Carathéodory periodic perturbations of degenerate systems

We study the structure of the set of harmonic solutions to T -periodically perturbed coupled differential equations on differentiable manifolds, where the perturbation is allowed to be of Carathéodory-type regularity. Employing degree-theoretic methods, we prove the existence of a noncompact connected set of nontrivial T -periodic solutions that, in a sense, emanates from the set of zeros of the unperturbed vector field. The latter is assumed to be “degenerate”: Meaning that, contrary to the usual assumptions on the leading vector field, it is not required to be either trivial nor to have a compact set of zeros. In fact, known results in the “nondegenerate” case can be recovered from our ones. We also provide some illustrating examples of Liénard- and ϕ-Laplacian-type perturbed equations.