Introduction To Chaos
Applications Of Chaos
An Introduction To Chaos
Many people believe that twentieth century science will be remebered
for three main theories: quantum mechanics, relativity, and chaos. Chaos
theory is a blanketing theory that covers all aspects of sciece, hence,
it shows up everywhere in the world today: mathematics, physics, biology,
finance, and even music. Where classical sciences end, chaos is only
The term chaos theory is used widely to describe an emerging
scientific discipline whose boundries are not clearly defined. The terms
complexity theory and complex systems theory provide a better description
of the subject matter, but the term chaos theory will be used throughout
this document as it is more widely accepted.
Chaos theory is a developing scientific discipline which is focused
on the study of nonlinear systems. To uderstand chaos theory, you must
first have a grasp upon its roots: systems and the term nonlinear.
The first term, system, can be defined as the understanding of
the relationship between things which interact. To better understand
this idea, we will examine the example of a pile of stones. The pile is
a system which interacts based upon how they were piled. If their
initial piling is not in balance, the interaction results in their
movement until they find a condition under which they are in balance. A
group of stones which do not touch each other, however, are not a system
because the interaction between is so minute, it can be considered
Another important aspect in dealing with systems is that systems can
be modeled. In other words, systems can be created which will
theoretically replicate the behavior of the original system. Following
the pile of stones example, one could take a second group of stones which
are identical to the first group, pile them in exactly the same way as
the first group, and predict that they will fall down into the exact same
configuration as the first group. Similiarly, a mathematical model,
based upon Newton's law of gravity, could be used to predict how piles
of same and different types will interact. Generally speaking,
mathematical modeleing is the key to modeling systems, although it is not
the only way.
The second term, nonlinear, has to do with the type of
mathematical model used to describe a system. Until the recent growth of
interest in chaos theory, hence nonlinear systems, most models were
analyzed as though they were linear systems. In other words, when the
mathematical models were draw in a graph format, the results appeared as
a straight line. Calculus was Netwon's mathematical method for showing
change in systems within the context of a straigt line and statistics,
regression analysis in particular, is a process of converting nonlinear
data into a linear format for further analysis and prediction.
Linear systems are easy to generate and simple to work with. That is
because they are very predictable. For example, you could think of a
factory as a linear system. We could predict that if we add a certain
number of people, or a certain amount of inventory to the factory, that
we will increase the number of pieces produced by the factory by a
comparable amount. As most managers know, factories don't operate this
way. Changing the number of people, inventory, or any other variable in
the factory and you receive widely differing results on a day to day
basis from what would be predicted from a linear model. This is true
because a factory is actually a nonlinear system, as are most systems
found in life. When systems in nature are modeled mathematically, we
find that their graphical representations are not straight lines and that
the system's behavior is not so easy to predict.
Prior to the devolepment of chaos theory, the majority of scientific
study involved attempting to understand the world using linear models.
Beginning with the work of Sir Isaac Newton, physics has been the has
provided the processes for modeling nature, and the mathematics
associated with them have been in a linear nature. Afterwards, when a
study resulted in strange answers, when a prediction usually held, but
not this one time, the failure was blamed on experimental error,
otherwise know as noise.
After research into complex systems, we now know that noise is
actually important information about the experiment. When noise is
inserted into the graph results, the graph no longer appears as a
straight line, nor are its points predictable. At one time, this noise
was referred to as the chaos in the experiment. The process of studying
the noise in an experiment is one of the major parts of the chaos theory.
Another important word which has been used repeatedly is complex.
What is the determining factor making one system more complex than
another? The complexity of a system is defined by the complexity of the
model necessary to effectively predict the behavior of the system. The
more the model must look like the actual system to predict the system's
results, the more complex the system is considered to be. The most
complex system example is the weather, which can only be modeled with an
exact duplicate of itself. On the other hand, a simple system to model
is to predict the amount of time it takes for a train to travel from
point A to point B, assuming there are no stops in between. To predict
the time we need only know the speed of the train and the distance
between the two points. The formula is simple: mph/miles. Our old
example of the pile of stones, which may look like a simple system, is
actually very complex. If your goal was to predict the eventuall
location of each indivdual stone, then you will have to know very
detailed information about each stone: their shape, weight, and the exact
starting location. If there is even a minor difference between the shape
of one stone in the model and the stone in reality, the modeled results
will likely be very different. Predictablity becomes very hard, making
the system highly complex.
Sensitive Dependance On Initial Conditions
One of the most essential elements in a complex system is
unpredictability. The generator of this unpredictability is what Lorenz
calls sensitivity to initial conditions, otherwise known to the
world as the butterfly effect. This concept means that with a
complex, nonlinear system, very (infinitely) small changes in the
starting conditions of a system will result in dramatically different
outputs for that system.
This phenomena is commonly known as the butterfly effect. Its name
is derived from the effect of a butterfly's actions upon the weather. As
was shown above, to acurately predict the weather, an exact replica of
the earth is needed. So, failing to take into acount a butterfly
flapping its wings in a distant country could cause the model to fail to
predict a thunderstorm over our home town in several weeks.
Because of extreme dependance on initial conditions, the general rule
for complex systems is that one cannot create a model that will
accurately predict outcomes. However, one can create models which
simulate the processes that the system will go through to create the
models. This realiztion is impacting many activities in business. For
example, it raises considerable questions relating to the value of
creating organizational visions and mission statements as have been and
are current practices.
The concept of sensitive dependance on initial conditions has strong
mathematical roots. Suppose we have to points
X0 and X1,
on a circle, which represent the correct starting value and a very close
to being correct value for a variable in the model of a system T.
