Analysing Couples (131, X) Under Temporally Stochastic Environment

Abstract

To understand the effect of temporal mixing in cellular automata, this study considers temporally stochastic couples (131, X) where elementary cellular automata 131 acts as a blind rule and X comes from the set of the rest elementary cellular automata rules. Overall, the result shows that periodic rule 131 is associated with a tendency to generate chaotic phenomena under the temporally stochastic environment. To understand the microscopic reason, here, we analyse the following situations (a) couple (131, 50) where two periodic rules together show chaotic dynamics for any mixing rate

(b) couple (131, 78) where it is possible to observe class change (from chaos to simplicity) for changing mixing rate

and (c) couple (131, 136) where this probabilistic system shows important phase transition dynamics. This study also classifies couples (131, X) based on these different interesting behaviours which guides towards overall dynamics of temporally stochastic elementary systems. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.