Chaos Theory

Chaos Theory: [ Introduction To Chaos | Sensitive Dependance | Critical Points | Strange Attractors | 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 beggining.

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 non-existant.

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 Tn(X1), is 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 defining features:

Critical Points

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

Strange Attractors

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 attractor.

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.
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