The logistic map xₙ₊₁ = r·xₙ·(1−xₙ) is a one-line population model that encodes the entire path from stability to chaos. The bifurcation diagram is its DNA: it shows exactly where period doubling cascades begin and chaos takes over.
For r < 3 the map settles to a fixed point. At r ≈ 3 the period doubles to 2, at r ≈ 3.449 to 4, then 8, 16 … accumulating at the Feigenbaum point r ≈ 3.56995 where chaos begins. The ratio of successive bifurcation widths converges to δ ≈ 4.669 — a universal constant.
Drag the zoom window to magnify any region. The diagram is self-similar — every chaotic window contains a miniature copy of the entire diagram. Toggle Orbit paths to trace individual trajectories alongside the full bifurcation structure.
Mitchell Feigenbaum discovered the constant δ in 1975 on an HP-65 pocket calculator. Remarkably, the same constant appears in any smooth one-dimensional map with a single quadratic maximum — from the Hénon map to the driven pendulum. It is a universal law of chaos.