Intrinsic stickiness in open integrable billiards: tiny border effects

TL;DR

This paper investigates how tiny rounded borders in open integrable billiards induce sticky motion and self-similar structures in escape times and emission angles, revealing significant effects even at very small border sizes.

Contribution

It demonstrates that minimal border rounding can cause stickiness and complex escape dynamics in integrable billiards, highlighting the sensitivity of such systems to boundary effects.

Findings

Tiny borders (~0.1%) induce stickiness and self-similar escape structures.

Larger borders (>10%) lead to chaotic escape time decay.

Border effects significantly influence the dynamical behavior of open billiards.

Abstract

Rounding border effects at the escape point of open integrable billiards are analyzed via the escape times statistics and emission angles. The model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form. Stickiness and self-similar structures for the escape times and emission angles are generated inside "backgammon" like stripes of initial conditions. These stripes are born at the boundary between two different emission angles but same escape times. As the rounding effects increase, backgammon stripes start to overlap and the escape times statistics obeys the power law decay and anomalous diffusion is expected. Tiny rounded borders (around $0.1\%$ from the whole billiard size) are shown to be sufficient to generate the sticky motion, while borders larger than $10\%$ are enough to produce escape times with chaotic decay.