Periodic solutions of differential equations with periodic constraints

We study the harmonic solutions of periodically perturbed differential equations subjected to a possibly time‐dependent constraint. We obtain a degree theoretic condition ensuring a “global branching” result for the nontrivial periodic solutions. The argument stems from a combination of techniques from the theory of topological degree and differential‐algebraic equations. As an application, the set of harmonic solutions of periodically perturbed implicit differential equations is investigated.