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: $$H(p,q,t)=h(p)+ϵf(p,q,t),(q,p)\in T^{\ast }{\mathbb{T}}^2,t\in {\mathbb{S}}^1=\mathbb{R}\mathrm{/}\mathbb{Z},$$ 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 $ϵ\ll 1$ sufficiently small and $f\in C^r(T^{\ast }{\mathbb{T}}^2\times {\mathbb{S}}^1,\mathbb{R})$ 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.