---
tag: TJ04Y2Z5Q3
kind: concept
name: Normal form
msc: [34C20, 37G05]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_form_(dynamical_systems)
same_as:
  wikidata: Q18354536
  wikipedia: Normal form (dynamical systems)
---

## Summary

A normal form of a differential equation near an [[equilibrium point|0N8NGKDZDZ]] is a simplest version of the equation.
It is obtained by a change of coordinates close to the identity.
The change removes, degree by degree, every nonlinear term that can be removed, and only the resonant terms remain.
The method goes back to Henri Poincaré.
It is a central tool in bifurcation theory, where normal forms describe the generic local behaviour, as in the Hopf bifurcation.

## Explanations

### L1 human

Near an equilibrium an equation looks like its [[linearisation|09ZWYRQ41J]] plus small nonlinear corrections.
Many of these corrections are only an artefact of the coordinates and disappear in better coordinates.
The normal form is what remains after removing them, and it shows the essential behaviour.
For $x' = x + x^2$ near $0$, the new coordinate $u = x / (1 + x)$ gives $u' = u$, so the quadratic term disappears completely.
The terms that this method cannot remove are called resonant; they come from special relations among the eigenvalues of the linear part.