The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation

In this paper, the mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation is developed, and the mapping structures for specified periodic motions are constructed. Based on the mapping structures, an analytical prediction of all possible, stable and unstable periodic motions is given. The symmetry for the stable, asymmetrical, periodic motions of such a system is observed. However, the unstable periodic motions do not have such symmetry. The methodology presented in this paper is applicable to other non-smooth systems such as friction-induced vibration, impact oscillator and power control systems. In addition, the mapping dynamics provides a useful and efficient tool for the co-existence of periodic motions and/or chaotic motions in nonlinear dynamical systems.