  Small non-autonomous perturbations around an equilibrium of a nonlinear
delayed system are studied. Under appropriate assumptions, it is shown that the
number of $T$-periodic solutions lying inside a bounded domain $\Omega\subset
\R^N$ is, generically, at least $|\chi \pm 1|+1$, where $\chi$ denotes the
Euler characteristic of $\Omega$. Moreover, some connections between the
associated fixed point operator and the Poincar\'e operator are explored.
