Practice Departmental Reading Test

Chaos Theory


by Loretta F. Kasper, Ph.D.

from Interdisciplinary English © 1998. pp. 70-73.


Directions: Read the following essay and type the answers to the questions into the computer screen. The questions are on this page below the reading text. When you are finished, click the button that says "Submit your answers." This will take you to a page that has the correct answers. To keep a copy of your answers, you will need to write them in your notebook as well as type them into the computer screen.

            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.

            Mathematicians, like the rest of us, have observed such events and the changes that result from them.   These scientists have tried to infuse order into the chaos by plotting equations that might allow them to predict the outcomes of certain small events.  These equations involve the principle of iteration.  Iteration means taking the answer to an equation and feeding it back into the equation, over and over.

            These iterative calculations are very long and boring.  They are virtually impossible to complete without the aid of a computer.  As you learned in the previous unit, computers are very good at performing mechanical, repetitive calculations.  So, mathematicians feed data into the computer and then instruct the computer about the required output.  Then they watch as the computer solves the equation and produces patterns of numbers.  Throughout this process, the scientists play with different equations, substituting different sets of numbers to see what happens.

            Through their work with computers, scientists have discovered that although chaotic systems appear to be random, they are not.   Rather, chaotic systems are deterministic, which means that these systems are ruled by a determining equation.  However because chaotic systems are also very sensitive to the initial conditions, the slightest change in the starting point can lead to very different outcomes.  The system and its outcome are determined by minor variations in starting conditions, and when these conditions cannot be stabilized, the system and its outcome are both fairly unpredictable.   Finally, while chaos theory states that order exists within random complex systems, it also asserts that seemingly simple systems can produce highly complex and unpredictable behavior.

            An example of a seemingly simple system which can produce complex behavior is the trajectory of a pool ball after it is hit.   In theory we should be able to calculate the exact path of a pool ball.  If we figure out the angles at which the ball strikes the sides of the table, the mass of the ball, and the force with which we hit it,  we should be able to predict the precise path that the ball will take.  However, this seemingly simple system is not so simple after all.  This is because the exact trajectory of the ball is determined by minor variations in the surface of the ball and in the surface of the table.  These minor variations play havoc with our careful calculations, and they make it impossible for us to predict the exact path that our pool ball will take after we strike it.

Comprehension Questions

Directions:  Refer to the reading passage to answer the following questions.  Be sure to use your own words in the answer.

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  1. What is "chaos theory"? When was it first developed?

  2. What do we mean by a linear system? Why are linear equations easily solved?

  3. Give one example of a linear system, and explain why that system is linear.

  4. What do we mean by a nonlinear system? Why are nonlinear equations difficult to solve?

  5. Give one example of a nonlinear system and explain why that system is nonlinear.

  6. In your own words, trace the history of chaos theory.

  7. What surprising finding has come out of research into the mathematics of chaos?

  8. How did scientists first learn about the order within the chaos?

  9. In your own words, describe the process of iteration in mathematical calculations.

  10. List three principles of chaos theory.

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