Center Manifold - The Center Manifold Theorem

The Center Manifold Theorem

The center manifold theorem states that if ƒ is Cr (r times continuously differentiable), then at every equilibrium point there is a unique Cr stable manifold, a unique Cr unstable manifold, and a (not necessarily unique) Cr−1 center manifold.

In example applications, a nonlinear coordinate transform to a normal form (mathematics) can clearly separate these three manifolds. A web service currently undertakes the necessary computer algebra for a range of finite dimensional systems.

There is theory for the existence and relevance of center manifolds in infinite dimensional dynamical systems. The general theory currently only applies when the center manifold itself is of finite dimension. However, some applications, such as to shear dispersion, can justify and construct an infinite dimensional center manifold.