Within the hybrid equations framework the concepts of hybrid time, hybrid arc and hybrid execution are discussion in the (now for already familiar) "HyEQ" resources, of which the paper:

is perfectly satisfactory for us.

Within the hybrid automata framework the same issues are also discussed in the references that we have already made. A particularly popular and recommendable are the (online available) lecture notes
 
or perhaps even better the updated version (obviously a draft of a book)

Transitions from one discrete state to another, even if not accompanied by a jump (or reset) of the continuous state variable can be modeled as discontinuities of the functions on right hand side of differential equations. Depending on circumstances, more or less peculiar phenomena can occur due to these discontinuities.These issues are discussed in quite some detail in the (fairly readable) paper 
 
  • Cortes, Jorge. “Discontinuous Dynamical Systems: A Tutorial on Solutions, Nonsmooth Analysis, and Stability.” IEEE Control Systems Magazine 28, no. 3 (June 2008): 36–73. https://doi.org/10.1109/MCS.2008.919306.

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