Transition to chaos with conical billiards

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.