One-dimensional inelastic collapse of four particles: asymmetric collision sequences and spherical billiard reduction

Abstract We consider a one-dimensional system of four inelastic hard spheres, colliding with a fixed restitution coefficient r , and we study the inelastic collapse phenomenon for such a particle system. We study a periodic, asymmetric collision pattern, proving that it can be realized, despite its instability. We prove that we can associate to the four-particle dynamical system another dynamical system of smaller dimension, acting on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , and that encodes the collision orders of each trajectory. We provide different representations of this new dynamical system, and study numerically its ω-limit sets. In particular, the numerical simulations suggest that the orbits of such a system might be quasi-periodic.