Chaotic or just complicated? Ball bouncing down the stairs

Abstract

The aim of this study is to investigate the bouncing dynamics of a small elastic ball on a rectangular stairway and to determine if its dynamics is chaotic. We derive a simple nonlinear recursion for the coordinates of the collisions from which the type of dynamics cannot be predicted. Numerical simulations indicate that stationary bouncing always sets in asymptotically, and is typically quasi-periodic. The dependence on the coefficient of restitution can be very complicated, yet the dynamics is found to be nonchaotic. Only elementary mathematics is required for the calculations, and we offer a piece of userfriendly demo software on our website, http://crnl.hu/stairway, to facilitate further understanding of this complex phenomenon.

Ball bouncing down rounded edge stairs: chaotic but tricky

“…In [2], a rectangular staircase was considered with step tread L and rise M, tilted from left to right. The main findings turn out to be reproducible by considering the bouncing of a ball on a slope but letting energy loss occur only in the vertical velocity component.…”

“…According to [2], these formulas provide good approximations for any k, down to k = 0.4 where their validity is lost because with such strong dissipation all balls might become trapped on a single step, stop bouncing, and start a sliding motion. The majority of the long lasting bouncing motions is quasi-periodic (i.e., they repeat themselves with some mismatch), and expressions ( 4) and ( 5) yield the average velocity and jump number over the quasiperiodic motion.…”

“…For y n = 0 these relations provide the exact description of the bouncing on rectangular stairs, as given in [2] where tildes are not used as the results always describe the state at the moment of a real collision.…”

“…[4][5][6]), a mere change in the geometry, namely the rounding of the edges (resulting in what we call rounded stairs in short), is sufficient to generate chaos. Intuitively, this is a natural expectation, and the authors of [2] conjectured indeed that the dynamics will be chaotic since the curvature at the edge will act as a magnifying mirror, which scatters parallel incoming beams (while the rectangular case contained only plane mirror-like surfaces).…”

“…In an Austrian high school textbook the bouncing motion of a ball down a staircase is given as one of the real life examples of chaotic dynamics [1]. This statement was investigated from a theoretical/numerical point of view in a previous paper [2] in the case of rectangular steps consisting of horizontal and vertical parts only. The authors, in order to keep the model as simple as possible, assumed the ball to be a non-rotating point, the air drag negligible, as in billiards (see e.g., [3]), and the bounces to occur elastically with some energy loss, described by a constant coefficient of restitution (COR), k < 1.…”