Feigenbaum Period Doubling

Logistic map xₙ₊₁ = r·xₙ(1−xₙ). The bifurcation diagram reveals universal period-doubling route to chaos. δ = 4.6692... appears in ALL unimodal maps.

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r min 2.50 r max 4.00 Iterations 300 Skip 200

δ = 4.6692016...

Bifurcation points:
r₁ ≈ 3.0000 (period 1→2)
r₂ ≈ 3.4495 (period 2→4)
r₃ ≈ 3.5441 (period 4→8)
r₄ ≈ 3.5644 (period 8→16)
r∞ ≈ 3.5699 (onset of chaos)

Feigenbaum constant:
δ = lim (rₙ-rₙ₋₁)/(rₙ₊₁-rₙ) ≈ 4.669

Lyapunov exponent:
λ = ⟨ln|r(1-2x)|⟩
λ<0: stable; λ=0: bifurcation; λ>0: chaos

Cobweb diagram (drag r slider)

Cobweb

r value 3.57 Steps 60

Period-doubling cascade:
r=2.9: fixed point
r=3.1: period 2
r=3.5: period 4
r=3.57: period 8
r=3.83: period 3 window
r=4.0: fully chaotic