The logistic map x_{n+1} = r·x_n(1−x_n). As r increases, period 2^n cascades appear. Feigenbaum's universal constants δ ≈ 4.669 (ratio of bifurcation intervals) and α ≈ 2.503 govern all unimodal maps.
Feigenbaum (1978): every smooth unimodal map has the same period-doubling cascade with δ = 4.6692... and α = 2.5029... These are transcendental numbers, derived from the renormalization fixed point equation g(x) = −αg(g(−x/α)). The accumulation point r_∞ ≈ 3.5699 is the onset of chaos. Drag the r slider to sweep through period-1, 2, 4, 8, chaos windows.