  We prove that if a certain entry in the map of the Hadamard-Perron theorem is
$T$-periodic in one of the variables, then the stable manifold guaranteed by
the Hadamard-Perron theorem is a graph of a $T$-periodic function. As an
application, we extend the classical Levinson's result about the occurrence of
an attracting closed invariant curve near a stable cycle of a system of
autonomous equations under periodic perturbations to hybrid differential
equations.
