Let M = j∈J M j and N = ∈L N be two hybrid manifolds. Note that if a map R : M → N is continuous, then for each j ∈ J there exists ∈ L such that R(M j ) ⊂ N and hence R| M j : M j → N . Using this observation, there is a natural definition of differentiability for continuous maps between hybrid manifolds. Namely, a map R : M → N is called smooth if R is continuous and R| M j : M j → N is smooth for each j ∈ J. In this case the pushforward DR : T M → T N is the smooth map defined piecewise as With these preliminaries established, we define the class of hybrid systems considered in this paper. This is a specialization of hybrid automata [5] that emphasizes the differential-geometric character of hybrid phenomena. Definition 1. A hybrid dynamical system is specified by a tuple H = (D, F, G, R) where: D = j∈J D j is a smooth hybrid manifold; F : D → T D is a smooth vector field; G ⊂ ∂D is an open subset of ∂D; R : G → D is a smooth map. As in [5], we call R the reset map and G the guard. When we wish to be explicit about the order of smoothness, we will say H is C r if D, F , and R are C r as a manifold, vector field, and map, respectively, for some r ∈ N.

### Language ###
English

### Assessment ###
We could ask about what makes a hybrid dynamical system, as it's clearly defined in the text.

### Query ###
Tell me all the components needed to specify a hybrid dynamical system.
Tell me all the components needed to specify a hybrid dynamical system.