---
tag: D9TW46P19Y
kind: concept
name: Bifurcation theory
msc: [37G10, 34C23]
sources:
  - url: https://en.wikipedia.org/wiki/Bifurcation_theory
same_as:
  wikidata: Q1154357
  wikipedia: Bifurcation theory
---

## Summary

Bifurcation theory studies how the qualitative behaviour of a family of [[dynamical systems|BBZP1R872N]] changes as parameters vary.
A bifurcation happens at a parameter value where the [[phase portrait|0T3GK0XCBC]] changes its topological type, for example when [[equilibria|0N8NGKDZDZ]] appear, disappear or change stability.
The theory classifies the typical bifurcations, such as the [[saddle-node|MAP5J6HK0T]] and [[Hopf|F5H8GW1XYS]] bifurcations, and gives normal forms for them.
It explains sudden transitions in models from physics, biology and engineering.

## Explanations

### L1 human

Many systems change smoothly when you turn a knob, but at certain settings their behaviour changes in kind.
A pendulum with friction comes to rest; push it with a motor and past some strength it starts to swing forever.
Bifurcation theory finds these critical settings and describes what happens there.
A simple example is $\dot x = \mu - x^2$: for $\mu < 0$ there is no resting state, and for $\mu > 0$ there are two, which are born together at $\mu = 0$.