View source for Normal forms

==Geometrical Consequences of the Normal Form==

It has already been pointed out that a normal form can decide
[[stability]] questions and establish hidden symmetries.  Computing the
normal form up to degree <math>k</math> also automatically computes (to degree
<math>k</math>) the stable, unstable, and center manifolds, the 
[[Center Manifold Theorem|center manifold reduction]], and the fibration of the center-stable and
center-unstable manifolds over the center manifold.  The common
practice of computing the center manifold reduction first, and then
computing the normal form only for this reduced system, seems to
save work but loses many of these results.  See chapter 5 of Murdock (2003). 

On occasion, the truncation of a normal form produces a simple
system that is [[Topological Equivalence|topologically equivalent]] to the original system in
a neighborhood of the equilibrium, called ''topological normal form''.  
For instance, in the example above, truncating after the first nonvanishing <math>\alpha_i</math> will accomplish
this, but if all <math>\alpha_i</math> are zero, the topological behavior is
probably determined by a transcendentally small effect that is not
captured by the normal form.

Normal forms are important for determining [[bifurcations]] of a
system, but this requires the inclusion of [[unfolding]] parameters.