Periodic motions in a class of discontinuous delayed systems with a parabolic boundary

Various periodic motions are investigated for a mass-spring-damper oscillator described by second-order discontinuous delayed systems with a parabolic boundary. The G-function related to a delayed variable is first developed to judge the crossing, sliding, and grazing motions at the boundary points. Then, basic and local mappings, together with their motion equations, are classified by current and delayed states of a local flow. It is shown that periodic motions can be predicted analytically in terms of the return mapping and the initial value function provided posteriorly. Finally, numerical simulations are given to illustrate the existence of slowly oscillating periodic solutions without or with sliding portions as well as a crossing periodic solution not slowly oscillating. These results not only enrich and extend switchability theory of discontinuous delayed systems, but also reveal some complicated periodic behaviors which cannot be found in continuous delayed systems and discontinuous systems without time delay.