Asymptotic trajectories of KAM torus

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 $\vec{\omega }$, we can find asymptotic orbits towards the KAM torus ${\mathcal{T}}_{\omega }$, which persists owing to the classical KAM theory, as long as $\epsilon \ll 1$ 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.

References in corpus (4)

• Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders
• A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems
• Improved stability for analytic quasi-convex nearly integrable systems and optimal speed of Arnold diffusion
• Interpolating vector fields for near identity maps and averaging