A hybrid dynamical system as an automaton on the fractal set

It is well known that a brain consists of many sub-modules, each of which performs a specific role. The experimental results many studies have indicated that prefrontal cortex appropriately switches each module. In this paper, we consider the brain as a hybrid dynamical system, which is composed of a higher module having discrete dynamics and a lower module having continuous dynamics. Two typical systems are investigated from the viewpoint of dynamical systems. When the higher module stochastically switches inputs to the lower module, i.e., a non-feedback system, the dynamics is characterizes by an attractive and invariant fractal set having hierarchical clusters addressed by input sequences. When the higher module switches according to the state of the lower module, i.e., a feedback system, various switching attractors correspond to infinite switching manifolds, which define each feedback control rule at the switching point. The switching attractors in the feedback system are the subsets of the fractal set in the non-feedback system. The system can be considered an automaton, which generated various sequences from the fractal, set by choosing the typical switching manifold.