Simulation complexities in the dynamics of a continuously piecewise-linear oscillator

Abstract

The vector field of a continuously piecewise-linear oscillator under periodic excitation is by nature nonsmooth. However, the nonsmoothness in studying the oscillator dynamics has drawn little attention. The paper presents the nonsmoothness effect on the differentiability of the Poincaré mapping, and then gives the dynamics comparison between the oscillator and a corresponding smoothed model. The numerical evidence, with the assistance from geometric concepts of dynamical systems, suggests that great care should be taken during locating a periodic orbit and tracing a branch of the periodic orbit when the orbit approaches a bifurcation of saddle-node type and relevant degenerated types or touches a switching plane at very low velocity. In these critical cases, the oscillator may behave quite different from an oscillator having smooth vector field.