Butterfly effect in an elliptic curve but not in an ellipse

Billiard comparison between the bounded region of the function y^2 = x^3 - x (elliptic curve; see https://www.wolframalpha.com/input?i=...) and x^2 + ay^2 = 1 (ellipse). For certain initial conditions, the elliptical curve system exhibits a chaotic behaviour, i.e., "butterfly effect" in contrast to the ellipse.

1. Ball fired off in a specific direction for both systems.

2. Ball fired off in another specific direction for both systems.

3. Varying starting directions (white = initial direction) spanning between vertical and horizontal direction, where the starting point is fixed.

4. 500 balls equidistantly arranged in the horizontal extension fired off upwards.

Heavily inspired by a mail conversation I had with Max Weinreich who is presently investigating the elliptical curve. Please take a look at https://arxiv.org/abs/2305.14287 for further mathematical insight.

Music by ‪@gpcbass‬; song called aLoop and used previously here.

Visuals in Python & FFmpeg.