---
tag: ACN92H5M6D
kind: concept
name: Bogdanov–Takens bifurcation
msc: [34C23, 37G10]
sources:
  - url: https://en.wikipedia.org/wiki/Bogdanov%E2%80%93Takens_bifurcation
same_as:
  wikidata: Q4937510
  wikipedia: Bogdanov–Takens bifurcation
---

## Summary

A Bogdanov–Takens bifurcation is a bifurcation of an [[equilibrium point|0N8NGKDZDZ]] in a family with two parameters, at which the linearisation has a double zero eigenvalue.
It is a point where a [[saddle-node bifurcation|MAP5J6HK0T]] curve and a [[Hopf bifurcation|F5H8GW1XYS]] curve meet.
Near it, the family also has a curve of homoclinic bifurcations.
It is one of the basic codimension-two bifurcations.
See the [[definition|H3GWX277EF]].

## Explanations

### L1 human

Most bifurcations happen when one parameter reaches a critical value.
A Bogdanov–Takens point needs two parameters to be tuned at once.
In the plane of the two parameters, three curves start from this point.
On one of them two equilibria collide, on another a small oscillation is born, and on the third this oscillation grows until it touches a saddle and disappears.
So a single point organizes several kinds of change in the behaviour of the system.