---
tag: P88X34MSVV
kind: concept
name: Pitchfork bifurcation
msc: [34C23, 37G10, 37G40]
sources:
  - url: https://en.wikipedia.org/wiki/Pitchfork_bifurcation
same_as:
  wikidata: Q1785924
  wikipedia: Pitchfork bifurcation
---

## Summary

A pitchfork bifurcation is a change in a family of dynamical systems with a symmetry, at which one [[equilibrium point|0N8NGKDZDZ]] splits into three.
It typically occurs when the system is unchanged under $x \mapsto -x$, so equilibria come in symmetric pairs.
In the supercritical case, a stable equilibrium becomes unstable and two new stable equilibria appear on either side.
See the [[definition|TRFDEJRPAZ]].

## Explanations

### L1 human

The model is $x' = \mu x - x^3$.
For $\mu \le 0$ the origin is the only equilibrium, and it attracts nearby solutions.
For $\mu > 0$ the origin repels, and two new stable equilibria appear at $\pm\sqrt{\mu}$.
Plotting the equilibria against $\mu$ gives a picture shaped like a pitchfork.
A ruler pressed at both ends that buckles to the left or to the right is a physical picture of this.