A note on Vishik's normal form

TL;DR

This paper proves that Vishik's normal form conjugation is analytic for analytic vector fields and manifolds, and explores the implications for the analyticity of Poincaré Half Maps near contacts.

Contribution

It extends Vishik's normal form to the analytic setting, showing the conjugation is also analytic, and applies this to the study of Poincaré Half Maps.

Findings

Conjugation in Vishik's normal form is analytic for analytic vector fields.

Analyticity of Poincaré Half Maps near contacts is established.

Provides a foundation for further analysis of local dynamics in analytic systems.

Abstract

The Vishik's Normal Form provides a local smooth conjugation with a linear vector field for smooth vector fields near contacts with a manifold. In the present study, we focus on the analytic case. Our main result ensures that for analytic vector field and manifold, the conjugation with the Vishik's normal form is also analytic. As an application, we investigate the analyticity of Poincar\'e Half Maps defined locally near contacts between analytic vector field and manifold.