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

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$\mathrm{.}Weassumethat$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.