Four collapsing one-dimensional particles: a dynamical system approach of the spherical billiard reduction

TL;DR

This paper analyzes the collision dynamics of four inelastic particles on a line using a dynamical system approach, revealing new periodic orbits and stability results for different restitution coefficients.

Contribution

It introduces a piecewise projective model for collision order analysis, discovering new periodic orbits and extending stability results in inelastic particle systems.

Findings

Identified three new families of periodic orbits.

Proved existence of stable periodic orbits for higher restitution coefficients.

Established the presence of quasi-periodic orbits in the system.

Abstract

We consider a system of four one-dimensional inelastic hard spheres evolving on the real line $\mathbb{R}$, and colliding according to a scattering law characterized by a fixed restitution coefficient $r$. We study the possible orders of collisions when the inelastic collapse occurs, relying on the so-called $\mathfrak{b}$-to-$\mathfrak{b}$ mapping, a two-dimensional dynamical system associated to the original particle system which encodes all the possible collision orders. We prove that the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping is a piecewise projective transformation, which allows one to perform efficient numerical simulations of its orbits. We recover previously known results concerning the one-dimensional four-particle inelastic hard sphere system and we support the conjectures stated in the literature concerning particular periodic orbits. We discover three new families of periodic orbits that coexist depending on the restitution coefficient, we prove rigorously that there exist stable periodic orbits for the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping for restitution coefficients larger than the upper bounds previously known, and we prove the existence of quasi-periodic orbits for this mapping.