“Chaos:
When the present determines the future, but the approximate present does not
approximately determine the future.” (Lorenz)

Chaos theory is a phenomenon found in
mathematics with applications in many other sciences, where initial conditions
of a system, without change of the system or its variables, can have extreme effects
on the outcome of the system. This effect is often referred to as the
“butterfly effect,” although it’s significance is much greater than what
popular culture might cause one to think. When discussing “chaos” in
mathematics, generally one is referring to mathematical chaos. Mathematical
Chaos is a behavior that looks random (chaotic) but is not. In order for “chaos”
to be referred to as mathematical chaos it must have the constraints of sensitivity,
determinism and recurrence. There are many important concepts of chaos theory
including fractals, strange attractors, sensitivity of initial conditions and bifurcation
or minimum chaos.

Chaos exists everywhere. It is found in
the weather outside, crowds of people, boiling water, governments and even in
the trees. This is also why there are so many applications for the things
learned by chaos theory. It is extremely
difficult to use two-dimensional models to study multi dimensional realities.
However, this is exactly what our traditional math attempts to do. Just because
someone makes his or her two-dimensional model more and more complex, does not
necessarily mean it will be more and more accurate. In fact the opposite may be
true if they are studying a chaotic system. By accepting that chaos does exist,
scientists have created several clever ways to identify, measure, analyze and
use it. The study of chaos has led to many new ways to analyze data many of
which also have application in linear (two-dimensional) analysis.

One of the most widely used applications
for chaos theory is in weather forecasting. As the science of meteorology, and
the tools used to analyze the weather (Doppler, satellites, computers, etc.) became
more and more advanced, scientists attempted to predict weather with increasing
complex models. The problem was that they didn’t work. While working on a weather-forecasting
model, Edward Lorenz noticed that a computer model could predict a vastly
different weather forecast from what was thought to be the same data. He
figured out that rounding some of the initial data to the third decimal point
changed the result significantly. He concluded that due to the sensitivity of
initial conditions, weather could never be forecasted more than about a week in
advance.

Chaos freed the scientists from the
obligation of exactness, but they did not give up. Instead of trying to figure
out an exact prediction of the weather, they now run iterations of multiple
models and then calculate a probability or “most likely scenario.” This kind of
forecasting is extremely more effective. However, because of the sensitivity to
initial conditions, and the complexity inherent in any weather condition, it is
still extremely difficult to predict the weather. A small error of any variable
can significantly change the forecast. Sometimes meteorologists don’t see the
storm coming until the day before. The study of chaos continues to make advances
in weather forecasting, and the science has come leaps and bounds since
Lorenz’s discovery in the 1960’s.

Another application of chaos, and one of
the most popular aspects of it, is fractals. A fractal is a an object who’s
irregularity is consistent over different scales. This is also known as
self-similarity. Sometimes something that seems random and chaotic has self-similar
properties. If you zoom in the image still looks the same. In that case, though
it may not be possible to measure the length of an infinitely complex curve, it
is possible to measure it’s “roughness.”

Made famous in the 1980’s by Benoit
Mandelbrot, and later clothing designers, fractals have many applications. In
1982, Mandelbrot published a book entitled “Fractal Geometry of Nature.”
Fractals are found everywhere in nature including in the mountains, forests,
sea shells, plants and bacteria. Mandelbrot could see fractals everywhere, when
most people only saw disorder and chaos. His book inspired many applications of
fractals including computer graphics from movies and flight simulators, antenna,
and even biologists studying the rain forest.

A good example of how fractals can bring
order to chaos is in measuring and studying rain forests. Scientists have
discovered that the distribution of branches on a single tree is self-similar
to the distributions of trees in the forest. By using this self-similarity they
can approximate the biomass of the trees in the forest and therefore the amount
of carbon dioxide the forest breaths. This allows scientists to more closely
measure the effect of the forest, including deforestation, on the global
climate.

