Lorenz's stranger, more ornate cousin
Three coupled equations, six dials, and a trajectory that folds itself into a basket-handled torus with a vertical spike through its heart. Aizawa lives in the same family as Lorenz and Rössler — but its geometry is uncannily sculptural, and a single parameter can melt it from basket to flower to chaos.
Yoji Aizawa and collaborators formulated this 3D system in the early 1980s; like Lorenz's butterfly it never settles and never repeats. But where Lorenz's attractor looks like two flat wings, Aizawa's looks like a torus threaded by a vertical spike — a geometry so distinctive it became a staple of strange-attractor galleries.
Pick the canonical parameters (a≈0.95, b≈0.7, c≈0.6, d≈3.5, e≈0.25, f≈0.1). Start anywhere. After a brief transient the orbit lies on a basket-handled torus pierced by a thin vertical column. Every visit traces a slightly different path along the basket's weave.
Aizawa is one point in a rich landscape of chaotic attractors — Rössler, Thomas, Halvorsen, Lorenz-84. Studying it teaches us what 3D chaos looks like geometrically, and how a small parameter change can rebuild an entire shape continuously.
Drag the canvas to rotate. Pull the «b» slider — watch the basket close, breathe, and bloom.
Each line says how fast one coordinate changes. The cross-terms (z·x, z·y, x²+y², z³, z·x³) are where the chaos lives; strip them and the x, y plane collapses to a damped spiral. Six parameters (a, b, c, d, e, f) give you a small kingdom of behaviours; the integrator step dt is a numerical knob, not a dynamical one.
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
At the canonical values the attractor looks like a doughnut threaded onto a vertical needle — a torus with a basket-style weave around its body and a tall spike piercing its centre. The trajectory winds many times around the torus before slipping up the spike and coming back down.
A schematic of the canonical shape: a thick toroidal weave around the z-axis, pierced by a thin column that the orbit visits between long episodes around the rim.
Lower b thins the torus toward a ring; higher b puffs it into a fuller basket. The vertical spike persists across the whole range.
Slide «b» from 0.3 to 1.2 and the basket morphs continuously: ribbon, weave, bowl, vase, chaos. There is no clean phase diagram — instead the parameter space is a smooth atlas of shapes, each a slightly different equilibrium between stretching and folding.
Same three equations, same five other dials. Only b changes, and the geometry breathes continuously.
Lorenz (1963), Rössler (1976), Aizawa (1982), Thomas, Halvorsen, Lorenz-84 — all are smooth 3D systems whose orbits fold onto fractal sets. Sprott's atlas catalogues hundreds. They share the same machinery: stretching that amplifies tiny differences, folding that keeps them bounded.
The leading Lyapunov exponent is positive (reported numerical estimates sit around 0.1), meaning nearby trajectories separate exponentially — the signature of chaos. The Kaplan-Yorke conjecture turns the exponent spectrum into a fractal dimension; for Aizawa it lands just above 2 — barely thicker than a surface.
Just above a surface, well below a solid. Computed from the Lyapunov spectrum (leading exponent positive, around 0.1; second near zero; third strongly negative).
Aizawa is a poster child for teaching 3D chaos — its shape reads at a glance. Control theorists use such systems as benchmarks for nonlinear stabilisation. And the trajectory's gentle, never-repeating undulations make beautiful raw material for procedural animation and analog-style music synthesis.
The Explorer integrates the equations live, lets you sweep all six parameters, and rotates the attractor freely in 3D. Switch between Aizawa, Rössler, Thomas and Halvorsen to feel the whole family.
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