Linearization Along Invariant Manifolds and a Generalization of the Hartman-Grobman Theorems for Degenerate Singularities of Vector Fields in R Cubed

摘要

The phase-portrait at a singularity of a vectorfield in R sup n when the singularity is nonhyperbolic is considered. The approach is to blow-up the singularity and apply linearization theorems along invariant manifolds which are not necessarily normally hyperbolic. Extending a result of Bonckaert, Dumortier, and van Strien on singularities of vector fields determined by their first nonvanishing jet slightly, a generalization of the Hartman-Grobman theorem in R cubed is given. This result holds for nonhyperbolic singularities and also gives nice smoothness of the homeomorphisms without requiring nonresonance conditions.