x_{n+1} = r·x_n·(1−x_n) — from order to chaos as r increases
Hover/click bifurcation diagram to set r, or drag slider
3.500
0.30
200
The logistic map x→rx(1−x) undergoes period-doubling bifurcations: stable fixed point for r<3, period-2 for r∈[3,3.449], period-4, period-8… until chaos around r≈3.57. The ratio of consecutive bifurcation intervals converges to Feigenbaum's δ≈4.669 — universal for any unimodal map! Cobweb diagram (left) shows iteration graphically; time series (right) shows periodicity or chaos.