Software

Matlab and Simulink and Stateflow

pure Simulink: we can build models of systems with hybrid dynamics using just Simulink, but then the relational operators and logical operators will be needed. We may also appreciate the extra options for the integrator block, such as the external reset, external initial conditions, and displaying the state port (to avoid algebraic loops if the resetting condition is derived from the integrator output).
pure Matlab: we can go relatively far just with Matlab – the key functionality is event location while calling the ODE solvers.
Stateflow: having been exposed to some basics of automata modeling in Stateflow, including continuous-time dynamics in Stateflow is relatively straightforward. Here we only single out one potential stumbling block: we must make sure that we set the Update method for the whole chart to Continuous (open the Property inspector pane in the Modeling menu, make sure the chart is selected).

HybridSystems.jl – the package essentially just defines some basic data types, currently it is mainly used by some other packages, such as ReachabilityAnalysis.jl.
ReachabilityAnalysis.jl – well-maintained and even award-winning (under its previous name) package that also supports hybrid systems.