2D Lorenz Attractor — Butterfly Projection & Lorenz Map

This is a 2D companion to the site's 3D Lorenz attractor. Instead of a rotatable 3D scene, it integrates the exact same three differential equations with real 4th-order Runge–Kutta stepping and draws two flat, information-dense views: the x–z "butterfly" projection of the trajectory itself, and the Lorenz map — the first-return map of successive z-peaks that Edward Lorenz used in his original 1963 paper to show that the attractor's apparent randomness hides a simple underlying rule. A second trajectory started just 0.00001 away in x runs alongside the first so the exponential divergence that gives chaos theory its "butterfly effect" name can be read as a number, not just seen as a visual fan-out.