ABSTRACT
The origin of the word chaos (χα′ os) is the Greek verb chasken, meaning “to yawn” or “to gape open,” referring either to the primeval emptiness of the universe before things came into being, or the abyss of Tartarus, the underworld (Encyclopedia Britannica, Vol. 5, p. 276). In modern dictionaries, chaos is defined as “total disorder and confusion.” The study of what we now call “chaotic systems” is due to Henri Poincare´
and certainly to the ergodic theorists Birkhoff [11] and Von Neuman [106] in the 1930s. The expression “chaos” became popularized through the paper of Li and Yorke [62], “Period three implies chaos.” Examples of chaotic systems include turbulent flow of fluids, population dynamics, irregularities in heartbeat, plasma physics, economic systems, weather forecasting, etc. These systems share the property of having a high degree of sensitivity to initial conditions. In other words, a very small change in initial values (due to error in measurement, noise, etc.) will multiply in such a way that the new computed system bears no resemblance to the one predicted. Meteorologist and mathematician Edward Lorenz [65] introduced one of
the most interesting examples of chaotic systems. In his study of weather forecasting, he concluded that weather is unpredictable, although it is deterministic. Thus, long-term weather forecasting would always elude science. This is due again to the fact that weather patterns are sensitive to initial conditions. Lorenz called this magnification of errors in weather forecasting the “butterfly effect.” The metaphor says the flapping of a butterfly’s wings in Brazil may cause a tornado in Texas several weeks later. Another simple example of a chaotic system is the ball in a two-well po-
tential as shown in Fig. 3.1. If the base vibrates with periodic motions of the
FIGURE 3.1 A pinball machine.
