Bifurcation & Feigenbaum δ
Iterate x ← r · x · (1−x). For each r ∈ [2.9, 4], plot the long-term values x visits. The single line splits to two, then four, then eight, then chaos. The ratio of successive splits is universal: 4.66920160910299…, the Feigenbaum constant δ — the same number appears for any smooth 1D map with a single maximum.
first split (1→2) at r₁
3.0000
universality — δ for any peaked map
The logistic map x ← r · x · (1−x) isn't special. Take any smooth function with a single quadratic maximum (sine, exponential, cosine, polynomial — anything peaked), iterate it, and look at the bifurcation diagram. The onset of chaos happens via period-doubling: 1 → 2 → 4 → 8 → 16 → … → ∞, and the ratio of distances between successive doublings is always δ ≈ 4.66920.
Mitchell Feigenbaum discovered this in 1975 at Los Alamos, working on a hand calculator. He could not have predicted that this number would also govern fluid dynamics, neural firing patterns, the geometry of dripping faucets, and the onset of turbulence in convection cells. It is a constant of universality — the same way π shows up wherever circles or oscillations live, δ shows up wherever a system undergoes period-doubling on its way to chaos.
Inside the chaotic region (r > 3.57) you'll see periodic windows — strips of stable cycles. The most famous is the period-3 window at r ≈ 3.83, where the chaos suddenly collapses to a three-cycle. Yorke's theorem says: period 3 implies chaos — any map with a 3-cycle has cycles of every period.