Automated computation of robust normal forms of planar analytic vector fields

TL;DR

This paper presents an auto-validated algorithm that computes a robust, analytic normal form transformation for saddle points in planar analytic vector fields, aiding in flow analysis near these points.

Contribution

It introduces a novel, validated method for computing normal forms of saddle points that is robust and applicable in a neighborhood of the saddle, with practical computational benefits.

Findings

Algorithm successfully computes normal forms for various examples.

Transformation is robust and analytic on a computable neighborhood.

Method facilitates flow enclosure and passage time estimation near saddles.

Abstract

We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighborhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.