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Bouncing Ball with a Uniformly Varying Velocity in a Metronome Synchronization Task

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Published on: September 21, 2017

Vertical chaos and horizontal diffusion in the bouncing-ball billiard.

Astrid S de Wijn¹, Holger Kantz

¹Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany. wijn@pks.mpg.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|May 16, 2007

Summary

The study of bouncing-ball billiards reveals a link between vertical motion and horizontal transport. Chaotic vertical motion leads to diffusion, while periodic motion results in localization, demonstrating a clear order-chaos transition.

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Area of Science:

Physics
Dynamical Systems
Statistical Mechanics

Background:

Bouncing-ball billiards are theoretical models for studying transport properties in physical systems.
Investigating systems with non-convex scatterers and small slopes is crucial for understanding complex dynamics.

Purpose of the Study:

To analyze the transport properties of a bouncing-ball billiard with non-convex scatterers and small slopes.
To investigate the relationship between vertical motion dynamics and horizontal transport.
To explore the transition from localization to delocalization based on the order-chaos transition in vertical motion.

Main Methods:

Application of the time-scale separation theory.
Theoretical analysis of the interplay between horizontal and vertical motion.
Numerical simulations to confirm theoretical predictions.

Main Results:

A separation of time scales exists between horizontal and vertical motion, controlled by the billiard's slope.
Chaotic vertical motion results in diffusive horizontal motion.
Quasi-periodic vertical motion leads to no diffusion, indicating localization.

Conclusions:

The order-chaos transition in the vertical degrees of freedom directly influences the horizontal motion.
This transition translates into a localization-delocalization transition for the horizontal movement.
The findings provide insights into the fundamental mechanisms governing transport in complex dynamical systems.