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The Lorenz attractor comes from a 1963 simplified model of atmospheric convection: three coupled ordinary differential equations relating a state (x, y, z) through three parameters σ, ρ, β:
dx/dt = σ(y−x) dy/dt = x(ρ−z)−y dz/dt = xy−βz
The classic values σ = 10, ρ = 28, β = 8/3 produce the iconic butterfly-shaped strange attractor: the trajectory never settles to a point or a repeating cycle, never crosses itself, and never leaves a bounded region — it spirals around one wing, unpredictably kicks over to the other, and repeats forever without exact repetition. Two trajectories starting a hair's breadth apart diverge exponentially, the sensitive dependence on initial conditions that gave chaos theory its "butterfly effect" name (Lorenz's own 1972 talk title). The attractor itself is a fractal: it has a non-integer dimension of roughly 2.06 — more than a surface, less than a solid.
Below ρ ≈ 24.74 (with σ = 10, β = 8/3) the chaos vanishes: trajectories spiral down to one of two stable fixed points instead. Try dragging ρ below that threshold to watch the butterfly collapse into a simple spiral.
The trajectory is integrated with fourth-order Runge–Kutta and rendered by the same point-cloud renderer built for the 3D IFS page — a dense point sequence along the curve, projected and density-shaded exactly like that chaos-game point cloud.
The whole page was created with Claude Code. These were the most significant prompts that shaped this page: