Feigenbaum Universality — Period-Doubling Cascade

As the control parameter r increases in the logistic map x→rx(1-x), period doublings occur at r₁, r₂, r₃,... with universal ratio δ = (rₙ-rₙ₋₁)/(rₙ₊₁-rₙ) → 4.66920... (Feigenbaum constant). This ratio is the same for ANY unimodal map — a profound universality explained by renormalization group fixed points in function space.

Map

r (parameter): 3.500 r display range min: 2.50 r display range max: 4.00 Iterations: 300

Current r: —

Period: —

δ₁ = 4.6692...

α = 2.5029...

Period-doubling bifurcations:
r₁ = 3.000
r₂ = 3.449
r₃ = 3.544
r₄ = 3.5644
r∞ = 3.5699... (onset of chaos)

δ = lim (rₙ-rₙ₋₁)/(rₙ₊₁-rₙ) = 4.6692
Universal for ALL unimodal maps