Asymptotic trajectories of KAM torus
Jianlu Zhang, Chong-Qing Cheng
TL;DR
The work advances the understanding of diffusion in near-integrable Hamiltonian systems with 2.5 degrees of freedom by constructing a self-similar, weakly coupled perturbation and proving the existence of asymptotic orbits toward a KAM torus T_{\omega} for sufficiently small \epsilon. It introduces a resonant skeleton and a unified stable normal form that handle an infinite cascade of resonances, establishes the persistence of weak NHICs, and locates Aubry sets within these cylinders. The combination of Mather theory (weak KAM), homogenization, and two diffusion mechanisms (time-step and space-step Lagrangians) yields a robust pathway to generate O(1) diffusion orbits and sharp estimates for the involved geometric structures, such as incomplete intersection annuli. This work lays groundwork for refining diffusion orbits and suggests potential applications to broader Hamiltonian contexts and PDE problems.
Abstract
In this paper we construct a certain type of nearly integrable systems of two and a half degrees of freedom: \[H(p,q,t)=h(p)+εf(p,q,t),\quad (q,p)\in T^{*}\mathbb{T}^2,t\in \mathbb{S}^1=\mathbb{R}/\mathbb{Z}, \] with a self-similar and weak-coupled $f(p,q,t)$ and $h(p)$ strictly convex. For a given Diophantine rotation vector $\vecω$, we can find asymptotic orbits towards the KAM torus $\mathcal{T}_ω$, which persists owing to the classical KAM theory, as long as $ε\ll1$ 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.