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

Roberto Castorrini, Théophile Dolmaire

TL;DR

This work analyzes a four-particle, one-dimensional inelastic system with fixed restitution $r$ and establishes a two-dimensional dynamical framework—the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping—through a spherical (projective) reduction. The authors prove that the reduced map $\widehat{\mathfrak{P}}$ is a piecewise projective transformation, enabling precise and efficient numerical exploration of collision orders and collapse patterns, including recovering known results and uncovering new periodic and quasi-periodic behaviors. They provide rigorous results for select patterns, notably the stable $132312$ orbit for $r>r_{\text{crit},132^J}\approx0.2200$, and show the nonexistence of stability for $13223122$, along with extensive spectral analyses of the constituent linear pieces $P_i$. The paper reveals rich period-adding structures and coexisting attractors, linking the dynamics to border-collision phenomena and suggesting a laminated statistical framework with multiple invariant measures. These insights advance understanding of inelastic collapse in granular-like systems and demonstrate how projective-dynamic methods can illuminate complex collision-order dynamics in low-dimensional reductions.

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.

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

TL;DR

This work analyzes a four-particle, one-dimensional inelastic system with fixed restitution and establishes a two-dimensional dynamical framework—the -to- mapping—through a spherical (projective) reduction. The authors prove that the reduced map is a piecewise projective transformation, enabling precise and efficient numerical exploration of collision orders and collapse patterns, including recovering known results and uncovering new periodic and quasi-periodic behaviors. They provide rigorous results for select patterns, notably the stable orbit for , and show the nonexistence of stability for , along with extensive spectral analyses of the constituent linear pieces . The paper reveals rich period-adding structures and coexisting attractors, linking the dynamics to border-collision phenomena and suggesting a laminated statistical framework with multiple invariant measures. These insights advance understanding of inelastic collapse in granular-like systems and demonstrate how projective-dynamic methods can illuminate complex collision-order dynamics in low-dimensional reductions.

Abstract

We consider a system of four one-dimensional inelastic hard spheres evolving on the real line , and colliding according to a scattering law characterized by a fixed restitution coefficient . We study the possible orders of collisions when the inelastic collapse occurs, relying on the so-called -to- mapping, a two-dimensional dynamical system associated to the original particle system which encodes all the possible collision orders. We prove that the -to- 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 -to- mapping for restitution coefficients larger than the upper bounds previously known, and we prove the existence of quasi-periodic orbits for this mapping.
Paper Structure (58 sections, 5 theorems, 129 equations, 26 figures, 3 tables)

This paper contains 58 sections, 5 theorems, 129 equations, 26 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Outline of the article.
  3. Notations.
  4. The model and the dimensional reductions
  5. The model
  6. The inelastic collision law with fixed restitution coefficient.
  7. A first parametrization of the evolution of the particle system.
  8. Time of existence of the trajectories.
  9. First dimensional reductions
  10. A first reduction: the discrete evolution between consecutive collisions.
  11. The $\mathfrak{b}$-to-$\mathfrak{b}$ mapping.
  12. The spherical reduction and its consequences
  13. The dimensional reduction to the projective sphere of DoHR025.
  14. Definition of the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping $\mathfrak{P}$.
  15. The mapping $\mathfrak{P}$ written as a piecewise projective transformation
  16. ...and 43 more sections

Key Result

Theorem 3.3

Let $X\subset \mathbb{R}^3$ be defined in EQUATDefinDomai_X_P_. Then the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping $\widehat{\mathfrak{P}}:X \to \mathbb{S}^2$ introduced in Definition DEFIN_Map_b_2_b__P__, defined on $X$, coincides with a piecewise projective linear transformation, whose associated p with the matrices $P_1$, $P_2$, $P_3$ and $P_4$ respectively defined as: In other words, we have:

Figures (26)

  • Figure 4.1: Plot of the tails (last $100$ iterations) of the $32$ orbits of $\widehat{\mathfrak{P}}_r$ for $0.0717 \leq r \leq 0.1717$ after $5000$ iterations, obtained with the algorithm of Appendix \ref{['SSECTAppenAlgoLinea']}.
  • Figure 4.2: Plot of the tails (last $100$ iterations) of the $32$ orbits of $\widehat{\mathfrak{P}}_r$ for $0.0717 \leq r \leq 0.1717$ after $5000$ iterations, in logarithmic scale.
  • Figure 4.3: Plot of the tails of the orbits of $\widehat{\mathfrak{P}}_r$ for $0.0717 \leq r \leq 0.0817$ after $5000$ iterations, in logarithmic scale.
  • Figure 4.4: Plot of the tails of the orbits of $\widehat{\mathfrak{P}}_r$ for $0.0717 \leq r \leq 0.0727$ after $5000$ iterations, in logarithmic scale.
  • Figure 4.5: Plot of the tails of the orbits of $\widehat{\mathfrak{P}}_r$ for $0.07179 \leq r \leq 0.07189$ after $5000$ iterations, in logarithmic scale.
  • ...and 21 more figures

Theorems & Definitions (21)

  • Definition 2.1: Action of the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping $\widehat{\mathfrak{P}}$ on $\mathbb{P}_2(\mathbb{R})$
  • Remark 2.2
  • Remark 3.1
  • Definition 3.2: Domain of the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping $\widehat{\mathfrak{P}}$
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Proposition 5.1: Eigenelements of $P_1$
  • ...and 11 more