On the classification of planar constrained differential systems under topological equivalence

TL;DR

This paper extends the topological classification of phase portraits to planar constrained differential systems, using resolution of singularities to analyze behavior near singular points and impasse sets.

Contribution

It introduces a method to classify constrained differential systems' phase portraits near singularities using weighted blow ups, expanding existing results for vector fields.

Findings

Extended classification results to constrained systems.

Analyzed phase portraits near impasse points.

Applied resolution of singularities techniques.

Abstract

This paper concerns the local study of analytic constrained differential systems (or impasse systems) of the form $A(x)\dot{x} = F(x)$, $x\in\mathbb{R}^{2}$, where $F$ is a vector field and $A$ is a matrix valued function. Using techniques of resolution of singularities with weighted blow ups, we extend well-known results in the literature on topological classification of phase portraits of planar vector fields to the context of such class of systems. We also study and classify phase portraits of constrained systems near singular points of the so called impasse set $\Delta=\{x:\det A (x) = 0\}$.