This paper studies various periodic oscillatory modes in second-order delayed equations with discontinuous right-hand sides. A delayed mass–spring–damper oscillator with an elliptic switching boundary is employed as the representative model for analysis. First, criteria for the traversability of a flow at the boundary point are provided using the delay-dependent Formula: see text-function. Then, local solution mappings’ linking current and historical flow states, as well as basic mappings connecting two adjacent switching points, are completely classified to form the return mapping in the phase space. After that, the procedure to predict analytically periodic solutions is established under some control laws and motion equations. The proposed approach demonstrates an equivalence between the existence of periodic solutions and the solvability of a system of algebraic equations. Finally, five numerical simulations are given to illustrate the existence of multimode periodic motions involving both the slow oscillation and the mixed oscillation with or without sliding segments. The sliding oscillatory behavior exhibits an inherent difference from smooth dynamical systems.
Chen et al. (Sat,) studied this question.