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Russell

c) the iterates exhibit sensitive dependence on initial conditions. That is, points in the attractor or repeller that are initially close together do not remain close together.

Simple Eiamples of Dynamical Systems

Real physical dynamical systems that are believed to exhibit chaotic behavior have been discovered. Some examples include: kneading ingredients into a bread dough, turbulent fluid flow (Spiegel, 1987), the time between drips of water from a faucet (Peterson, 1987), Brownian motion, the spinning of a water wheel, transmission errors on phone lines, and the beating of a heart (Glassetal., 1987).

Time senes obviously could be generated from any of the above mentioned physical dynamical systems. These systems might be thought of, in a modeling situation, in terms of standard ARIMA (autoregressive integrated moving average, see Box and Jenkins 1976) models with morę or less "noise" tossed in. However, from the point of view of physics, the randomness enters systems, such as the Quincunx to be discussed below, in the uncertainty in the initial conditions only. Ali successive time points are purely deterministic.

The concept of deterministic chaos is conceptually challenging at first glance. The idea implies that entirely deterministic systems can appear to exhibit random behavior. Even morę challenging is that relatively simple deterministic systems can exhibit apparent randomness. It should be noted however, that given a data stream from one of these deterministic, chaotic time series, that statistical methods are still of great value. This is in part due to the property of "sensitive dependence on initial conditions" of these dynamical series.

This entire concept of deterministic chaos should not be a surprise to statisticians and Computer scientists because of the use of such systems in generating pseudo-random numbers. For example, one of the better systems for generating uniform pseudo-random numbers is the Super-Duper algorithm which has been shown to hołd up in five-dimensions (Walker, 1985):

Seed, = ((Seed_M *69069) +1) mod 2”

_ Seed,

For another familiar, simple, physical example of chaotic behavior, consider the Quincunx in Figurę 3. This simple device has been used by