---
tag: F5H8GW1XYS
kind: concept
name: Hopf bifurcation
msc: [34C23, 37G15]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_bifurcation
same_as:
  wikidata: Q531126
  wikipedia: Hopf bifurcation
---

## Summary

A Hopf bifurcation is the birth of a [[periodic orbit|0AA97XA70K]] from an [[equilibrium point|0N8NGKDZDZ]] as a parameter crosses a critical value.
It happens when a pair of complex conjugate eigenvalues of the linearisation crosses the imaginary axis.
It is the main way in which a steady state of a system turns into a small oscillation.
See the [[definition|1JPGWFMAGK]].

## Explanations

### L1 human

Think of a system resting at an equilibrium, where small disturbances spiral back to rest.
As a parameter changes, the spiral becomes slower, and at a critical value it turns outward.
Then the motion no longer returns to rest but settles on a small loop, a [[limit cycle|0AAQ73WRFB]].
In polar coordinates, the model $r' = \mu r - r^3$, $\theta' = 1$ shows this.
For $\mu \le 0$ every solution spirals into the origin, and for $\mu > 0$ solutions approach the circle $r = \sqrt{\mu}$.