Period-doubling cascade

Summary

A period-doubling cascade is an infinite sequence of period-doubling bifurcations in a family of maps. In it, stable cycles of periods $1, 2, 4, 8, \dots$ appear one after another. The parameter values of the bifurcations increase to a finite limit, beyond which the dynamics can be chaotic. For the logistic map $x \mapsto r x (1 - x)$ the cascade starts at $r = 3$ and accumulates near $r \approx 3.5699$. For typical smooth unimodal maps, the gaps between successive bifurcation values shrink at a universal geometric rate. This rate is the Feigenbaum constant $\delta \approx 4.6692$.