Lorenz Attractor & Butterfly Effect Simulator

Lorenz Attractor

σ (sigma) 10.0

ρ (rho) 28.0

β (beta) 2.67

Simulation speed 1.0×

Elapsed time

–

Divergence

–

Point A pos

–

Trail points

–

How it works

Two points start almost exactly together in a 3D phase space governed by the Lorenz equations, then get integrated forward one tiny timestep at a time. Because the system is chaotic, any minuscule difference between the two starting points is amplified exponentially — the trajectories look identical for a while, then peel apart and roam independently over the same butterfly-shaped attractor. Neither trajectory ever crosses itself or repeats, yet both stay confined to two lobes forever: order and unpredictability coexisting in one bounded shape.

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

σ (sigma) — the Prandtl-like ratio of momentum to thermal diffusivity; controls how fast x chases y.
ρ (rho) — the Rayleigh-like driving parameter; above ≈24.74 (with σ=10, β=8/3) the fixed points become unstable and the butterfly attractor appears.
β (beta) — a geometric damping factor; reshapes how tightly the two lobes of the attractor are wound.
Twin trajectory — a second point started only 0.0001 away from the first; watch how quickly the two paths diverge — the "butterfly effect".

Real-world relevance: this is the original 1963 Lorenz model of atmospheric convection. It's why weather forecasts are fundamentally limited to about two weeks — no matter how good the model, any tiny uncertainty in today's measurements grows exponentially until the forecast is worthless.

Trajectory A Trajectory B (+0.0001)

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.