Localisation of Critical Transition Phenomena in Cellular Automata Rule-Space

Several studies have shown the existence of critical transition phenomena in cellular automata rule-space (i.e. a sharp transition from order to chaos), but the problem of precisely localising such transitional regime in the rule-space has not been solved yet. Previous parametric approaches, such as classic lambda parameter, failed to individuate a precise parameter value at which the transition is found. In this work we localise the critical transition in a parameterised survey of multi-valued cellular automata rule-space, showing that there exists a hyperplane that precisely localises the critical transition and cuts the rule-space into two separate regions: the region of ordered rules and the region of chaotic rules. We also found that near this hyperplane, complex rules neither ordered nor chaotic, are likely to be found. Moreover, considering the normal vector of the hyperplane, we define a new critical parameter that is able to localise the critical transition.