Two trajectories that start 1e-5 apart on the Lorenz butterfly. Change rho to move from calm to chaos.
The science
Equations. dx/dt = sigma (y - x), dy/dt = x (rho - z) - y, dz/dt = x y - beta z. Lorenz (1963) derived them from a truncated model of convection in a fluid layer heated from below; rho scales with the temperature difference.
Steady states. For rho > 1 there are two non-zero fixed points x = y = +/- sqrt(beta (rho - 1)), z = rho - 1 (the yellow dots). For sigma = 10, beta = 8/3 they lose stability at rho = 24.74 and the orbit wanders chaotically between them.
Divergence. The flow contracts volume at the constant rate -(sigma + 1 + beta), so every orbit lands on a zero-volume strange attractor. The largest Lyapunov exponent (about 0.906 for the classic values) means two nearby copies separate like exp(0.9 t) until they reach the size of the attractor; the numerical value is measured in the checks panel with periodic renormalisation of the gap.
Simplifications. Three modes only: this is not weather prediction. Fixed-step RK4 at dt about 4 ms, which keeps the shadowing error far below the plotted scale but a single trajectory is never exact beyond tens of time units.