large, more precisely it is of order $\exp\left(r^{-a}\right)$ with the exponent $a=(1+\tau)^{-1}$. A similar result holds true for Gevrey smooth systems (with the exponent $\alpha^{-1}a$ instead of $a$, where $\alpha\geq 1$ is the Gevrey exponent). Moreover, in the real-analytic case and under some convexity assumption (which is non-generic), it was proved in [@MG95a] that $T(r)$ is at least doubly exponentially large, that is, it is of order $\exp\exp\left(r^{-a}\right)$. Our first theorem extends this last result: for real-analytic Hamiltonians, or more generally Gevrey smooth Hamiltonians, under a generic condition, $T(r)$ is at least doubly exponentially large; see Theorem \[Th1\] below and its corollary for precise statements. This result is the counterpart to a result we previously obtained in the context of elliptic equilibrium points (see [@BFN15]).

In a second setting, we consider a small $\varepsilon$-perturbation of an integrable Hamiltonian system in action-angle coordinates, namely $$\label{H2}\tag{H$_\varepsilon$}
H_\varepsilon(\theta,I)=h(I)+\varepsilon f(\theta,I), \quad (\theta,I)\in \T^n \times \bar{D},$$ where $\bar{D}$ is the closure of some open bounded domain $D \subseteq {\mathbb{R}}^n$, and $\varepsilon\geq 0$ is a small parameter. When $\varepsilon=0$, letting $\omega_*=\nabla h(I_*) \in {\mathbb{R}}^n$ for $I_* \in D$, the Lagrangian tori $\mathcal{T}_{\omega_*}:=\{I=I_*\}$ are obviously invariant by the Hamiltonian flow of $H_0=h$, and they are quasi-periodic with 