Center Manifold - Definition

Definition

Let

be a dynamical system with equilibrium point .

The linearization of the system at the equilibrium point is

The matrix defines three subspaces:

• the stable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues λ with Re λ < 0;
• the unstable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues λ with Re λ > 0;
• the center subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues λ with Re λ = 0.

These spaces are all invariant subspaces of the linearized equation.

Corresponding to the linearized system, the nonlinear system has invariant manifolds, consisting of orbits of the nonlinear system. There is an invariant manifold tangent to the stable subspace and with the same dimension; this manifold is the stable manifold. Similarly, the unstable manifold is tangent to the unstable subspace, and the center manifold is tangent to the center subspace. If, as is common, the eigenvalues of the center subspace are all precisely zero, rather than just real part zero, then a center manifold is often called a slow manifold.