Stickiness in mushroom billiards.

Area of Science:

• Physics
• Dynamical Systems
• Chaos Theory

Background:

• Mushroom billiards exhibit distinct regular and chaotic regions separated by a clear boundary.
• Understanding the dynamics near this boundary is crucial for characterizing the system's overall behavior.

Purpose of the Study:

• To investigate the dynamical properties of chaotic trajectories in mushroom billiards.
• To identify the mechanisms responsible for trajectory 'stickiness' near the regular region's border.
• To explain the observed periodicity and power-law distribution of recurrence times.

Main Methods:

• Analysis of chaotic trajectories within the mushroom billiard model.
• Identification and characterization of periodic orbits, particularly those near the regular-chaotic border.
• Examination of the statistical properties of recurrence times.

Main Results:

• Chaotic trajectories exhibit 'stickiness' near the regular region's border due to an infinite number of marginally unstable periodic orbits.
• These orbits, though of zero measure, dictate the system's primary dynamical properties.
• The distribution of recurrence times shows periodicity and a power-law behavior with exponent gamma=2, explained by these orbits.

Conclusions:

• Marginally unstable periodic orbits are key to understanding the complex dynamics of mushroom billiards.
• These orbits govern the long-time behavior and statistical properties of chaotic trajectories near regular regions.
• The findings provide insights into the interplay between regular and chaotic dynamics in polygonal billiards.