---
tag: MAP5J6HK0T
kind: concept
name: Saddle-node bifurcation
msc: [34C23, 37G10]
sources:
  - url: https://en.wikipedia.org/wiki/Saddle-node_bifurcation
same_as:
  wikidata: Q740190
  wikipedia: Saddle-node bifurcation
---

## Summary

A saddle-node bifurcation is a change in a family of dynamical systems at which two [[equilibrium points|0N8NGKDZDZ]] meet and disappear.
On one side of the critical parameter there are two equilibria, one stable and one unstable, and on the other side there are none nearby.
It is the most common way in which equilibria are created or destroyed as a parameter varies.
It is also called a fold bifurcation or a tangent bifurcation.
See the [[definition|TB7NBCCXJ1]].

## Explanations

### L1 human

The model is $x' = \mu - x^2$ on the line.
For $\mu < 0$ the right side is always negative, so there is no equilibrium.
At $\mu = 0$ the single equilibrium $x = 0$ appears.
For $\mu > 0$ it splits into a stable equilibrium at $\sqrt{\mu}$ and an unstable one at $-\sqrt{\mu}$.
Read backwards in $\mu$, the two equilibria collide and annihilate each other.