A note on topological methods for a class of Differential-Algebraic Equations

TL;DR

This paper introduces a simple formula for computing the degree of tangent vector fields on manifolds related to a class of Differential-Algebraic Equations, aiding in analyzing harmonic solutions under perturbations.

Contribution

It provides a novel, easy-to-apply formula for the degree calculation without explicit manifold knowledge, advancing the analysis of DAE solutions on manifolds.

Findings

Derived a straightforward degree formula for tangent vector fields

Applied the formula to study harmonic solutions of perturbed equations

Presented two classes of practical applications

Abstract

We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the computation of the degree of the associated tangent vector field that does not require any explicit knowledge of the manifold. We use this formula to study the set of harmonic solutions to periodic perturbations of our equations. Two different classes of applications are provided.