# Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions

## TL;DR

This paper develops normal form theorems for manifolds with normally hyperbolic singularities and analyzes the asymptotic behavior of transition maps, providing explicit computation methods and revealing structural similarities to planar saddle singularities.

## Contribution

It introduces formal and $C^k$ normal form theorems for normally hyperbolic singularities and applies them to study the asymptotic properties of Dulac maps between transverse sections.

## Key findings

- Normal form theorems for normally hyperbolic singularities established.
- Explicit methods for computing Dulac maps provided.
- Dulac maps exhibit asymptotic structures similar to planar saddle singularities.

## Abstract

This paper contains theory on two related topics relevant to manifolds of normally hyperbolic singularities. First, theorems on the formal and $ C^k $ normal forms for these objects are proved. Then, the theorems are applied to give asymptotic properties of the transition map between sections transverse to the centre-stable and centre-unstable manifolds of some normally hyperbolic manifolds. A method is given for explicitly computing these so called Dulac maps. The Dulac map is revealed to have similar asymptotic structures as in the case of a saddle singularity in the plane.