Chaotic Behaviour of a Ball Bouncing on a Rough Inclined Line

Abstract

We present a simple theoretical model which describes the motion of ball bouncing on a rough inclined line. The rough line consists of micro-facets whose orientation can be different from the line inclination. We examine the behaviour of the ball as a function of the orientation of the micro-facets and determine the conditions under which the jumps of the ball are decreasing or increasing in their amplitude. In particular, we show that when the facet inclination varies along the line with a well-defined spatial periodicity, the ball can reach a steady bouncing regime which leads ultimately to chaotic behaviour via a period-doubling scenario. Furthermore, we analyze the ball dynamics in presence of stochastic fluctuations associated to the inclination of the facets.