Chaos Theory Tamed

Chaos Theory Tamed

Chaos Theory Tamed

Chaos Theory Tamed

Synopsis

This book presents an accessible introduction to chaos theory for the scientist who is not a mathematician but would like to bring this theory into focus as something conceptually and operationally useful. It bridges the gap between non-mathematical popular treatments and the distinctly mathematical publications that nonmathematicians find so difficult to penetrate. With only basic algebra, trigonometry, geometry and statistics assumed as a background, this book provides clearly understandable derivations and explanations of many key concepts, such as Kolmogrov-Sinai entropy, dimensions, Fourier analysis, and Lyapunov exponents.

Excerpt

The concept of chaos is one of the most exciting and rapidly expanding research topics of recent decades. Ordinarily, chaos is disorder or confusion. In the scientific sense, chaos does involve some disarray, but there's much more to it than that. We'll arrive at a more complete definition in the next chapter.

The chaos that we'll study is a particular class of how something changes over time. In fact, change and time are the two fundamental subjects that together make up the foundation of chaos. The weather, Dow-Jones industrial average, food prices, and the size of insect populations, for example, all change with time. (In chaos jargon, these are called systems. A “system” is an assemblage of interacting parts, such as a weather system. Alternatively, it is a group or sequence of elements, especially in the form of a chronologically ordered set of data. We'll have to start speaking in terms of systems from now on.) Basic questions that led to the discovery of chaos are based on change and time. For instance, what's the qualitative long-term behavior of a changing system? Or, given nothing more than a record of how something has changed over time, how much can we learn about the underlying system? Thus, “behavior over time” will be our theme.

The next chapter goes over some reasons why chaos can be important to you. Briefly, if you work with numerical measurements (data), chaos can be important because its presence means that long-term predictions are worthless and futile. Chaos also helps explain irregular behavior of something over time. Finally, whatever your field, it pays to be familiar with new directions and new interdisciplinary topics (such as chaos) that play a prominent role in many subject areas. (And, by the way, the only kind of data we can analyze for chaos are rankable numbers, with clear intervals and a zero point as a standard. Thus, data such as “low, medium, or high” or “male/female” don't qualify.)

The easiest way to see how something changes with time (a time series) is to make a graph. A baby's weight, for example, might change as shown in Figure 1.1 a; Figure 1.1b is a hypothetical graph showing how the price of wheat might change over time.

Even when people don't have any numerical measurements, they can simulate . . .

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