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 \[ H_0(p)+\epsilon H_1(\theta,p,t),\quad \ \theta\in \mathbb{T}^2,\ p\in B^2,\ t\in \mathbb{T}, \] 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}.