---
tag: 2XDRZV3GSA
kind: concept
name: Bifurcation diagram
msc: [37G35, 37G10]
sources:
  - url: https://en.wikipedia.org/wiki/Bifurcation_diagram
same_as:
  wikidata: Q2595915
  wikipedia: Bifurcation diagram
---

## Summary

A bifurcation diagram is a picture of how the long-term behaviour of a one-parameter family of dynamical systems depends on the parameter.
The parameter runs along the horizontal axis, and above each value one plots the [[equilibria|0N8NGKDZDZ]], periodic orbits or [[attractor|7FWX24X5MW]] of the system.
Branches that split, merge or end mark the bifurcations studied by [[bifurcation theory|D9TW46P19Y]].
The best known example is the diagram of the [[logistic map|QN9X9BNKTX]], with its cascade of period doublings.

## Explanations

### L1 human

Run the system for many steps at one setting of the parameter and mark where it ends up.
Do this for many settings side by side, and you get a bifurcation diagram.
A single line means the system settles to one value; two lines mean it alternates between two values; a smeared band means chaos.
For the logistic map $x \mapsto r x (1 - x)$ the single line splits into two near $r = 3$, then into four, and so on, until chaos sets in.