We assume that the difference between the two numbers is represented by
the distance between the points on the circle, given by the variable
d. To demonstrate the importance of infinate accuracy of intial
conditions, we interate T. After only one iteration, d, or
the distance between T(X0) and
T(X1), has doubled. Iterating again, we
find that the distance between the two points, already twice its intial
size, doubles again. In this pattern, we find that the distance between
the two points, Tn(X0) and
2nd. Clearly, d is expanding quite
rapidly, leading the model further and further astray. After only ten
iterations, the distance between the two points has grown to a wopping
210d = 1024d.
Figure 1. Mathematical Roots Of Sensitive Dependance
The example given above demonstrates that no matter how close two
conditions start out, after only a few iterations, minor differences will
be blown way out of proportion. The two points will seperate from each
other at an exponential rate. For a more relative example, we'll explore
what happens when a computer conducts the same experiment with the
circle. The particular point on the circle can only be specified with a
finite number of decimal places, and the remaining decimal places are
simple discarded. This means that even if the intial numbers are entered
into the computer with precision, there will still be a certain amount of
decimal error. After iterating, we quickly notice that the very small
error is magnified so that the computed location of the point is actually
very far away from the actual location of the point. A very tiny error
in the initial conditions makes a very large diffence in the outcome.
After discussing sensitive dependance, we are ready to summarize the
qualities of a chaotic system. A chaotic system has these simple
- Chaotic systems are deterministic. This means they have some
determining equation ruling their behavior.
- Chaotic systems are sensitive to initial conditions. Even a very
slight change in the starting point can lead to significant different
- Chaotic systems are not random, nor disorderly. Truly random systems
are not chaotic, chaos has a sense of order and patter.
For the most part, critical points are a subject better left to a
mathematical couse, however they have an application in chaos, and
fractals in particular. Critical points are used as the starting value
for specific variables in a function while calculating fractal sets. For
example, in the Mandelbrot set, the initial value of z in the
function z = z2 + c, is always zero
because zero is the critical point of the function.
A critical point is defined as the point where the graph of the
function changes from increasing to decreasing. Calculating the critical
points of a function requires a working knowledge of differentiation, or
finding the slope of the tangent line to a funtion. To calculate the
critical points of a function, you must find the point at which a
function changes from increasing to decreasing. The first derivative can
be used to determine, at any given point, wether a function is increasing
or decreasing with this simple rule: where the function is increasing the
first derivative is positive and where the function is decreasing, the
first derivative is negative. In almost all cases, the critical points
will be located where the first derivative is zero, although in some
cases, they may be at a point in which the function is not defined.
Figure 2. Critical Points
A strange attractor is the limit set of a chaotic trajectory. A
strange attractor is an attractor that is topologically distinct from a
periodic orbit or a limit cycle. It can also be considered a fractal
Consider a volume in phase space defined by all the initial
conditions a system may have. For a dissipative system, this volume will
shrink as the system evolves in time. If the system is
sensitive to initial conditions, the trajectories of the points defining
the initial conditions will move apart in some directions, closer in others,
but there will be a net shrinkage in volume. Ulitmately, all points will
lie along a fine line of zero volume. This is the strange attractor.
All initial points in phase space which ultimately land on the attractor
form a Basin of Attraction. A strange attractor results if a system is
sensitive to initial conditions and is not conservative.
Applications Of Chaos Theory To Real-Life Situations
Much like physics, chaos theory provides a foundation for the study
of all other scientific disciplines. It is acually a tool box of methods
for incorporating nonlinear dynamics into the study of science. For many
people, the work in chaos represents the reunification of the sciences.
In mathematics, the use of strange attractors, fractals, and cellular
automata, and other nonlinear, graphical models are used for studying data
that was previously thought of as random. Mathematical applications of
chaos theory actually began being developed 100 years ago by the French
mathematician Henre Poincare.
In biology, chaos is used in the identification of new evolutionary
processes leading to understanding the genetic algorithim, artificial
life simulations, better understanding of learning processes in systems
including the brain, and studies of such previously unresearchable areas
as consciousness and the mind. This strain can be traced back to the
work of Charles Darwin, and is a significant new understanding of
evolutionary processes. Darwin's work also appears in direct conflict
with Newton's because it changes our understanding of the nature of time,
demonstrating that some time is not reversible.
In physics, thermodynamics in particular, chaos is applied in the
study of turbulence leading to the understanding of self-organizing
systems and system states (equilibrium, near equilibrium, the edge of
chaos, and chaos). Prigogine explains that the concept of entropy is
actually the physicists application of the concept of evolution to
physical systems. The greater the entropy of a system, the more highly
evolved the system is. Chaos theory is also having a major impact on
quantum physics and attempts to reconcile the chaos of quantum physics
with the predictability of Newton's universe. The push for such
unification cam from Einstein. Chaos theory is causing most quantum
physicists to accept what Einstein rejected, that God probably did play
dice with the universe.
Chaos theory is already affecting the critical aspects of our lives.
It greatly impacts all sciences. For example, it is answering previously
unsolvable problems in quantum mechanics and cosmology. The
understanding of heart arrhthmias and brain function has been
revolutionized by chaos research. There have been games and toys
developed from chaos research, such as the SimAnt, SimLife, SimCity, etc.
series of computer games. Fractal mathematics are critiical to improved
information compression and encryption schemese needed for computer
networking and telecommunications. Genetic algorithims are being applied
to economic research and stock predictions. Engineering applications
range from factory scheduling to product design, with pioneering work
being done at places such as DuPont and Deere & Co.
Glossary Of Terms And Phrases
Appendix I - Documentation
Appendix II - Source Code