Properties of some dynamical systems for three collapsing inelastic particles
Théophile Dolmaire, Juan J. L. Velázquez
TL;DR
This work rigorously analyzes the nearly-linear inelastic collapse of three identical inelastic hard spheres in $d\ge2$, reducing the full high-dimensional dynamics to a two-dimensional two-collision mapping in the regime where the two normal relative velocities do not vanish at collapse. It identifies the Zhou-Kadanoff regime as a stable, attracting region of the reduced phase space and proves its local stability, while numerical evidence and a formal low-energy limit support a global separatrix structure dividing well-defined trajectories from ill-defined ones. In the symmetric case $a=b$, the study yields a complete fixed-point picture, including an unstable Zhou-Kadanoff equilibrium, and extends the analysis to a formal low-energy limit where the phase-space structure becomes tractable and confirms the separatrix scenario. Overall, the results establish the Zhou-Kadanoff regime as the generic long-time outcome for nearly-linear collapses and lay groundwork for proving broader continuation properties beyond collapse, with future work aimed at addressing triangular collapse and global dynamics via perturbative or computer-assisted methods.
Abstract
In this article we continue the study of the collapse of three inelastic particles in dimension $d \geq 2$, complementing the results we obtained in its companion paper. 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.