Existence of periodic orbits for piecewise-smooth vector fields with sliding region via Conley theory

The Conley theory has a tool to guarantee the existence of periodic trajectories in isolating neighborhoods of semi-dynamical systems. We prove that the positive trajectories generated by a piecewise-smooth vector field $Z=(X,Y)$ defined in a closed manifold of three dimensions without the scape region produces a semi-dynamical system. Thus, we have built a semiflow that allows us to apply the classical Conley theory. Furthermore, we use it to guarantee the existence of periodic orbits in this class of piecewise-smooth vector fields.