Introduction

As is well known, a variety of complex behaviour, including chaos, is exhibited by some nonlinear mappings when iterated. In the spirit of the theme of Complex'96, the work reported here vividly demonstrates that in nonlinear systems the global behaviour which emerges from coupling locally chaotic processes is not intuitively apparent, and that experimental computer studies are an important tool in gaining insight.

The behaviour of coupled 2 to 6-dimensional arrays of nonlinear mappings, known as coupled map lattices, has been reported in [1]. These authors have used insights from statistical mechanics in their analysis, revealing some interesting results about the overall behaviour without uncovering the detailed dynamics.

In this paper, we consider the simplest possible collection of coupled mappings --- a one-dimensional array of them --- which we refer to as a coupled map chain (CMC) and define below. Our motivation for this is twofold:

1. We believe this to be the simplest possible extended system in which interesting behaviour can be observed.
2. The CMC studied here can be looked upon as a neural network with several layers, each containing only one element, with the elements being both self-coupled and implementing a non-monotonic function.  

In relation to item 2 above, a neural network (NN), like a coupled map lattice, consists of a coupled array of nonlinear elements (perceptrons) that form a weighted sum of their inputs and produce a single output [2]. It is important to note, however, that in all NNs studied to date, either (a) the output function is monotonic ( e.g. ) and a perceptron is allowed to be coupled to itself (a Hopfield NN), or (b) the output function is non-monotonic ( e.g. Gaussian) and no self-coupling is allowed (a Radial Basis NN). The classical NN is monotonic and not self-coupled. It can be shown that a single element of any of these neural networks can never exhibit chaos.

We now define a CMC and present some results on its behaviour.