Stable manifolds in the method of averaging

Consider the differential equation $\dot z = \varepsilon f(z,\,t,\,\varepsilon )$, where $f$ is $T$periodic in $t$ and $\varepsilon > 0$ is a small parameter, and the averaged equation $\dot z = \overline f (z): = (1/T)\,\int_0^T {\,f(z,\,t,\,0)\,dt}$. Suppose the averaged equation has a hyperbolic equilibrium at $z = 0$ with stable manifold $\overline W$. Let ${\beta _\varepsilon }(t)$ denote the hyperbolic $T$-periodic solution of $\dot z = \varepsilon f(z,\,t,\,\varepsilon )$ near $z \equiv 0$. We prove a result about smooth convergence of the stable manifold of ${\beta _\varepsilon }(t)$ to $\overline W \times {\mathbf{R}}$ as $\varepsilon \to 0$. The proof uses ideas of Vanderbauwhede and van Gils about contractions on a scale of Banach spaces.