Center manifold

Summary

A center manifold of an equilibrium point is an invariant 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.