---
tag: X45STCVFZR
kind: concept
name: Center manifold
msc: [34C45, 37D10, 37G10]
sources:
  - url: https://en.wikipedia.org/wiki/Center_manifold
same_as:
  wikidata: Q4504202
  wikipedia: Center manifold
---

## Summary

A center manifold of an [[equilibrium point|0N8NGKDZDZ]] is an [[invariant|WWZTW11664]] manifold through the equilibrium.
It is tangent to the directions where the linearisation neither grows nor decays.
The center manifold theorem says that such a manifold exists, though it need not be unique.
For an equation of class $C^k$ with finite $k$, it can be taken of class $C^k$.
The interesting dynamics near a non-hyperbolic equilibrium happens on the center manifold.
So the study of stability and bifurcations reduces to a system of lower dimension.

## Explanations

### L1 human

At an equilibrium, some directions pull solutions in fast, some push them out fast, and some do neither at the linear level.
The neutral directions are where the nonlinear terms decide what happens.
The center manifold is a curved surface along these neutral directions that solutions near the equilibrium follow.
For $x' = x^2$, $y' = -y$, the $y$-direction contracts and the $x$-axis is a center manifold, on which the motion is $x' = x^2$.
So all the delicate behaviour can be read from a one-dimensional equation.