We have investigated certain aspects of the behaviour of one-dimensional arrays of coupled nonlinear maps. We have given examples of the determination of global behaviour (of the CMC) by local interactions (the individual cells).
There is clearly a great deal of work yet to be done in order to understand fully the behaviour of even a simple extended system such as this. However, we have numerically illustrated behaviour that is both surprising and likely to be of interest in the area of nonlinear signal processing as this is where we see the main applications of this work. We have shown that the CMC considered can perform at least three `signal processing'-like functions: thresholding, low-pass filtering and differentiation.
A theory that explains this behaviour is missing. This is not surprising given that even the apparently simple problem of determining the amplitude of oscillation for the smooth solutions to the logistic mapping with delay [4] has not yet been solved. Along with some theoretical analysis, much further work is needed to determine, for instance,
and 
and other coupling schemes.
and 
.
In relation to item 2 above, an electronic implementation of a CMC would suffer from noise; how would this affect performance? Referring to item 4, it would be interesting to know if the mapping in equation 4 is a discretised version of a PDE, and if so if anything can be said about its solutions.
The insights into signal processing which we have from understanding linear systems and linear filters are of little use to us in this work. Here we see both integration (low pass filtering or smoothing) and differentiation (high pass filtering, edge enhancement) when interpreted in the light of linear system theory. However, these effects are frequency independent in our non-linear CMC. We expect there to be amplitude dependent effects in a non-linear processor; here we see how the CMC can act as a threshold detector on a random input. It appears from the experiments that the only way to find out what the effect on a particular waveform will be is to run a simulation; common concepts of superposition and transfer function analysis in the frequency domain are of no use. An underlying theory of how CMCs process signals would be a desirable goal for future work. It is apparent that without such a theory, speculation about the collective behaviour of coupled non-linear systems may be very wide of the mark.