Characterizing critical rules at the edge of chaos

This paper aims to provide a deeper understanding of the `edge of chaosʹ phenomenon. This is an important concept in some strands of work on complex systems, but exactly what it means remains unclear. Broadly, it may be seen as a region between rigid order and chaos. In cellular automata which show the edge of chaos phenomenon, this is indicated by the occurrence of smooth transitions in certain variables, such as the average Shannonian entropy, where generally one finds sudden jumps. The abrupt jump is between an ordered realm and a chaotic one, so the smoothness of a transition holds out the promise of the existence of a realm between order and chaos. Since living systems may also be described as lying between order and chaos, then the edge of chaos may be the key to understanding the form of organization which characterizes life. The smooth transitions in 2-D cellular automata were therefore investigated in detail to see whether it is possible to characterize the cellular automata rules which give rise to them. It was found that the transitions were smoother the less that there was a marked jump in the number of rule table entries applied. But, the bare fact of this correspondence does not reveal the cause of the smoothness. Further work was therefore carried out to try to understand more about what led to the smooth behaviour. However in two further studies in which `smooth rulesʹ were slightly altered and in which the dominant parts of a `smooth ruleʹ were imposed on other rules, it was found that the rules which led to a smooth transition were very sensitive to small changes. A change of less than 0.05% of the rule table entries of a `smooth ruleʹ was more than likely to lead to an abrupt rather than smooth transition. This suggests that it may prove difficult to understand what it is about the rules which leads to the edge of chaos.