Thomas Cyclically Symmetric Attractor

The Thomas system has exact cyclic symmetry: each equation is the same function shifted to the next variable. A single dissipation parameter b controls the transition from chaos to limit cycles to fixed points. The attractor has fractal structure with Lyapunov dimension ≈ 2.08.

Dissipation b: 0.18 Rotation X: 25° Rotation Z: -30° Trail length: 8000

x: —

y: —

z: —

λ₁ ≈: —

dx/dt = sin(y) − b·x
dy/dt = sin(z) − b·y
dz/dt = sin(x) − b·z

b≈0.18: chaos (strange attractor)
b≈0.33: intermittency / transition
b≈0.50: stable limit cycle / fixed point

The cyclic symmetry (x→y→z→x) gives a 3-fold rotational structure. Poincaré section shows self-similar return map.