What is a strange attractor?

A strange attractor is a fractal subset of phase space toward which a chaotic dynamical system converges. Unlike fixed points (0D) or limit cycles (1D), strange attractors have non-integer (fractal) Hausdorff dimension. Trajectories are confined to the attractor, never repeat (quasi-periodic), and display sensitive dependence on initial conditions — the butterfly effect. The Lorenz attractor has fractal dimension ≈ 2.06.

What is the butterfly effect?

The butterfly effect, formalised by Edward Lorenz in 1963, is sensitive dependence on initial conditions: two trajectories that start infinitesimally close will diverge exponentially, characterised by the largest Lyapunov exponent λ₁ > 0. For the Lorenz system λ₁ ≈ 0.906, meaning errors roughly double every 0.76 time units. Despite long-term unpredictability, the attractor's shape is deterministic — its structure is identical for all initial conditions that land on it.

How does drag-to-rotate work?

Click and drag the canvas horizontally to rotate around the vertical (Y) axis, and vertically to rotate around the horizontal (X) axis. This applies Euler angle rotations to the 3D coordinates before projecting to 2D using a simple perspective transform. Releasing the drag restores auto-rotation. The projection uses a depth field of view of 4 to give a slight perspective effect.

Why are the attractors coloured differently?

Each preset uses a different colour mapping strategy. Lorenz, Halvorsen, and Chen colour by z-coordinate: low z gives blue-green, high z gives red-orange. Aizawa and Dadras colour by angular position arctan(y/x) — the hue rotates as the trajectory orbits, revealing the toroidal / gyrating structure. Thomas colours by instantaneous speed |ẋ|, so fast segments appear warm and slow segments appear cool.

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🧪 Strange Attractors

Interactive 3D strange attractor gallery: Lorenz butterfly, Thomas cyclically-symmetric, Halvorsen cyclic, Aizawa, Chen, and Dadras attractors. RK4 integration, drag-to-rotate, colour-coded by z-value or angular position, adjustable trail length and speed. Drag to rotate, click presets.

Chaos & Dynamics2DModerate60 FPS

⚙ Under the hood

This simulation demonstrates the concept of strange attractors in dynamical systems theory. It showcases examples like the Lorenz attractor, the Halvorsen attractor, and the Chen attractor, illustrating how chaotic systems can exhibit complex patterns despite deterministic rules.

Strange AttractorsLorenz

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install