Lorenz Attractor

A real-time WebGL simulation of the Lorenz system, the classic chaotic ODE that traces out the butterfly-shaped strange attractor. Drag the canvas to rotate, scroll to zoom, and tweak every parameter below.

drag to rotate · scroll to zoom

Auto-rotate

σ (sigma) 10.0

ρ (rho) 28.0

β (beta) 2.667

Time step (dt) 0.0080

Trail length 2500

Speed (steps/frame) 3

Rotation speed 0.15

Initial state (applied on Reset trail)

x₀

y₀

z₀

dx/dt = σ(y − x) dy/dt = x(ρ − z) − y dz/dt = xy − βz

Devised by Edward Lorenz in 1963 as a simplified model of atmospheric convection, this system is deterministic yet aperiodic: two trajectories starting a hair's breadth apart diverge exponentially, the signature of chaos. The classic values σ=10, ρ=28, β=8/3 produce the familiar two-lobed "butterfly". Try lowering ρ below 1 to see the trail collapse onto the single fixed point at the origin instead.