Dynamics of perturbed elliptical billiard tables

TL;DR

This work studies the dynamics of billiard maps on perturbed elliptical tables, motivated by the Birkhoff conjecture about integrability. It develops an implicit real-analytic framework to iterate billiard maps on analytic, convex boundaries and couples it with the parameterization method to compute high-order local stable and unstable manifolds of hyperbolic periodic orbits; analytic continuation to the complex domain enables efficient composition via DFT/FFT. Through multiple shooting and spectral parameterizations, the authors obtain local and global invariant manifolds, observe transverse intersections indicative of chaos, and demonstrate robust numerical convergence across several perturbations of ellipses. The results reveal organized chaotic structures (homoclinic tangles) in perturbed elliptical billiards and provide data that align with chaos indicators like weighted Birkhoff averages, offering a path toward computer-assisted proofs and providing open-source tools for reproducibility. Overall, the paper advances numerical techniques for analytic billiard maps and sheds light on the transition from integrable to chaotic dynamics in smooth convex billiards, with significant implications for the Birkhoff conjecture and Hamiltonian dynamics more broadly.

Abstract

Dynamical billiards consist of a particle on a two-dimensional table, bouncing elastically off a boundary curve. The state of the system is given by two numbers: one describing the location along the curve where the bounce occurs, and another describing the incoming angle of the bounce. Successive bounces define a two-dimensional area preserving map, and iterating this map gives a dynamical system first studied by Birkhoff. One of the simplest smooth table shapes is that of an ellipse, in which case the dynamics of the billiard map is completely integrable. The longstanding Birkhoff conjecture is that elliptical tables are the only smooth convex table for which complete integrability occurs. In this spirit, we present an implicit real analytic method for iterating billiard maps on perturbed elliptical tables. This method allows us to compute local stable and unstable manifolds of periodic orbits using the parameterization method. Globalizing these local manifolds numerically provides insight into the dynamics of the table.

Figures (16)

• Figure 1: Top Left: Orbits on the physical table which is an unperturbed ellipse with eccentricity $0.4583$. Each trajectory has a caustic, i.e. a curve which is tangent to every line. Top Middle: Phase space orbit corresponding to the trajectory on the top left. The value of $\theta$ indicates the location of bounce, and $r$ measures the angle of the bounce. Top Right: The full phase space of orbits are quasiperiodic, colored by their frequency. Bottom Left, Middle: The trajectories and orbits for the same orbits, but for a small perturbation of the ellipse (Table A described in \ref{['eq:pert_ell']} and Table \ref{['table:coeff']}). Some caustics remain. However, the red trajectory no longer has a caustic; its corresponding orbit is chaotic in phase space. Bottom Right: For the full phase space, many quasiperiodic orbits persist and are colored by frequency, but some orbits are now chaotic, colored in gray.
• Figure 2: This figure shows the stable and unstable manifolds and their intersections for periodic orbits with a variety of periods. This data is for the same perturbed elliptical billiard table as shown in Fig. \ref{['fig:TableOrbits']}, (Table A in Tbl. \ref{['table:coeff']}).
• Figure 3: The figure shows stable and unstable manifolds for periodic orbits computed for Tables B and C (cf. Tbl. \ref{['table:coeff']}), larger perturbations of the same ellipse as in Figs. \ref{['fig:TableOrbits']} and \ref{['fig:PS1']}. The phase space in the background is colored by frequency for regular orbits and gray for chaotic orbits.
• Figure 4: The figure shows stable and unstable manifolds for periodic orbits computed for Tables D and E (cf. Tbl. \ref{['table:coeff']}). These tables are perturbations of an ellipse with larger eccentricity. The phase space in the background is colored by frequency for regular orbits and gray for chaotic orbits.
• Figure 5: Shape of billiard tables used in the numerics. (Left) Billiard Tables A-C are perturbations of an ellipse with eccentricity $0.4583$. (Right) Billiard Tables D and E are perturbations of an ellipse with eccentricity of $1.7321$.

Theorems & Definitions (13)

• Definition 2.1
• Lemma 2.1: Circular Table
• Lemma 2.2: Ellipse
• Theorem 2.1: Birkhoff Conjecture
• Lemma 4.1: Multipliers via the multiple shooting eigenvalues
• Lemma 4.2: Parameterization Lemma
• proof
• Lemma 4.3: Functional equation, stable case
• Lemma 4.4: Functional equation, unstable case
• Remark 4.1: Generalizations