Stable manifolds for periodically perturbed maps

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.