types of manifolds are three cases of an invariant manifold. Algebraically, let be a dynamical system with equilibrium point . The linearization of the system near the equilibrium point is The Jacobian matrix defines three main subspaces: the center subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues with (more generally, ); the stable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues with (more generally, ); the unstable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues with (more generally, ). Depending upon the application, other subspaces of interest include center-stable, center-unstable, sub-center, slow, and fast subspaces. These subspaces are all invariant subspaces of the linearized equation. Corresponding to the linearized system, the nonlinear system has invariant manifolds, each consisting of sets of orbits of the nonlinear system. An invariant manifold tangent to the stable subspace and with the same dimension is the stable manifold. The unstable manifold is of the same dimension and tangent to the unstable subspace. A center manifold is of the same dimension and 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. Center manifold theorems The center manifold existence theorem states that if the right-hand side function is ( times continuously differentiable), then at every equilibrium point there exists a neighborhood of some finite size in which there is at least one of a unique stable manifold, a unique unstable manifold, and a (not necessarily unique) center manifold. In example applications, a nonlinear coordinate transform to a normal form can clearly separate these three manifolds. A web service currently undertakes the necessary computer algebra for a range of finite-dimensional systems. In the case when the unstable manifold does not exist, center manifolds are often relevant to modelling. The center manifold emergence theorem then says that the neighborhood may be chosen so that all solutions of the system staying in the neighborhood tend exponentially quickly to some solution on the center manifold. That is, for some rate . This theorem asserts that for a wide variety of initial conditions the solutions of the full system decay exponentially quickly to a solution on the relatively low dimensional center manifold. A third theorem, the approximation theorem, asserts that if an approximate expression