Feigenbaum Constants

The period-doubling cascade to chaos is universal: the ratio of successive bifurcation intervals converges to δ≈4.669 (Feigenbaum's first constant). The ratio of successive branch widths converges to α≈2.502 (second constant). They apply to ANY smooth unimodal map.

Map
r range start2.5
r range end4.0
Iterations300
Feigenbaum δ4.669201609...
Feigenbaum α2.502907875...
Onset of chaos (logistic)r∞ ≈ 3.56995...
Click canvas to zoom into a region. Bifurcations annotated with δ ratios.