May's Ecological Chaos — Logistic Map Bifurcation

x_{n+1} = r·x_n(1−x_n). Click the bifurcation diagram to select r and see the time series. Period-doubling route to chaos with Feigenbaum constant δ ≈ 4.669.

Parameters

Growth rate r: 3.50

← or click bifurcation diagram

Current Dynamics

Period—

Lyapunov λ—

Regime—

Key Values

r=1: stable 0
r=2: stable (r-1)/r
r=3: period-2 onset
r≈3.449: period-4
r≈3.544: period-8
r≈3.5644: period-16
r≈3.569: onset chaos
r=4: fully chaotic

δ = (r_{n}-r_{n-1})/(r_{n+1}-r_{n})
δ → 4.6692... (Feigenbaum)