The Aizawa Attractor

The animation in the background is the Aizawa attractor, a strange attractor: a chaotic system whose trajectories never repeat or settle down, yet stay confined to this sphere-like shape forever. Each particle follows three coupled differential equations:

dx/dt = (z − b)·x − d·y

dy/dt = d·x + (z − b)·y

dz/dt = c + a·z − z³/3 − (x² + y²)(1 + e·z) + f·z·x³

with constants a = 0.95, b = 0.7, c = 0.6, d = 3.5, e = 0.25, f = 0.1. The first two equations spin points around the vertical axis while pushing them outward or inward depending on height; the third squeezes the flow vertically, folding it back into the tube you see.

The system is chaotic: two particles that start almost together drift apart exponentially fast, which is why the swarm smears into a cloud instead of marching in lockstep. Every dot here is just these three equations stepped forward in time, thousands of times per second.

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