Randomness, chaos, and structure

We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex dynamics regularities and patterns emerge. 1 Symbolic dynamics distinguishing chaotic from random dynamics 1.1 Random sequences and dynamical iterates According to many popular accounts, chaotic dynamics seem to blur the distinction between determinism and randomness. While following a fixed rule, it is characteristic of chaotic dynamics that in the longer term no prediction of the iterates of given initial values is possible, and it therefore seems that sequences of points generated by chaotic dynamics are difficult, if not impossible, to distinguish from random sequences. Of course, this is not so, and one may exploit regularities in the relationships between subsequent points in the sequence to extract useful information about the underlying dynamics. By now, very sophisticated methods have been successfully developed, and we refer to [10] for a good account of the state of the art, describing both the older linear and the more recent non-linear tools, in particular phase space and other embedding methods, together with a rich spectrum of applications. It is the purpose of the present article to analyze the relationship between randomness and chaos in an elementary manner using simple symbolic dynamics, and to utilize this to elucidate the formation of higher level structures through the coordination of lower level non-linear dynamics, as initiated in our earlier contribution [2]. ∗also: Santa Fe Institute, USA; [email protected]