Pattern formation of generalized fuzzy elementary cellular automaton

We propose a general method for constructing a fuzzy cellular automaton from a given cellular automaton. Unlike previous approaches that use fuzzy distinctive normal form, whose update function is restricted to third-order polynomials, our method accommodates a wide range of fuzzification functions, enabling the generation of diverse and complex time-evolution patterns that are unattainable with simpler heuristic models. We demonstrate that phase transitions in pattern formation can be observed by changing the parameters of the fuzzification function or the mixing ratio between two distinct evolution rules of elementary cellular automata. Remarkably, the resulting generalized fuzzy elementary cellular automata exhibit rich dynamical properties, including stable manifolds and chaos, even in minimal systems composed of just three cells.