Explore the famous Lorenz strange attractor in 3D. Two trajectories starting infinitesimally apart diverge exponentially — the butterfly effect. Adjust σ, ρ, β parameters and discover bifurcations. Drag to rotate.
dx/dt = σ(y−x); dy/dt = x(ρ−z)−y; dz/dt = xy−βz. Classic parameters: σ=10, ρ=28, β=8/3. The Lorenz attractor has a fractal dimension D≈2.06 and positive Lyapunov exponent λ₁≈0.906, confirming chaotic behaviour. Trajectories never repeat yet stay within a bounded region of phase space.
Two trajectories starting ε=10⁻⁵ apart diverge exponentially: |δ(t)| ≈ |δ₀|·e^(λ₁t). With λ₁≈0.9, predictability horizon T_pred ≈ (1/λ₁)·ln(1/ε) ≈ 13 time units. In atmospheric modelling (where Lorenz discovered this), this limits weather forecasting to ~2 weeks regardless of how accurate the initial measurements are.
For ρ < 1: stable origin (all trajectories → 0). For 1 < ρ < 24.74: two stable fixed points (C⁺ and C⁻). At ρ=24.74: period-doubling bifurcation cascade begins. For ρ > 24.74: chaotic strange attractor. At ρ≈99.96: a periodic window within chaos. Drag the ρ slider to explore these regimes.