Logistic Map — Feigenbaum Universality

Period-doubling cascade xₙ₊₁ = r·xₙ(1-xₙ); Feigenbaum constant δ ≈ 4.669

2.50
4.00
3.500
300
r = 3.500 Lyapunov λ = - Orbit period: -
δ ≈ 4.66920... (universal for unimodal maps) α ≈ 2.50291... (rescaling) r₁ = 3.0000, r₂ = 3.4495, r₃ = 3.5441
The logistic map f(x) = rx(1−x) undergoes a period-doubling route to chaos. Bifurcation points rₙ converge geometrically: (rₙ−rₙ₋₁)/(rₙ₊₁−rₙ) → δ ≈ 4.6692... (Feigenbaum 1978). This constant is universal across all unimodal maps with a quadratic maximum — proven by renormalization group theory (Lanford 1982). The Lyapunov exponent λ = lim(1/N)Σ ln|f'(xₙ)| is negative (stable) in periodic windows and positive (chaos) in the chaotic band. At r=4: fully chaotic (λ = ln 2), conjugate to tent map. Drag slider to explore.