Properties of some dynamical systems for three collapsing inelastic particles

Abstract

In this article we continue the study of the collapse of three inelastic particles in dimension $d\ge 2$, complementing the results we obtained in the companion paper (Dolmaire and Velázquez, 2024). We focus on the particular case of the nearly-linear inelastic collapse, when the order of collisions becomes eventually the infinite repetition of the period ⓪-①, ⓪-②, under the assumption that the relative velocities of the particles (with respect to the central particle ⓪) do not vanish at the time of collapse. Taking as starting point the full dynamical system that describes two consecutive collisions of the nearly-linear collapse, we derive formally a two-dimensional dynamical system, called the two-collision mapping. This mapping governs the evolution of the variables of the full dynamical system. We show in particular that in the so-called Zhou–Kadanoff regime, the orbits of the two-collision mapping can be described in full detail. We study rigorously the two-collision mapping, proving that the Zhou–Kadanoff regime is stable and locally attracting in a certain region of the phase space of the two-collision mapping. We describe all the fixed points of the two-collision mapping in the case when the norms of the relative velocities tend to the same positive limit. We establish conjectures to characterize the orbits that verify the Zhou–Kadanoff regime, motivated by numerical simulations, and we prove these conjectures for a simplified version of the two-collision mapping.