In 1963, Ed Lorenz simplified atmospheric convection to three coupled ODEs. The solution wraps around two saddle points and never repeats — the original "butterfly effect." Two trajectories starting 10⁻⁵ apart diverge to fill the entire attractor within a few thousand steps. The watch is real chaos: deterministic, bounded, mixing.
three equations, infinite complexity
The system: dx/dt = σ(y − x), dy/dt = x(ρ − z) − y, dz/dt = xy − βz. Three real variables, three quadratic terms. Standard values: σ = 10, ρ = 28, β = 8/3.
What's visualized: two orbits (purple, gold) started at almost-identical conditions. They overlap perfectly at first, then their tiny separation grows exponentially and they diverge into different lobes of the butterfly. This is the textbook demonstration that sensitive dependence on initial conditions can coexist with topological stability — the trajectories never leave a bounded region (the attractor) but their futures become uncorrelated.
Lorenz's discovery launched modern chaos theory. The same equations model thermal convection, laser physics, electronic circuits, and (oddly) the dynamics of population balance in some ecological models. The hidden mathematics — strange attractors with non-integer dimension, the geometry of fractal basin boundaries — was developed almost entirely after the picture above became famous.