Chaos Theory


Loretta F. Kasper, Ph.D.

from Interdisciplinary English, 2nd edition


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         Chaos theory is a mathematical theory first developed in the 1970's.  Chaos theory deals with the breakdown of ordered systems into chaotic, or disordered, ones.  The word, CHAOS, means "utter confusion or random disorder." .  It is relatively easy to describe the behavior of certain kinds of events mathematically.    For example pendulums appear to follow a regular pattern of movement which can be mathematically described by linear equations; equations which are easily solved.   

         Although the discipline of mathematics is very good at dealing with regularity, it is not so good at dealing with turbulence.  In the 1960's, scientists were trying to use computers to understand and to predict the weather.  THE WEATHER is an example of an event whose behavior is turbulent.  Turbulent events are described by nonlinear equations.  Unfortunately, nonlinear equations are difficult, if not impossible, to solve. 

         So that we may better understand this concept, let's contrast a linear with a nonlinear system.   In a linear system, a small change produces a small and easily quantifiable, or measurable, change.  However, a nonlinear system is highly sensitive to and dependent upon its INITIAL CONDITIONS.   Small or virtually immeasurable differences in initial conditions can lead to tremendously differing results.  

         Because the behavior of weather systems is so dependent upon initial conditions, it is virtually impossible TO PREDICT THE WEATHER.    Tiny differences in initial conditions lead to wide differences in outcomes.    That is the essence of nonlinear dynamics, which is another name for chaos theory.   So, chaos theory originally grew out of attempts to make computer models of the weather. 

         Chaotic, or turbulent, systems can be found IN MANY OTHER ASPECTS OF OUR LIVES. Mathematics, science, and society itself are all in a constant state of seemingly random change.  The study of the dynamics of such systems is an essential part of the effort to understand the principles of order that form the basis of all real systems, from ecosystems to social systems to the universe as a whole. 

         The possibility of chaos in a natural system was first proposed by the French mathematician, HENRI POINCARE, in the late 19th century, in his work on planetary orbits.  Many years later, in 1963, the American meteorologist, EDWARD LORENZ, demonstrated that thermal convection in the Earth's atmosphere was a chaotic system.  In the mid-1970's the American physicist, Mitchell Feigenbaum, predicted that when an ordered system begins to break down into chaos, there would be a consistent sequence of doubling transitions.  Feigenbaum then went on to calculate a numerical constant that governs the doubling process.  A numerical constant is a quantity whose value, under specified conditions, does not change.   Feigenbaum showed that his results could be applied to a wide range of chaotic systems, and the numerical constant he calculated is called Feigenbaum's number.  

         On the surface, the movement of chaotic systems appears to be totally random, unpredictable, and without form.  Yet, cutting edge research into the mathematics of chaos has demonstrated that the motion of a chaotic system is not completely random, and that events within chaotic systems do operate according to a definite form and pattern, evolving in predictably unpredictable ways  

         So that we may better understand the order within the chaotic system, let's consider a simple example.  To the outside observer, your desk appears to be a chaotic mess with papers and books scattered all over in a seemingly haphazard fashion.  It appears that you will be unable to find anything amid all the clutter.  Yet when asked to produce a specific piece of paper, you plow through the chaos to the exact spot and find it easily.  You are able to do this time and again, until one day, when someone decides to "help" you clean up your messy desk and straightens everything out.  Now the order and the pattern of your chaotic system has been changed and as a result you cannot find anything.

         We now know that the natural world, like your desk, is characterized by AN ORDERLY FORM OF CHAOS   Because of the great calculating power of the computer, scientists are now able to discover and to analyze patterns in events that were previously believed to be purely random.  Nevertheless, although we now know that all natural events do follow a certain pattern, the outcome of these events is still unpredictable.



Page last updated on November 29, 2002