---
tag: JV45S0KQE6
kind: concept
name: Bifurcation set
msc: [58K35, 58K05]
sources:
  - url: https://en.wikipedia.org/wiki/Catastrophe_theory
---

## Summary

The bifurcation set of a smooth family of functions is a set of parameter values.
It contains the values at which some function of the family has a degenerate critical point.
It is the central object of [[catastrophe theory|6FKJ6SXP3G]].
Off this set the critical points move smoothly with the parameters, and across it they are created or destroyed.
For the cusp family $x^4/4 + a x^2/2 + b x$ it is the cusp curve $4a^3 + 27b^2 = 0$ in the $(a, b)$-plane.
For a family of potentials it describes where sudden jumps of the equilibrium can start.

## Explanations

### L1 human

Imagine a landscape that changes shape as you turn some knobs.
For most knob settings every valley and every hilltop is a clean, rounded one.
At special settings a valley and a hilltop merge into a flat spot and then vanish.
The bifurcation set is the map of all such special settings in the space of knobs.
Take the family $x^3 + a x$ with one knob $a$.
A valley and a hilltop exist for $a < 0$, they merge at $a = 0$, and none exist for $a > 0$.
So the bifurcation set is the single point $a = 0$.