Controlling grazing-induced multistability in a piecewise-smooth impacting system via the time-delayed feedback control

Grazing events may create coexisting attractors and cause complex dynamics in piecewise-smooth dynamical systems. This paper studies the control of grazing-induced multistability in a soft impacting oscillator by using the time-delayed feedback control. The control switches from one of the coexisting attractors to a desired one to suppress complex dynamics near grazing events. We use path-following (continuation) techniques for non-smooth dynamical systems to investigate robustness of the controller and the parameter dependence of the controlled system. In particular, several newly developed computational methods are used, including a numerical method for analysing non-smooth delay equations and a method for calculating the Lyapunov exponents and the grazing point estimation. Numerical simulations demonstrate that the delayed feedback controller is effective, and a proper selection of the control gain and delay time can simplify the complex dynamics of the system near grazing.