  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}.
