Visualization
The Lorenz attractor. Where deterministic chaos was born.
Edward Lorenz discovered this system in 1963 while simplifying a model of atmospheric convection. Three ODEs. No randomness. Yet the trajectory is aperiodic, sensitive to initial conditions, and confined to a strange attractor with fractal dimension ≈ 2.06.
The Lorenz system is the classical example of deterministic chaos — infinitesimally close starting points diverge exponentially, yet the trajectory is confined to a strange attractor of fractal dimension ≈ 2.06.
The system
Three coupled ordinary differential equations:
dx/dt = σ (y − x)
dy/dt = x (ρ − z) − y
dz/dt = x y − β zWith Lorenz's classical parameters σ = 10, ρ = 28, β = 8/3, all trajectories approach a butterfly-shaped attractor. The two "wings" correspond to the two saddle-focus equilibria at ±(√(β(ρ−1)), √(β(ρ−1)), ρ−1). Trajectories orbit one wing, then suddenly jump to the other — apparently at random.
Try ρ < 24: you'll see convergence to a stable equilibrium. Between ρ = 24 and ρ ≈ 24.74 the system has two coexisting stable limit cycles. Beyond that, chaos — and the classic butterfly shape at ρ = 28. For much larger ρ, new periodic windows open up.
The fractal dimension of the attractor was one of the first precise quantitative results on chaotic dynamics. It is also the system from which "the butterfly effect" takes its name.