Chaos in the square billiard with a modified reflection law

Area of Science:

• Dynamical systems
• Mathematical physics
• Chaos theory

Background:

• Standard square billiards exhibit integrable dynamics.
• Non-standard reflection laws can introduce complex behaviors.
• Understanding billiard dynamics is crucial in various scientific fields.

Purpose of the Study:

• To investigate the dynamical properties of a square billiard with a linear contraction reflection law.
• To analyze the structure of the nonwandering set under this non-standard condition.
• To contrast the system's behavior with the integrable standard square billiard.

Main Methods:

• Numerical simulations to observe particle trajectories.
• Analytical arguments to support theoretical findings.
• Topological analysis of the nonwandering set.

Main Results:

• The nonwandering set decomposes into three distinct invariant sets: a parabolic attractor, a chaotic attractor, and a set of several horseshoes.
• The system exhibits positive topological entropy.
• This behavior contrasts sharply with the integrability of standard square billiards.

Conclusions:

• The non-standard reflection law fundamentally alters the dynamics of the square billiard.
• The system demonstrates characteristics of chaos, indicated by positive topological entropy.
• The decomposition of the nonwandering set provides insight into the complex behavior of this modified billiard system.