In this paper, periodic motions in a discontinuous dynamical systems are studied. The discontinuous dynamical system has three domains partitioned through two circular boundaries. On the three domains, there are three distinct dynamical systems. From the G-functions, the switchability conditions of a flow from one domain to anther domain at the boundary are developed. The flow mappings from a boundary to a bounbary are developed for each domain and boundary. From the mapping structures, periodic motions in the discontinuous dynamical system are predicted. Numerical simulations of periodic motions and motion switchability at boundaries are presented in the discontinuous dynamical system.