The Lorenz attractor — two trajectories starting 10⁻⁴ apart diverge into the iconic butterfly shape.
The Discovery
In 1963, Edward Lorenz was running a simplified model of atmospheric convection on a computer. He wanted to re-run part of a simulation and, rather than starting from scratch, he typed in the numbers from a printout — but only to three decimal places instead of six. The second run diverged wildly from the first over time. That tiny rounding error exposed something profound: deterministic chaos — a system governed by precise equations whose long-term behaviour is fundamentally unpredictable.
The System
The three equations are deceptively simple:
dx/dt = σ (y − x)
dy/dt = x (ρ − z) − y
dz/dt = x y − β z
- σ (Prandtl number) relates diffusion rates. Classic value: 10.
- ρ (Rayleigh number) controls the strength of convection. At ρ = 28, the system enters the chaotic regime — the attractor becomes a strange attractor.
- β is a geometric factor. Classic value: 8/3.
For ρ < 1, the system has a single stable fixed point at the origin — the fluid is motionless. At ρ = 1, convection begins. By ρ ≈ 24, the system is fully chaotic.
Why It Matters
The Lorenz system is the canonical example of deterministic chaos. The "butterfly effect" — the idea that a butterfly flapping its wings in Brazil could set off a tornado in Texas — comes directly from this system. Two trajectories starting arbitrarily close together diverge exponentially fast, meaning any measurement error, no matter how small, eventually makes the forecast useless.
Rotate the view by watching for a moment — the attractor's 3D structure becomes clear. Two lobes connected by a narrow bridge, with trajectories orbiting one lobe a few times before crossing to the other. The pattern never repeats.