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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 $\rho$-denseorbit $(\theta_{\epsilon},p_{\epsilon},t)(t)$ 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. Thefundamental elements of the construction are usage of crumpled normallyhyperbolic invariant cylinders from \cite{BKZ}, flower and simple normallyhyperbolic invariant manifolds from as well as their kissing property at astrong double resonance. This allows us to build a "connected" net of$3$-dimensional normally hyperbolic invariant manifolds. To construct diffusingorbits along this net we employ a version of Mather variational method\cite{Ma2} proposed by Bernard in \cite{Be}. This version is equipped with weakKAM theory \cite{Fa}.

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