Chaotic dynamics of a bouncing coin

TL;DR

This paper analyzes the chaotic behavior of a bouncing coin modeled as a billiard system, demonstrating the existence of Smale horseshoe structures and the realization of any collision sequence through initial conditions.

Contribution

It provides an analytical proof of chaos in the bouncing coin system and shows how arbitrary collision sequences can be achieved.

Findings

Existence of Smale horseshoe structure in the system

Any collision sequence can be realized by initial conditions

Chaotic dynamics confirmed through analytical methods

Abstract

We study the dynamics of a bouncing coin whose motion is restricted to the two-dimensional plane. Such coin model is equivalent to the system of two equal masses connected by a rigid rod, making elastic collisions with a flat boundary. We first describe the coin system as a point billiard with a scattering boundary. Then we analytically verify that the billiard map acting on the two disjoint sets produces a Smale horseshoe structure. We also prove that any random sequence of coin collisions can be realized by choosing an appropriate initial condition.