  By using topological methods, mainly the degree of a tangent vector field, we
establish multiplicity results for $T$-periodic solutions of parametrized
$T$-periodic perturbations of autonomous ODEs on a differentiable manifold $M$.
In order to provide insights into the key notion of $T$-resonance, we consider
the elementary situations $M = \mathbb{R}$ and $M = \mathbb{R}^2$. So doing, we
provide more comprehensive analysis of those cases and find improved
conditions.
