Branches of forced oscillations for a class of constrained ODEs: a topological approach

TL;DR

This paper uses topological methods to analyze harmonic solutions of periodically perturbed ODEs on manifolds, providing global continuation results and formulas for computing degrees of tangent vector fields.

Contribution

It introduces a topological approach to study harmonic solutions on manifolds and derives a formula linking tangent vector field degrees to Brouwer degrees in Euclidean space.

Findings

Derived a formula reducing tangent vector field degree to Brouwer degree.

Established global continuation results for harmonic solutions.

Applied methods to periodic semi-explicit differential-algebraic equations.

Abstract

We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a $k$-dimensional differentiable manifold $M \subseteq \mathbb{R}^m$. We assume that $M$ is globally defined as the zero set of a smooth map and, as a first step, we determine a formula which reduces the computation of the degree of a tangent vector field on $M$ to the Brouwer degree of a suitable map in $\mathbb{R}^m$. As further applications, we study the set of harmonic solutions to periodic semi-esplicit differential-algebraic equations.