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

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 $$H_0(p)+ϵH_1(\theta ,p,t),\theta \in {\mathbb{T}}^2,p\in B^2,t\in \mathbb{T},$$ with a strictly convex $H_0$, there exists a $\rho$-dense orbit $({\theta }_{ϵ},p_{ϵ},t)(t)$ in $$\mathbb{T}$^2 \times B^2 \times $\mathbb{T}$$,namely,a$\rho$-neighborhoodoftheorbitcontains$$\mathbb{T}$^2 \times B^2 \times $\mathbb{T}$$. 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}.