Ball bouncing down rounded edge stairs: chaotic but tricky

Abstract

Abstract The aim of this study is to investigate the bouncing dynamics of a small elastic ball on a staircase consisting of rounded edge steps, as an example of a dissipative gravitational billiard, and to determine if its dynamics is chaotic. We derive a nonlinear recursion for the coordinates of the collisions, completed with numerical simulations, which indicate that the bouncing dynamics is chaotic, as also follows from elementary considerations regarding the Lyapunov exponent. It is, however, surprising that instead of permanent chaos, only the transient form is present. The main reason behind this is that a collision with the rounded edge of the step enhances the horizontal velocity leading to larger and larger jumps. Not even the introduction of a tangential coefficient of restitution (COR) on the curvature can hinder the flying away of some trajectories. There is also a chance for remaining trapped on a single step in the form of sliding, representing another possibility for escape. Therefore, chaoticity holds for long trajectories before any kind of escape takes place. We also show that an impact-velocity-dependent COR converts the dynamics to permanently chaotic with an underlying fractal attractor. Only elementary mathematics is required for the analytic calculations used, and we offer a set of problems to solve, as well as a user-friendly demo software on our website: https://theorphys.elte.hu/fiztan/stairs to facilitate experimentation and further understanding of this complex phenomenon.