Asymptotic trajectories of KAM torus

TL;DR

This paper constructs asymptotic trajectories approaching KAM tori in nearly integrable Hamiltonian systems with small perturbations, using new methods inspired by Arnold diffusion to better understand diffusion orbits.

Contribution

It introduces a novel approach to find asymptotic orbits towards KAM tori in weakly coupled nearly integrable systems, expanding on methods from Arnold diffusion research.

Findings

Existence of asymptotic orbits towards KAM tori for small perturbations.

Development of new techniques based on Chong-Qing Cheng's methods.

Insights into the search for more precise diffusion orbits.

Abstract

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.