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Transition to chaos with conical billiards.

Lara Braverman1, David R Nelson1,2

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Particle trajectories on cones exhibit complex dynamics, transitioning from predictable paths to chaotic behavior influenced by cone shape and boundary tilt. This study reveals distinct regions governing trajectory patterns and ergodicity.

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Area of Science:

  • Mathematical Physics
  • Dynamical Systems
  • Geometric Optics

Background:

  • Classical billiard dynamics and geometrical optics offer frameworks for analyzing particle motion.
  • Conical geometry presents unique challenges due to its concentrated Gaussian curvature at the apex.

Purpose of the Study:

  • To investigate particle trajectories on a cone with specular reflections off an elliptical boundary.
  • To explore the influence of cone deficit angle (χ) and plane tilt angle (γ) on trajectory dynamics and ergodicity.
  • To identify conditions leading to chaotic dynamics and analyze the transition pathways.

Main Methods:

  • Adapting concepts from geometrical optics and classical billiard dynamics.
  • Analyzing particle trajectories with constant velocity on a cone with an elliptical boundary.
  • Utilizing Poincaré maps to visualize the transition to chaos.

Main Results:

  • Identified distinct trajectory regimes (a, b, c) based on initial conditions, cone parameters (γ, χ).
  • Observed a transition to chaotic dynamics for larger χ and γ, characterized by uniform sampling (ergodicity).
  • The untilted cone case results in ring caustics, while tilted cases show complex behavior.

Conclusions:

  • The interplay between cone geometry and boundary conditions dictates particle trajectory patterns, including ergodicity.
  • Poincaré maps effectively illustrate the transition to chaos in this conical billiard system.
  • The study provides insights into chaos formation in conservative dynamical systems.