In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of the slow manifold—center manifold theory rigorously justifies the modelling. For example, some global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics, and is thus crucial to forecasting with a climate model. In some cases, a slow manifold is defined to be the invariant manifold on which the dynamics are slow compared to the dynamics off the manifold. The slow manifold in a particular problem would be a sub-manifold of either the stable, unstable, or center manifold, exclusively, that has the same dimension of, and is tangent to, the eigenspace with an associated eigenvalue (or eigenvalue pair) that has the smallest real part in magnitude. This generalizes the definition described in the first paragraph. Furthermore, one might define the slow manifold to be tangent to more than one eigenspace by choosing a cut-off point in an ordering of the real part eigenvalues in magnitude from least to greatest. In practice, one should be careful to see what definition the literature is suggesting. Definition Consider the dynamical system for an evolving state vector and with equilibrium point . Then the linearization of the system at the equilibrium point is The matrix defines four invariant subspaces characterized by the eigenvalues of the matrix: as described in the entry for the center manifold three of the subspaces are the stable, unstable and center subspaces corresponding to the span of the eigenvectors with eigenvalues that have real part negative, positive, and zero, respectively; the fourth subspace is the slow subspace given by the span of the eigenvectors, and generalized eigenvectors, corresponding to the eigenvalue precisely (more generally, corresponding to all eigenvalues with separated by a gap from all other eigenvalues, those with ). The slow subspace is a subspace of the center subspace, or identical to it, or possibly empty. Correspondingly, the nonlinear system has invariant manifolds, made of trajectories of the nonlinear system, corresponding to each of these invariant subspaces. There is an invariant manifold tangent to the slow subspace and with the same dimension; this manifold is the slow manifold. Stochastic slow manifolds also exist for