One of the most widely used applications
of fractals is in antenna. When cell phones were first made popular they were
big and bulky. In order make the antenna strong enough, they had to loop wire
back and forth hundreds of times, and it still didn’t work very well. After
attending a presentation by Mandelbrot, and during and antenna dispute with his
landlord, Nathan Cohen decided to try and make an antenna in the shape of a
fractal. Not only did it work better, but it was also smaller and it could receive
more frequencies simultaneously. He continued to develop this idea and the
technology soon found use in CB radios, cell phones and many other radio
devices.

A typical smart phone receives Bluetooth,
Wi-Fi, data (3G, 4G, etc.), calls, texts, and sometimes radio signals. These
signals are all on different wavelengths, but because of fractal antenna a
smart phone does not need multiple giant antennas. It only needs one small
fractal antenna. In fact, the multi-functionality of modern smart phones would
be extremely difficult, if not impossible, without fractal antennas.

Another application of Chaos theory is in
organizational behavior. Lynn Adams, then the Governor of Heber City, Utah,
observed principles of chaos, including strange attractors, in local
governments. His observations during specific political efforts to reorganize
local governments led to work in a model of revolution he called “The ARM.” Dr.
Adams was in the right place at the right time to observe patters from within
the organization during revolution. He was able to isolate certain aspects of
the organization that led to success and or failure. This led to a detailed explanation
of the cycle of revolution and the elements within.

Future applications of Dr. Adams work
might include social analysis. If one was able to measure the level of chaos in
different stages of Dr. Adam’s model, then it could help give an idea of where society
is in the cycle and where it might go next. This kind of analysis could
possibly have applications in political policies including economic, monetary
and taxation policies. However, one must consider the ethical consequences of
such a model. Not only could it help lead better policies to cope with chaos in
society, but it can also be used to thwart an ethical revolution and or enact
an unethical or false revolution. A model that allows policy makers to
understand the chaos within their country can also help a dictator maintain
power.

This same research could possibly be
expanded and adapted to analyzing business cycles of individual companies, markets
or entire economies. A business cycle of an individual company is similar to a
sector’s and an economy’s business cycle. Therefore in some ways economies may
be self-similar. By analyzing the business cycle of an individual company one can
learn things about the entire economy. If it is possible measure the level of
chaos within the organization, one can estimate what stage of the business
cycle the organization is in and where it might go next. This can be added to
other analysis to better valuate companies.

Another widely used application of
chaos theory is in financial markets and financial analysis. Security prices
seam completely chaotic. For years statistical (linear) analysis has been
applied to try and predict where the price will go next and “beat” the market.
Linear analysis has also been used for years for investing strategies. A
perfect example of this is Modern Portfolio Theory, which uses statistical
analysis to minimize risk and maximize gain. The problem is it doesn’t work
because security prices are not linear.

The Prediction Company, or PredCo,
mentioned page 146 of “Chaos: A Very Short Introduction” (Smith), was founded
with the intent of finding a better way to analyze financial markets. They
theorized that if there was chaos in the financial markets, then non-linear
analysis could create better strategies. A confidentiality agreement keeps
secret exactly what they are doing, but as Smith mentions, “ The continued
profitability of the company indicates that whatever it is doing, it is doing
well” (Smith).

This world is not a linear world,
and yet society continues to teach linear math. Chaos theory and non-linear
analysis will eventually dominate linear math and continue to lead to
innovations in mathematics and all scientific disciplines. The more that is
learned about chaos the more it is understood, and the more applications are found
for it. Science has only begun to scratch the service of chaos, and advances in
medicine, finance, physics and biology will continue to come forth. Chaos is an
entirely different way to look at the world. It is more effective and more
productive to study a multidimensional system in a multidimensional way.

Bibliography

Lorenz, Edward N. (1963).
"Deterministic non-periodic flow". *Journal of the Atmospheric
Sciences* **20** (2): 130–141.

Adams,
L. Lynn; Adams, Nathanael;
Cortez, Jiar (2002) “PATTERNS OF REVOLUTION:
COMPREHENDING, CREATING, AND LEADING PERMANENT CHANGE”, Department of
Finance and Economics, Utah Valley University, Woodbury School of Business

Smith, Leonard (2007).
“Chaos: A Very Short I
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