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; this tool is the main result of \cite{Mccord}. We prove that the positive trajectories generated by a piecewise-smooth vector field $Z=(X, Y)$ defined in a closed manifold of dimension three without the scape region produce a semi-dynamical system. Thus, supported by \cite{Mccord}, we have a mechanism to warrant the existence of periodic orbits in this class of piecewise-smooth vector fields.