📈 Population Chaos — Logistic Map & Bifurcation
The logistic map xn+1 = r·xₙ·(1−xₙ) models population growth with limited resources. For small r the population reaches a fixed point; as r increases it undergoes a period-doubling cascade — 1 → 2 → 4 → 8 → … fixed points — until around r ≈ 3.57 it enters chaos. The ratio of successive period-doublings converges to the Feigenbaum constant δ ≈ 4.669. The bifurcation diagram (top) reveals this entire structure; the time series (bottom) shows individual trajectories.
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Time series
Stats
xₙ₊₁ = r·xₙ·(1−xₙ)
Lyapunov exp.:
λ = (1/N)·Σ ln|r(1−2xₙ)|
λ < 0 → stable
λ = 0 → period-doubling
λ > 0 → chaos
δ = 4.6692… (Feigenbaum)
Route to Chaos
Population models were the first systems where deterministic chaos was clearly demonstrated. Robert May's 1976 paper showed that the simple logistic recurrence — meant to model insect population dynamics — produces complexity rivalling genuinely random processes. The Feigenbaum constants (δ ≈ 4.669 and α ≈ 2.502) are universal: they appear in every one-dimensional map with a quadratic maximum, from the logistic map to real experiments in fluid convection and laser physics. This universality is explained by the renormalization group theory of Feigenbaum (1978).