A strong form of Arnold diffusion for two and a half degrees of freedom

Abstract

In the present paper we prove a strong form of Arnold diffusion. Let ${\mathbb{T}}^2$ be the two torus and $B^2$ be the unit ball around the origin in ${\mathbb{R}}^2$. Fix $\rho >0$. Our main result says that for a "generic" time-periodic perturbation of an integrable system of two degrees of freedom \[ H_0(p)+εH_1(θ,p,t),\quad \ θ\in \mathbb{T}^2,\ p\in B^2,\ t\in \mathbb{T}, \] with a strictly convex $H_0$, there exists a $\rho$-dense orbit $({\theta }_{\epsilon },p_{\epsilon },t)(t)$ in ${\mathbb{T}}^2\times B^2\times \mathbb{T}$, namely, a $\rho$-neighborhood of the orbit contains ${\mathbb{T}}^2\times B^2\times \mathbb{T}$. Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are usage of crumpled normally hyperbolic invariant cylinders from \cite{BKZ}, flower and simple normally hyperbolic invariant manifolds from as well as their kissing property at a strong double resonance. This allows us to build a "connected" net of $3$-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of Mather variational method \cite{Ma2} proposed by Bernard in \cite{Be}. This version is equipped with weak KAM theory \cite{Fa}.

arXiv comment185 pages with many figures. This version is a thorough rewrite of the earlier version of this paper circa 2013. The main structure of the paper has been adjusted to improve readablity, and new concept of Aubry-Mather type is introduced to keep the proof more modular

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• Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders