Physics / Math
Chaos and attractors
The Lorenz attractor: sensitivity and the Lyapunov exponent.
What happens
The Lorenz system, born from a weather model: trajectories orbit a strange attractor shaped like butterfly wings, never repeating and never leaving. Two points started a whisper apart separate exponentially, as the log-scale chart shows; that growth rate is the Lyapunov exponent, the fingerprint of chaos.
Equations used in the module
Equations x′ = σ (y − x), y′ = x (ρ − z) − y, z′ = x y − β z (Lorenz) σ = 10, ρ = 28, β = 8/3 δ(t) ∝ e^(λt), λ ≈ 0.906
