Introduction

Abstract

Vibro-impact systems had been known to laymen and children before they were considered by the scientific community. A skipping stone on the water surface and woodpecker toy are common examples. According to Wikipedia: “Stone skipping is a pastime which involves throwing a stone with a flattened surface across a lake or other body of water in such a way that it bounces off the surface of the water. The object of the game is to see how many times a stone can be made to bounce before sinking.” It is amazing that this game has attracted many researchers to provide analytical and physical insight (see, e.g., [685], [358], [963], [126], [183], [706], [895], [914], [535]). The woodpecker toy operates by self-excited vibrations due to the combined effects of friction, impact and weight. The motion of this toy is greatly influenced by simultaneous impacts, which cause discontinuous bifurcations ([836], [364], [837], [838], [579], [581], [365], [578]). The dynamics of the woodpecker toy can be analyzed with a one-dimensional Poincaré map.