Geometrical Structures in Normal Forms

Abstract

After a system has been placed into normal form, an immediate question arises: What does the normal form tell us about the dynamics of the system? Sometimes the normalized system is simple enough to be integrable (“by quadrature,” that is, its solution can be reduced to the evaluation of integrals), but this is not the usual case (except in two dimensions). We approach this question in Section 5.1 by establishing the existence of geometrical structures, such as invariant manifolds and preserved foliations, for systems in truncated normal form (that is, polynomial vector fields that are entirely in normal form, with no nonnormalized terms). These geometrical structures explain why some truncated normal forms are integrable, and give partial information about the behavior of others. In particular, the computation of a normal form up to degree k simultaneously computes the stable, unstable, and center manifolds, the center manifold reduction of the system, the stable and unstable fibrations over the center manifold, and various preserved foliations. These concepts will be defined (to the extent that they are needed) when they arise, but the reader is expected to have some familiarity with them. We do not prove the existence of these structures, except in the special case of truncated systems in normal form, where each structure mentioned above takes a simple linear form. For the full (untruncated) systems the relevant existence theorems will be stated without proof, and the normal form will be used to compute approximations.