Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case

TL;DR

This paper develops a parametric normal form for perturbations of the Bogdanov--Takens singularity's generalized saddle-node case, providing explicit formulas and a Maple program for automated computation.

Contribution

It introduces a simplified, explicit parametric normal form for the generalized saddle-node case of Bogdanov--Takens singularity, facilitating practical computation.

Findings

Explicit formulas for the normal form are derived.

A Maple program for automatic computation is provided.

Practical formulas are presented to simplify implementation.

Abstract

We obtain a parametric normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov--Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer algebra system. A Maple program is prepared for an automatic parametric normal form computation. A section is devoted to present some practical formulas which avoid technical details of the paper.