Normal form

Summary

A normal form of a differential equation near an equilibrium point 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.