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.
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Definition
Consider the dynamical system
for an evolving state vector
The matrix
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.
Simple case with two variables
The coupled system in two variables
has the exact slow manifold
Slow dynamics among fast waves
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
Linearized about the origin the eigenvalue zero has multiplicity three, and there is a complex conjugate pair of eigenvalues,
Eliminate an infinity of variables
In modeling we aim to simplify enormously. This example uses a slow manifold to simplify the 'infinite dimensional' dynamics of a partial differential equation to a model of one ordinary differential equation. Consider a field
with Robin boundary conditions
Parametrising the boundary conditions by
Now for a marvelous trick, much used in exploring dynamics with bifurcation theory. Since parameter
Then in the extended state space of the evolving field and parameter,
Here one can straightforwardly verify the slow manifold to be precisely the field
That is, after the initial transients that by diffusion smooth internal structures, the emergent behavior is one of relatively slow decay of the amplitude (
Notice that this slow manifold model is global in
Perhaps the simplest nontrivial stochastic slow manifold
Stochastic modeling is much more complicated—this example illustrates just one such complication. Consider for small parameter
One could simply notice that the Ornstein–Uhlenbeck process
and then assert that
Alternatively, a stochastic coordinate transform extracts a sound model for the long term dynamics. Change variables to
then the new variables evolve according to the simple
In these new coordinates we readily deduce
A web service constructs such slow manifolds in finite dimensions, both deterministic and stochastic.