the Lorenz attractor

In 1963 the meteorologist Edward Lorenz stripped atmospheric convection down to just three equations and integrated them on an early computer, expecting a simple result. Instead the solution wove endlessly around two spirals, never settling and never repeating, tracing a shape that famously resembles a butterfly's wings. It became the first and most iconic strange attractor and the birthplace of modern chaos theory.

The Lorenz system is three coupled ordinary differential equations for variables x, y, z: dx/dt = sigma (y - x), dy/dt = x (rho - z) - y, dz/dt = x y - beta z. Here x is proportional to the convective overturning, y and z to horizontal and vertical temperature variations, and sigma (the Prandtl number), rho (a scaled Rayleigh number), and beta are parameters. At the classic values sigma = 10, beta = 8/3, rho = 28, the two symmetric fixed points representing steady convection are both unstable, and trajectories are pulled onto a bounded fractal set, the Lorenz attractor, of dimension about 2.06, on which the motion is chaotic with a positive Lyapunov exponent. The system is dissipative: phase-space volume contracts everywhere (the divergence of the flow is the constant -(sigma + 1 + beta) < 0), so the attractor has zero volume yet trajectories never converge to a point.

The Lorenz attractor is the canonical demonstration that a simple, low-dimensional, fully deterministic system can be inherently unpredictable, and it gave the butterfly effect both its name and its emblem. Honest caveats: the three equations are a drastic truncation of the fluid dynamics (a severe Galerkin approximation keeping only three modes), so the model is a conceptual paradigm, not a quantitative model of real convection; and the rigorous mathematical proof that this set really is a strange attractor was not completed until Warwick Tucker did so in 2002, decades after its discovery.