Reduction and decomposition of differential automata: Theory and applications

The paper considers an important class of hybrid dynamical systems called differential automata. A differential automaton is said to be reducible if its dynamics can be described by some discrete automaton with a finite number of states. Our main results show that, under certain general assumptions, any differential automaton is reducible. Furthermore, we prove that any reducible differential automaton can be represented as a union of a finite number of differential automata with simple cyclic dynamics. Moreover, we show that the differential automaton has a periodic trajectory corresponding to each of this cyclic automata. For planar differential automata, we derive an analog of the classic Poincare-Bendixon theorem.