René Thomas's 1999 attractor is the mathematical equivalent of a marble rolling on a bumpy 3D landscape: driven by the sine of each coordinate, it wanders forever in a ghostly lattice pattern, never repeating, never escaping.
The system ẋ = sin(y) − bx, ẏ = sin(z) − by, ż = sin(x) − bz has cyclic symmetry: it maps to itself under x→y→z→x. For b ≈ 0.208 the attractor is chaotic. The parameter b controls dissipation — near zero the attractor fills 3D space; larger b compresses it to lower dimension.
Drag to rotate the 3D attractor. Adjust b near 0.208 for peak chaos, below 0.1 for hyperchaos, above 0.32 for period-3 orbit. Launch many particles simultaneously from a tight cluster to visualise the Lyapunov divergence.
Thomas designed the system to model the minimal conditions for chaos: a 3D autonomous system needs at least one cubic term or two quadratic terms — or, as he showed, three sine terms. His interest was not physics but the minimal grammar required for biological regulatory networks.