Chaos Theory: A Brief Introduction
What exactly is chaos? Put simply, it is the idea that it is possible to
get apparently random results from normal equations. The techniques of
chaos theory also cover the reverse: finding the order in what appears
to be 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, but it did predict the weather of a hypothetical
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
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 out of sync from the original. 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)
, 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
, background noise
, or an inaccuracy
equipment. Such things are impossible to avoid in even the most isolated
lab. With a starting number of 2, the final result in a chaotic system
can be entirely different from the same system with a starting value of
2.000001. In many experiments, it is simply impossible to achieve this
level of accuracy - just try and measure something to the nearest
millionth of an second!
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
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 in mathematical
systems: 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
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. In one model of this system, 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. A simple model that takes this into
account is the following:
next year's population = r * this year's population * (1 - this
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, all hell broke loose. As soon as the growth rate passed
3, the line broke in two. Instead of settling down to a single size, the
population would jump between two different sizes. 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, or 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, of the graph of final solutions, it is possible
to see white strips where the population never reaches certain sizes.
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
Market dynamics and Coastlines
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
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
(James Gleick, Chaos - Making a New Science, pg. 86)
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
bays in between grains of sand. No matter how much a
coastline is magnified, there will be more bays visible if it is
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. 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 difficult to conceive, but it is possible to understand
what's going on to some extent. 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 sideline research area for scientists working on
problems that were related to 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. 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
typically, different colors represent how quickly a point diverges from
the set. One interesting feature of the Mandelbrot set is that the
circular humps match up to the bifurcation graph of the population
growth equation. 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 periodic intervals. 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.
Other 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 scientific research. 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
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. Many musicians who hear the new
sounds believe that the variations are very musical and creative.
The philosophical implications of chaos have already had a lasting
effect on science. Now it is recognized that even when it is possible
to reduce a system to a simple mathematical model, the question is more
complex than if the system has a stable or unstable solution. Now
there's something in between: the chaotic attractor, a bit more stable
than complete randomness, but more dynamic than a simple periodic
The first draft of this essay was for my final project in a
high school advanced composition class. It has been available online at
my web site since 1996, most recently at the address
http://www.imho.com/grae/chaos (and that one has
pictures...). This version has been edited for use on E2.
- "Bach to Chaos: Chaotic Variations on a Classical Theme", Science
News, Dec. 24, 1994, pg. 428.
- Gleick, James, Chaos - Making a New Science, Penguin Books Ltd,
Harmondsworth, Middlesex, 1987.
- Lowrie, Peter, personal interview over the Internet, May 17,
- Rae, Kevin, "Chaos", unpublished paper, submitted to Professor
Gould, Modern Physics class, Claremont McKenna College, December 5,
- Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, Penguin
Books Ltd, Harmondsworth, Middlesex, 1989.