Period-doubling route to chaos: xn+1 = r·xn·(1 − xn)
The logistic map is one of the simplest chaotic systems. As the growth rate r increases from 0 to 4, the long-term behavior of the population undergoes a period-doubling cascade: stable fixed point → period-2 cycle → period-4 → … → chaos. The ratio of successive bifurcation intervals converges to the Feigenbaum constant δ ≈ 4.669, universal across a whole class of 1D maps.
Drag on the canvas to zoom into a region and explore self-similar structure — the bifurcation pattern repeats at every scale.
chaosdynamical systemsuniversalityFeigenbaum