THE LOGISTIC MAP

Period-doubling cascade to chaos — Feigenbaum universality

xn+1 = r · xn · (1 - xn)
2.800
0.50
Period:1
Regime:FIXED POINT
Lyapunov:-0.693
Zoom:1.0x
Bifurcation Diagram (click+drag to zoom, right-click to reset)
Time Series — xn vs n
Cobweb Diagram
Lyapunov Exponent vs r
Feigenbaum Constants
nrn (bifurcation)deltanratio → delta
13.000000——
23.4494900.449490—
33.5440900.0946004.7514
43.5644070.0203174.6562
53.5687590.0043524.6684
63.5696920.0009334.6686
∞3.569946...—4.66920...
delta = 4.66920... appears in ALL period-doubling cascades: fluid convection, electronic circuits, dripping faucets, heart fibrillation, laser instabilities. It is as universal as pi.
edge_of_chaos · Logistic Map Explorer · xn+1 = rxn(1-xn)