On periodic solutions of a second-order, time-delayed, discontinuous dynamical system

This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.