Bifurcation diagram
One equation: x → rx(1−x). One parameter: r. Below r = 3, every starting point converges to the same fixed value — a stable equilibrium. At r = 3, that equilibrium splits: the system oscillates between two values. Split again at r ≈ 3.449. Again at 3.544. The splits accelerate, governed by the Feigenbaum constant δ ≈ 4.6692, until around r ≈ 3.57 the pattern gives way to chaos — yet order persists within it, islands of periodicity scattered through the noise.
xₙ₊₁ = r·xₙ·(1−xₙ) · δ = lim (rₙ−rₙ₋₁)/(rₙ₊₁−rₙ) ≈ 4.6692 · Mitchell Feigenbaum, 1975
first bifurcation — period-1 splits to period-2
second bifurcation — period-4 begins
third bifurcation — period-8
onset of chaos — accumulation point
The ratio of successive bifurcation intervals converges to δ ≈ 4.6692 (Feigenbaum constant) — universal across all unimodal maps. The period-3 window around r ≈ 3.83 implies all other periods by the Sharkovskii theorem.