Recurrence time distribution in mushroom billiards with parabolic hat

Area of Science:

• Mathematical Physics
• Dynamical Systems
• Chaos Theory

Background:

• Billiard systems are classical models for studying dynamical systems and chaos.
• Recurrence time distributions often follow algebraic laws in integrable and chaotic systems.
• Mushroom billiards with parabolic hats present unique dynamical features.

Purpose of the Study:

• Investigate the recurrence time distribution in parabolic mushroom billiards.
• Analyze the impact of marginally unstable periodic orbits on dynamics.
• Understand deviations from universal behavior in recurrence time distributions.

Main Methods:

• Numerical simulations of particle trajectories.
• Theoretical analysis of phase space structure.
• Examination of periodic orbit families and their stability.

Main Results:

• Classical dynamics show a sharply divided phase space.
• Recurrence time distribution generally obeys algebraic laws.
• Sticky motion deviates from simple drift due to orbit crossings.
• Exponent for cumulative recurrence time distribution approaches 2 asymptotically.
• Significant deviations observed at smaller timescales due to slowdown effects.

Conclusions:

• Parabolic mushroom billiards exhibit complex dynamics influenced by specific periodic orbits.
• Deviations from universal recurrence time distributions are explained by orbit crossing phenomena.
• The study highlights the interplay between periodic orbits and chaotic dynamics.