Slow manifold

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 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. Consider the dynamical system for an evolving state vector x → ( t ) {displaystyle {vec {x}}(t)} and with equilibrium point x → ∗ {displaystyle {vec {x}}^{*}} . Then the linearization of the system at the equilibrium point is The matrix A {displaystyle A} defines four invariant subspaces characterized by the eigenvalues λ {displaystyle lambda } 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 λ {displaystyle lambda } 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 λ = 0 {displaystyle lambda =0} precisely. 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 noisy dynamical systems (stochastic differential equation), as do also stochastic center, stable and unstable manifolds. Such stochastic slow manifolds are similarly useful in modeling emergent stochastic dynamics, but there are many fascinating issues to resolve such as history and future dependent integrals of the noise. The coupled system in two variables x ( t ) {displaystyle x(t)} and y ( t ) {displaystyle y(t)} has the exact slow manifold y = x 2 {displaystyle y=x^{2}} on which the evolution is d x / d t = − x 3 {displaystyle dx/dt=-x^{3}} . Apart from exponentially decaying transients, this slow manifold and its evolution captures all solutions that are in the neighborhood of the origin. The neighborhood of attraction is, roughly, at least the half-space y > − 1 / 2 {displaystyle y>-1/2} . Edward Norton Lorenz introduced the following dynamical system of five equations in five variables to explore the notion of a slow manifold of quasi-geostrophic flow