Chaos Theory
Chaos theory is the study of non-linear dynamic systems, that is, systems of activities (weather, turbulence in fluids, the stock market) that cannot be visualized in a graph with a straight line. Although dictionaries usually define "chaos" as "complete confusion," scientists who study chaos have discovered deep patterns that predict global stability in dynamic systems in spite of local instabilities.
Isaac Newton and the physicists of the 18th and 19th centuries who built upon his work showed that many natural phenomena could be accounted for in equations that would predict outcomes. If enough was known about the initial states of a dynamic system, then, all things being equal, the behavior of the system could be predicted with great accuracy for later periods, because small changes in initial states would result in small changes later on. For Newtonians, if a natural phenomenon seemed complex and chaotic, then it simply meant that scientists had to work harder to discover all the variables and the interconnected relationships involved in the physical behavior. Once these variables and their relationships were discovered, then the behavior of complex systems could be predicted.
But certain kinds of naturally-occurring behaviors resisted the explanations of Newtonian science. The weather is the most famous of these natural occurrences, but there are many others. The orbit of the moon around the Earth is somewhat irregular, as is the orbit of the planet Pluto around the sun. Human heartbeats commonly exhibit minor irregularities, and the 24-hour human cycle of waking and sleeping is also irregular.
In 1961, Edward N. Lorenz discovered that one of the crucial assumptions of Newtonian science is unfounded. Small changes in initial states of some systems do not result in small changes later on. The contrary is sometimes true: small initial changes can result in large, completely random changes later. Lorenz's discovery is called the butterfly effect: a butterfly beating its wings in China creates small turbulences that eventually affect the weather in New York.
Lorenz, of MIT, made crucial discoveries in his research on the weather in the early 1960s. Lorenz had written a computer program to model the development of weather systems. He hoped to isolate variables that would allow him to forecast the weather. One day he introduced an extremely small change into the initial conditions of his weather prediction program: he changed one variable by one one-thousandth of a point. He found that his prediction program began to vary wildly in later stages for each tiny change in the initial state. This was the birth of the butterfly effect. Lorenz proved mathematically that long-term weather predictions based upon conditions at any one time would be impossible.
Mitchell Feigenbaum was one of several people who discovered order in chaos. He showed mathematically that many dynamic systems progress from order to chaos in a graduated series of steps known as scaling. In 1975 Feigenbaum discovered regularity even in orderly behavior so complex that it appeared to human senses as confused or chaotic. An example of this progression from order to chaos occurs if you drop pebbles in a calm pool of water. The first pebble that you drop makes a clear pattern of concentric circles. So do the second and third pebbles. But if the pool is bounded, then the waves bouncing back from the edge start overlapping and interfering with the waves created by the new pebbles that you drop in. Soon the clear concentric rings of waves created by dropping the first pebbles are replaced by a confusion of overlapping waves.
Feigenbaum and others located the order in chaos: apparently chaotic activities occur around some point, called an attractor because the activities seem attracted to it. Figure 1 illustrates an attractor operating in three-dimensional space. Even though none of the curving lines exactly fall one upon the other, each roughly circular set of curves to the left and right of the vertical line seems attracted to an orbit around the center of the set of circles. None of the curved lines in Figure 1 are perfectly regular, but there is a clear, visual structure to their disorder, which illustrates the structure of a simple chaotic system.
James Yorke applied the term "chaos" to non-linear dynamic systems in the early 1970s. But before Yorke gave non-linear dynamical systems their famous name, other scientists had been describing the phenomena now associated with chaos.
Chaos theory has a variety of applications. One of the most important of these is the stock market. Some researchers believe that they have found non-linear patterns in stock indexes, unemployment patterns, industrial production, and the price changes in Treasury bills. These researchers believe that they can reduce to six or seven the number of variables that determine some stock market trends. However, the researchers concede that if there are non-linear patterns in these financial areas, then anyone acting on those patterns to profit will change the market and introduce new variables which will make the market unpredictable.
Population biology illustrates the deep structure that underlies the apparent confusion in the surface behavior of chaotic systems. Some animal populations exhibit a boom-and-bust pattern in their numbers over a period of years. In some years there is rapid growth in a population of animals, followed by a bust created when the population consumes all of its food supply and most members die from starvation. Soon the few remaining animals have an abundance of food because they have no competition. Since the food resources are so abundant, the few animals multiply rapidly, and some years later, the booming population turns bust again as the food supplies are exhausted from overfeeding. This pattern, however, can only be seen if many data have been gathered over many years. Yet this boom-and-bust pattern has been seen elsewhere, including disease epidemics. Large numbers of people may come down with measles, but in falling ill, they develop antibodies which protect them from future outbreaks. Thus, after years of rising cases of measles, the cases will suddenly decline sharply because so many people are naturally protected by their antibodies.. After a period of reduced cases of measles, the outbreaks will rise again and the cycle will start over, unless a program of inoculation is begun.
Chaos theory can also be applied to human biological rhythms. The human body is governed by the rhythmical movements of many dynamical systems: the beating heart, the regular cycle of inhaling and exhaling air that makes up breathing, the circadian rhythm of waking and sleeping, the saccadic (jumping) movements of the eye that allow us to focus and process images in the visual field, the regularities and irregularities in the brain waves of mentally healthy and mentally impaired people as represented on electroencephalograms. None of these dynamic systems is perfect all the time, and when a period of chaotic behavior occurs, it is not necessarily bad. Healthy hearts often exhibit brief chaotic fluctuations, and sick hearts can have regular rhythms. Applying chaos theory to these human dynamic systems provides information about how to reduce sleep disorders, heart disease, and mental disease.
This is the complete article, containing 1,166 words
(approx. 4 pages at 300 words per page).
