Lorenz Attractor (3D)

Lorenz System (1963): dx/dt = σ(y−x), dy/dt = x(ρ−z)−y, dz/dt = xy−βz. Originally derived as a simplified model of atmospheric convection. At σ=10, ρ=28, β=8/3, the system is chaotic — trajectories never repeat yet are confined to the butterfly-shaped strange attractor.

Sensitive dependence on initial conditions (butterfly effect): two trajectories starting 10⁻¹⁰ apart diverge exponentially with Lyapunov exponent λ₁ ≈ 0.906. This makes long-term weather prediction fundamentally impossible — not just practically hard.

The attractor has fractal dimension ≈ 2.06 (Hausdorff) — it's a 2D surface with holes, embedded in 3D. The system is dissipative (phase volume contracts at rate σ+1+β) but the attractor has zero volume yet positive measure in the flow direction.