AUTOMATA ON FRACTAL SETS OBSERVED IN HYBRID DYNAMICAL SYSTEMS

Abstract

We studied a hybrid dynamical system composed of a higher module with discrete dynamics and a lower module with continuous dynamics. Two typical examples of this system were investigated from the viewpoint of dynamical systems. One example is a nonfeedback system whose higher module stochastically switches inputs to the lower module. The dynamics was characterized by attractive and invariant fractal sets with hierarchical clusters addressed by input sequences. The other example is a feedback system whose higher module switches in response to the states of the lower module at regular intervals. This system converged into various switching attractors that correspond to infinite switching manifolds, which define each feedback control rule at the switching point. We showed that the switching attractors in the feedback system are subsets of the fractal sets in the nonfeedback system. The feedback system can be considered an automaton that generates various sequences from the fractal set by choosing the typical switching manifold. We can control this system by adjusting the switching interval to determine the fractal set as a constraint and by adjusting the switching manifold to select the automaton from the fractal set. This mechanism might be the key to developing information processing that is neither too soft nor too rigid.