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Updated: May 24, 2026
05:04
Bouncing Ball with a Uniformly Varying Velocity in a Metronome Synchronization Task
Published on: September 21, 2017
Coupling of bouncing-ball modes to the chaotic sea and their counting function
Steffen Löck1, Arnd Bäcker, Roland Ketzmerick
1Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany.
Summary
We analyzed how bouncing-ball modes couple with chaotic modes in 2D billiards. Our analytical predictions for decay rates matched numerical results for stadium and cosine billiards, revealing new asymptotic behaviors.
Area of Science:
- Mathematical Physics
- Quantum Chaos
- Statistical Mechanics
Background:
- Bouncing-ball modes describe particle motion in specific billiard geometries.
- Chaotic modes represent complex, unpredictable trajectories within these systems.
- Understanding mode coupling is crucial for predicting system behavior.
Purpose of the Study:
- To investigate the coupling between bouncing-ball and chaotic modes in 2D billiards.
- To analytically predict decay rates using the fictitious integrable system approach.
- To determine the asymptotic behavior of the counting function N_{bb}(E).
Main Methods:
- Utilizing the fictitious integrable system approach for analytical predictions.
- Employing numerical simulations to determine decay rates.
- Comparing analytical predictions with numerical results for validation.
Main Results:
- Analytical predictions for decay rates show agreement with numerical data for stadium and cosine billiards.
- The asymptotic behavior of the counting function N_{bb}(E)∼E^{δ} was predicted.
- For the stadium billiard, δ=3/4 was confirmed; for the cosine billiard, δ=5/8 was derived and numerically validated.
Conclusions:
- The fictitious integrable system approach accurately predicts decay rates in 2D billiards.
- A novel asymptotic behavior (δ=5/8) was identified for the cosine billiard, differing from previous upper bounds.
- This study advances the understanding of mode coupling and spectral properties in chaotic systems.
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