Properties of Chaotic Phenomena

Abstract

Two time series are illustrated in Fig. 6.1. They look similar. They have approximately the same statistical properties, approximately the same mean, and approximately the same variance. They both look random, but we now know that not everything that looks random is random. These two time series are very different. The time series on the left is random; it was generated by choosing random numbers. However, the time series on the right is not random at all; it is completely deterministic. The next value, x _n+1, was computed from the previous value x _n by the relationship x _n+1=3.95x _n(1−x _n). Systems, like the one on the right, that are deterministic but whose output is so complex that it mimics random behavior, are now known by the jargon word “chaos.” Chaos is perhaps a poor word to describe this phenomenon. In normal usage chaos means disordered. Here it means just the opposite, namely a highly ordered and often simple system whose output is so complex that it mimics random behavior. It is quite important that we distinguish between chaos and noise in a given experimental time series, because how we subsequently process the data is determined by this judgment.

Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties.

R.M. May (1976)