Bifurcation & Chaos
The logistic map xn+1 = r·xn(1−xn) is a single equation with one parameter. As r increases past 3, the fixed point splits into two, then four, then eight — a cascade of period doublings converging at the Feigenbaum point r∞ ≈ 3.5699. Beyond it: chaos, with islands of order buried inside. Drag to zoom. Click to inspect.
The logistic map
Originally a population model (Verhulst, 1845), the map xn+1 = r·xn(1−xn) compresses the richness of nonlinear dynamics into one line. For r < 1 the population collapses. For 1 < r < 3 it settles to a fixed point. At r = 3 the fixed point loses stability and the orbit bounces between two values — a period-2 cycle. This doubles again near r ≈ 3.449, then r ≈ 3.544, approaching r∞ ≈ 3.5699 geometrically with ratio δ ≈ 4.6692 — the Feigenbaum constant.
Why δ ≈ 4.6692 matters
Feigenbaum showed in 1978 that the same ratio appears in any smooth one-humped map — the quadratic, sin(x), x·e1−x, all of them. δ is universal in the way π is. It has since been measured experimentally in dripping faucets, Rayleigh-Bénard convection, and electronic circuits. This is why chaos theory belongs in the same conversation as renormalisation group theory in statistical mechanics.
The Lyapunov exponent
λ = limN→∞ (1/N) Σ ln|f′(xn)| where f′(x) = r(1−2x). Negative λ means the orbit is attracted to a stable cycle; positive λ is chaos. λ = 0 marks bifurcation points and the Feigenbaum accumulation point. The diagram brightness encodes orbit density — dense regions glow bright cyan, sparse regions fade to dark.
The period-3 window
Around r ≈ 3.828, a stable period-3 cycle appears from nothing, surrounded by its own bifurcation cascade. By Sharkovskii's theorem, period-3 implies all periods exist — but the chaotic sea swallows most of them. Zoom into the period-3 preset and drag to explore the self-similar structure inside: you are looking at a miniature copy of the full diagram.