essentially, it
is a line of
infinite length
surrounding a
finite area.
To get around
this difficulty,
mathematicians
invented fractal
dimensions.
Fractal comes
from the word
fractional. The
fractal
dimension of the
Koch curve is
somewhere around
1.26. A
fractional
dimension is
impossible to
conceive, but it
does make sense.
The Koch curve
is rougher than
a smooth curve
or line, which
has one
dimension. Since
it is rougher
and more
crinkly, it is
better at taking
up space.
However, it's
not as good at
filling up space
as a square with
two dimensions
is, since it
doesn't really
have any area.
So it makes
sense that the
dimension of the
Koch curve is
somewhere in
between the two.
Fractal has
come to mean any
image that
displays the
attribute of
self-similarity.
The bifurcation
diagram of the
population
equation is
fractal. The
Lorenz Attractor
is fractal. The
Koch curve is
fractal.
During this
time, scientists
found it very
difficult to get
work published
about chaos.
Since they had
not yet shown
the relevance to
real-world
situations, most
scientists did
not think the
results of
experiments in
chaos were
important. As a
result, even
though chaos is
a mathematical
phenomenon, most
of the research
into chaos was
done by people
in other areas,
such as
meteorology and
ecology. The
field of chaos
sprouted up as a
hobby for
scientists
working on
problems that
maybe had
something to do
with it.
Later, a
scientist by the
name of
Feigenbaum was
looking at the
bifurcation
diagram again.
He was looking
at how fast the
bifurcations
come. He
discovered that
they come at a
constant rate.
He calculated it
as 4.669. In
other words, he
discovered the
exact scale at
which it was
self-similar.
Make the diagram
4.669 times
smaller, and it
looks like the
next region of
bifurcations. He
decided to look
at other
equations to see
if it was
possible to
determine a
scaling factor
for them as
well. Much to
his surprise,
the scaling
factor was
exactly the
same. Not only
was this
complicated
equation
displaying
regularity, the
regularity was
exactly the same
as a much
simpler
equation. He
tried many other
functions, and
they all
produced the
same scaling
factor, 4.669.
This was a
revolutionary
discovery. He
had found that a
whole class of
mathematical
functions
behaved in the
same,
predictable way.
This
universality
would help other
scientists
easily analyze
chaotic
equations.
Universality
gave scientists
the first tools
to analyze a
chaotic system.
Now they could
use a simple
equation to
predict the
outcome of a
more complex
equation.
Many
scientists were
exploring
equations that
created fractal
equations. The
most famous
fractal image is
also one of the
most simple. It
is known as the
Mandelbrot set (pictures
of the
mandelbrot set).
The equation is
simple:
z=z2+c .
To see if a
point is part of
the Mandelbrot
set, just take a
complex number
z. Square it,
then add the
original number.
Square the
result, then add
the original
number. Repeat
that ad
infinitum, and
if the number
keeps on going
up to infinity,
it is not part
of the
Mandelbrot set.
If it stays down
below a certain
level, it is
part of the
Mandelbrot set.
The Mandelbrot
set is the
innermost
section of the
picture, and
each different
shade of gray
represents how
far out that
particular point
is. One
interesting
feature of the
Mandelbrot set
is that the
circular humps
match up to the
bifurcation
graph. The
Mandelbrot
fractal has the
same
self-similarity
seen in the
other equations.
In fact, zooming
in deep enough
on a Mandelbrot
fractal will
eventually
reveal an exact
replica of the
Mandelbrot set,
perfect in every
detail.
Fractal
structures have
been noticed in
many real-world
areas, as well
as in
mathematician's
minds. Blood
vessels
branching out
further and
further, the
branches of a
tree, the
internal
structure of the
lungs, graphs of
stock market
data, and many
other real-world
systems all have
something in
common: they are
all
self-similar.
Scientists at
UC Santa Cruz
found chaos in a
dripping water
faucet. By
recording a
dripping faucet
and recording
the periods of
time, they
discovered that
at a certain
flow velocity,
the dripping no
longer occurred
at even times.
When they
graphed the
data, they found
that the
dripping did
indeed follow a
pattern.
The human
heart also has a
chaotic pattern.
The time between
beats does not
remain constant;
it depends on
how much
activity a
person is doing,
among other
things. Under
certain
conditions, the
heartbeat can
speed up. Under
different
conditions, the
heart beats
erratically. It
might even be
called a chaotic
heartbeat. The
analysis of a
heartbeat can
help medical
researchers find
ways to put an
abnormal
heartbeat back
into a steady
state, instead
of uncontrolled
chaos.
Researchers
discovered a
simple set of
three equations
that graphed a
fern. This
started a new
idea - perhaps
DNA encodes not
exactly where
the leaves grow,
but a formula
that controls
their
distribution.
DNA, even though
it holds an
amazing amount
of data, could
not hold all of
the data
necessary to
determine where
every cell of
the human body
goes. However,
by using fractal
formulas to
control how the
blood vessels
branch out and
the nerve fibers
get created, DNA
has more than
enough
information. It
has even been
speculated that
the brain itself
might be
organized
somehow
according to the
laws of chaos.
Chaos even
has applications
outside of
science.
Computer art has
become more
realistic
through the use
of chaos and
fractals. Now,
with a simple
formula, a
computer can
create a
beautiful, and
realistic tree.
Instead of
following a
regular pattern,
the bark of a
tree can be
created
according to a
formula that
almost, but not
quite, repeats
itself.
Music can be
created using
fractals as
well. Using the
Lorenz
attractor, Diana
S. Dabby, a
graduate student
in electrical
engineering at
the
Massachusetts
Institute of
Technology, has
created
variations of
musical themes.
("Bach to Chaos:
Chaotic
Variations on a
Classical
Theme", Science
News, Dec. 24,
1994) By
associating the
musical notes of
a piece of music
like Bach's
Prelude in C
with the x
coordinates of
the Lorenz
attractor, and
running a
computer
program, she has
created
variations of
the theme of the
song. Most
musicians who
hear the new
sounds believe
that the
variations are
very musical and
creative.
Chaos has
already had a
lasting effect
on science, yet
there is much
still left to be
discovered. Many
scientists
believe that
twentieth
century science
will be known
for only three
theories:
relativity,
quantum
mechanics, and
chaos. Aspects
of chaos show up
everywhere
around the
world, from the
currents of the
ocean and the
flow of blood
through fractal
blood vessels to
the branches of
trees and the
effects of
turbulence.
Chaos has
inescapably
become part of
modern science.
As chaos changed
from a
little-known
theory to a full
science of its
own, it has
received
widespread
publicity. Chaos
theory has
changed the
direction of
science: in the
eyes of the
general public,
physics is no
longer simply
the study of
subatomic
particles in a
billion-dollar
particle
accelerator, but
the study of
chaotic systems
and how they
work.
Bibliography
Chaos Theory
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