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.
[1][2] 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,[3] and is thus crucial to forecasting with a climate model.
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.
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.
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
Correspondingly, the nonlinear system has invariant manifolds, made of trajectories of the nonlinear system, corresponding to each of these invariant subspaces.
[5] 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.
Apart from exponentially decaying transients, this slow manifold and its evolution captures all solutions that are in the neighborhood of the origin.
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[9] Linearized about the origin the eigenvalue zero has multiplicity three, and there is a complex conjugate pair of eigenvalues,
Hence there exists a three-dimensional slow manifold (surrounded by 'fast' waves in the
Lorenz later argued a slow manifold did not exist!
[10] But normal form[11] arguments suggest that there is a dynamical system that is exponentially close to the Lorenz system for which there is a good slow manifold.
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.
empowers us to cover the insulating Neumann boundary condition case
Now for a marvelous trick, much used in exploring dynamics with bifurcation theory.
is constant, adjoin the trivially true differential equation 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
evolves according to That is, after the initial transients that by diffusion smooth internal structures, the emergent behavior is one of relatively slow decay of the amplitude (
) at a rate controlled by the type of boundary condition (constant
Notice that this slow manifold model is global in
may be taken, but the theory assures us the results do hold for some finite parameter
the two variable dynamics of this linear system forced with noise from the random walk
However, this solution then inappropriately contains fast time integrals, due to the
Alternatively, a stochastic coordinate transform extracts a sound model for the long term dynamics.
where then the new variables evolve according to the simple In these new coordinates we readily deduce
undergoing a random walk to be the long term model of the stochastic dynamics on the stochastic slow manifold obtained by setting
A web service constructs such slow manifolds in finite dimensions, both deterministic and stochastic.