---
tag: 6J921PEDN2
kind: concept
name: Period-doubling cascade
msc: ["37E05", 39A28, 39A33]
sources:
  - url: https://en.wikipedia.org/wiki/Period-doubling_bifurcation
---

## Summary

A period-doubling cascade is an infinite sequence of [[period-doubling bifurcations|NQDTTZ59X8]] in a [[family of maps|EE9X3NZW81]].
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|F1EHT2CPHC]] $\delta \approx 4.6692$.

## Explanations

### L1 human

Turn up the growth rate in a simple population model.
First the population settles to one value, then it alternates between two values, then it cycles through four, then eight, and so on.
Each doubling happens sooner than the last, and the doublings pile up at a finite setting.
Past that point the population can jump around in a way that never repeats.
This road from order to chaos is seen in many models and in experiments with fluids and electronic circuits.