Lorenz Attractor: Butterfly Effect Simulator

Lorenz Attractor

σ (sigma) 10.0

ρ (rho) 28.0

β (beta) 2.67

Simulation speed 1.0×

Time elapsed

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Separation Δ

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Trail A points

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Speed |v|

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How it works

The Lorenz system is three coupled nonlinear ordinary differential equations, originally derived from a truncated model of atmospheric convection. Each point in the 3D scene is integrated forward in time with a small fixed step (RK4), tracing a continuous trajectory through phase space that never crosses itself yet never repeats — settling onto the fractal, butterfly-shaped strange attractor. A second point (cyan) starts just 0.0001 away from the first (violet) on every axis. Because nearby trajectories on the attractor diverge exponentially, the two curves track each other closely at first, then visibly peel apart after a few dozen time units — the butterfly effect made visible.

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

σ (sigma) — the Prandtl-number-like ratio of momentum to thermal diffusivity; controls how fast x chases y.
ρ (rho) — proportional to the Rayleigh number; above ρ≈24.74 (with the classic σ, β) the two fixed points become unstable and the system turns chaotic.
β (beta) — a geometric factor from the box the convection cell was truncated to; the classic value is 8/3 ≈ 2.67.
Simulation speed — how many integration steps run per rendered frame, i.e. how fast the trajectories are drawn.
Twin trajectory — toggle the second, near-identical starting point to see (or hide) the sensitivity-to-initial-conditions demonstration.

Real-world relevance: this exact three-equation structure is why weather forecasts are fundamentally limited to about two weeks — no matter how precise the model, any tiny uncertainty in the initial atmospheric state grows exponentially until the forecast is no better than a guess.

Trajectory A Trajectory B (Δ0 = 0.0001)

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