Pendulum Basin 3D:
Guess the magnet, then zoom in

The pendulum’s equation of motion combines a restoring spring, friction, and an attraction toward each magnet that grows sharply as the bob approaches. Small differences in the initial position are amplified by the passes near each magnet, which is the defining property of a chaotic system.

Amplification means that arbitrarily close to a starting point that ends on the red magnet there are starting points that end on the blue one. Colouring by outcome therefore cannot produce clean regions with smooth borders; it produces regions whose boundary has structure at every scale.

That is why the zoom is a genuine third act rather than a flourish. Magnifying the boundary is not showing a more detailed picture of the same thing; it is showing new interleaving that was not visible before and that continues as far as the computation is willing to go.

The attraction term includes the bob’s height above the plane, and it is there for a practical reason: without it, the inverse-square attraction blows up directly over a magnet and the simulation breaks.

It also turns out to be the most expressive parameter in the mode. A high bob feels a soft, broad pull from each magnet, so trajectories are gentle and outcomes are fairly predictable, which produces a map with large simple regions.

A low bob feels sharp, localised pulls, so trajectories whip between magnets and tiny differences compound quickly. The map becomes almost entirely boundary. That is the setting worth exploring, because it is where the mode’s two payoffs, the unpredictable guess and the endless zoom, are both at their strongest.