Ball bouncing down rounded edge stairs: chaotic but tricky

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.

Subtle dynamics of the chaotic torsion pendulum: a detailed comparison between experiments and numerical simulations

“…The seeming contradiction between the deterministic process and the unpredictable behavior of chaos offers an interesting and valuable general insight into the nonlinear nature of the world. Research on the topic of chaos is flourishing since its discovery, not only from the perspective of advanced science [2][3][4][5][6][7], but also from an educational point of view [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Many excellent introductory monographs that explain basic concepts of chaos exist [8][9][10][11][12][13][14][15], as well as various elaborate devices [16][17][18][19][20][21] that vividly depict the dynamical feature of a chaotic system.…”

“…Research on the topic of chaos is flourishing since its discovery, not only from the perspective of advanced science [2][3][4][5][6][7], but also from an educational point of view [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Many excellent introductory monographs that explain basic concepts of chaos exist [8][9][10][11][12][13][14][15], as well as various elaborate devices [16][17][18][19][20][21] that vividly depict the dynamical feature of a chaotic system. In college-level physics courses, one of the standard devices for exhibiting chaotic behavior is the chaotic torsion pendulum (CTP) [22][23][24][25].…”