Asymptotic trajectories of KAM torus

In this paper we construct a certain type of nearly integrable systems of two and a half degrees of freedom: $$H(p,q,t)=h(p)+\epsilon f(p,q,t),\quad (q,p)\in T^{*}\mathbb{T}^2,t\in \mathbb{S}^1=\mathbb{R}/\mathbb{Z},$$ with a self-similar and weak-coupled $f(p,q,t)$ and $h(p)$ strictly convex. For a given Diophantine rotation vector $\vec{\omega}$, we can find asymptotic orbits towards the KAM torus $\mathcal{T}_{\omega}$, which persists owing to the classical KAM theory, as long as $\epsilon\ll1$ sufficiently small and $f\in C^r(T^{*}\mathbb{T}^2\times\mathbb{S}^1,\mathbb{R})$ properly smooth. The construction bases on the new methods developed in {\it a priori} stable Arnold Diffusion problem by Chong-Qing Cheng. As an expansion of that, this paper sheds some light on the seeking of much preciser diffusion orbits.