---
tag: JQYCTY3Q8V
kind: concept
name: Homoclinic bifurcation
msc: [37G20, 34C37, 37C29]
sources:
  - url: https://en.wikipedia.org/wiki/Bifurcation_theory
---

## Summary

A homoclinic bifurcation is a global bifurcation in which, at one parameter value, an orbit leaves a saddle equilibrium and returns to it.
This orbit is a [[homoclinic orbit|6DBSQKZK4W]].
As the parameter crosses this value, the loop breaks, and periodic orbits can be created or destroyed near it.
In the plane, under nondegeneracy conditions, a single [[limit cycle|0AAQ73WRFB]] is born from the loop.
In three dimensions, near a saddle-focus, the bifurcation can produce chaos, as the [[Shilnikov theorem|Q134D9EMAZ]] shows.

## Explanations

### L1 human

A saddle point has a path leading out of it and a path leading into it.
Usually these paths miss each other, but at a special parameter value the outgoing path loops around and comes back in.
That loop is fragile and breaks as soon as the parameter moves.
When it breaks, a closed orbit that circles around near the old loop can appear, with a period that becomes infinite as the parameter approaches the special value.