Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards

Area of Science:

• Physics
• Wave Propagation
• Chaos Theory

Background:

• Electromagnetic wave propagation in closed domains with reflecting boundaries can be modeled as ray propagation in a billiard system.
• The symplectic reflection map describes polygonal ray paths in uniform media.
• Understanding trajectory stability is crucial for analyzing wave behavior.

Purpose of the Study:

• To analyze the stability of trajectories in electromagnetic wave propagation using billiard dynamics.
• To introduce and test Lyapunov and reversibility error indicators for chaos detection.
• To evaluate the effectiveness of these indicators in a family of chaotic billiards.

Main Methods:

• Eikonal approximation for wave propagation.
• Billiard dynamics to model ray paths.
• Linear response theory for trajectory stability analysis.
• Lyapunov and reversibility error indicators for sensitivity estimation.

Main Results:

• Symplectic reflection map accurately describes ray paths in uniform media.
• Lyapunov and reversibility error indicators provide estimates of sensitivity to initial and reflection deviations.
• These indicators were tested on chaotic billiards to assess their chaos detection capabilities.

Conclusions:

• The study demonstrates the utility of billiard dynamics for analyzing electromagnetic wave propagation.
• Lyapunov and reversibility error indicators are effective tools for detecting chaos in such systems.
• The findings contribute to a deeper understanding of wave behavior in complex, confined environments.