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.

Consider the dynamical system

for an evolving state vector ${\displaystyle {\vec {x}}(t)}$ and with equilibrium point ${\displaystyle {\vec {x}}^{*}}$. Then the linearization of the system at the equilibrium point is

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