I am interested in studying periodic perturbations of autonomous differential systems having attractive hyperbolic periodic solutions. As normally hyperbolic invariant manifold is a kind of new topic to me, I would really appreciated if you help me to answer the following question:
Let $x\in \Omega\subset \mathbb{R}^n$, $\theta \in \mathbb{S}^1$ and $f \in \mathcal{C^p}(\mathbb{R}^n)$ $(p\geq 2)$. Consider $\varphi(t)$ a non trivial periodic solution of the differential system $$ \begin{align} \dfrac{d x}{dt}&= f(x),\\ \dfrac{d \theta}{dt}&=1. \end{align} $$ Assume that the (characteristic)Floquet multipliers of the variational system $$ \dfrac{dy}{dt}=D_xf(\varphi(t))y, $$
i.e., $\mu_1$, ... $\mu_{n-1}$, $1$, satisfies the relation $|\mu_i|<1$. I would like to know if the invariant torus passing through this periodic orbit is a normally hyperbolic invariant manifold? It seems to be true but I am not sure yet.