  This paper studies conditions for invariance of dynamical systems on
stratified do- mains as originally introduced by Bressan and Hong. We establish
Hamiltonian conditions for both weak and strong invariance of trajectories on
systems with non-Lipschitz data. This is done via the identification of a new
multifunction, the essential velocity multifunction. Properties of this
multifunction are investigated and used to establish the relevant invariance
criteria.
