Geometry-induced chaos and scaling near two singular limits of an eye-shaped billiard
Abstract
We investigate how boundary geometry controls the coexistence of regular and chaotic motion in a static, eye-shaped billiard. A single control parameter continuously deforms the smooth, strictly convex boundary between two singular geometric limits. Across the sampled parameter range, chaotic layers coexist with regular islands and invariant curves that restrict phase-space transport. We derive analytical conditions for the existence of diagonal period-two orbits and determine the stability transitions of the axial period-two trajectories. We also establish the existence of invariant caustics near the boundary for every finite positive value of the control parameter. Numerical simulations reveal a nonmonotonic dependence of the average maximal Lyapunov exponent on the control parameter, withthe strongest sampled instability occurring at an intermediate geometry. The mean squared-angle envelope width associated with the sampled chaotic component decreases toward both singular limits, following power laws with fitted exponents of opposite signs and magnitudes close to two thirds. We further characterize a geometric response function and discuss the resulting analogy with continuous critical behavior. Our findings show how geometric deformation governs orbital stability, barriers to chaotic transport, and scaling properties in conservative billiard dynamics.
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BibTeX
@misc{fonseca2026geometryinduced,
title = {{Geometry-induced chaos and scaling near two singular limits of an eye-shaped billiard}},
author = {Anne Kétri P. da Fonseca and Edson D. Leonel and Diego F. M. Oliveira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29128},
eprint = {2610.29128},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29128},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometry-induced chaos and scaling near two singular limits of an eye-shaped billiard
AU - Anne Kétri P. da Fonseca
AU - Edson D. Leonel
AU - Diego F. M. Oliveira
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29128
UR - https://arxiv.org/abs/2610.29128
AB - We investigate how boundary geometry controls the coexistence of regular and chaotic motion in a static, eye-shaped billiard. A single control parameter continuously deforms the smooth, strictly convex boundary between two singular geometric limits. Across the sampled parameter range, chaotic layers coexist with regular islands and invariant curves that restrict phase-space transport. We derive analytical conditions for the existence of diagonal period-two orbits and determine the stability transitions of the axial period-two trajectories. We also establish the existence of invariant caustics near the boundary for every finite positive value of the control parameter. Numerical simulations reveal a nonmonotonic dependence of the average maximal Lyapunov exponent on the control parameter, withthe strongest sampled instability occurring at an intermediate geometry. The mean squared-angle envelope width associated with the sampled chaotic component decreases toward both singular limits, following power laws with fitted exponents of opposite signs and magnitudes close to two thirds. We further characterize a geometric response function and discuss the resulting analogy with continuous critical behavior. Our findings show how geometric deformation governs orbital stability, barriers to chaotic transport, and scaling properties in conservative billiard dynamics.
ER -