The Lorenz Attractor
In 1963 Edward Lorenz reduced atmospheric convection to three nonlinear ODEs. Round-off in the fourth decimal produced a different weather. The trajectory below integrates the same system in real time using 4th-order Runge–Kutta. Pull on σ, ρ, and β and the butterfly deforms — sometimes gracefully, sometimes catastrophically.
The system
Three coupled ODEs model idealised Rayleigh–Bénard convection:
dx/dt = σ (y − x)
dy/dt = x (ρ − z) − y
dz/dt = x y − β z
What to watch for
With the classical parameters (σ=10, ρ=28, β=8/3) the trajectory is attracted to a bounded set with fractal
structure — the Lorenz manifold. Decrease ρ below ≈ 24.74 and the chaos collapses onto
one of two stable fixed points. Push ρ higher and the orbits grow more tangled. The three coloured
trails are seeded with initial conditions that differ by one part in ten thousand. Left alone, they diverge
exponentially — the signature of a positive Lyapunov exponent.
Numerical note
Integration uses classical RK4 with Δt = 0.005. Each rendered frame advances the state by
integration speed × Δt. Trail fade is a per-frame alpha overlay — higher fade means shorter memory.