Chaotic Evolution via Generalized Probabilistic Automata (Probabilistic Arrays)

An n-state generalized probabilistic automaton/array maps a list of d states stochastically into a next state, resulting in a degree-d polynomially nonlinear transformation of the n-component state-probability vector, in contrast with the linear transformation given by the conventional Markovian model. This nonlinearity introduces the possibility of chaotic behaviour as time (iteration) progresses. It is shown that for two-state systems, the d = 2 case is nonchaotic, while chaotic behaviour is found for degree d as low as 5. Examples, and open issues for other values of n and/or d, are also noted.