Dimension Reduction:A Key Concept in Dynamics
The main idea in dimension reduction is the separation of the variables into
essential or active modes and into ignorable or passive modes. However,
it would be a mistake
to assume that the ignorable modes, can be completely
ignored in the derivation of the reduced system. Due to nonlinear
coupling the elimination of these passive modes
is a delicate matter. If these modes are completely ignored
in the calculation of the reduced system the
so-called standard (linear or flat) Galerkin method, mostly
used in engineering, is obtained. However, this approach
sometimes gives qualitatively incorrect results and hence
better strategies must be looked for. Hence the
representation of the passive modes by means
of the active modes is the essential step in applying
more sophisticated dimension reduction methods. This is done
in almost all different approaches by invariant manifolds.