On the flat remainder in normal forms of families of analytic planar saddles

TL;DR

This paper provides an explicit formula for the flat remainder after normal form reduction of planar saddle families, considering rational or irrational eigenvalue moduli ratios, enhancing understanding of local dynamics.

Contribution

It introduces a precise expression for the flat remainder in normal forms of planar saddle families, differentiating cases based on eigenvalue ratio rationality.

Findings

Explicit formula for flat remainder derived

Distinction between rational and irrational eigenvalue ratios

Improved understanding of local saddle dynamics

Abstract

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.