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

Vadim Kaloshin, Ke Zhang

Published 2012-12-05, updated 2013-01-28Version 2

In the present paper we prove a strong form of Arnold diffusion. Let $\T^2$ be the two torus and $B^2$ be the unit ball around the origin in $\R^2$. Fix $\rho>0$. Our main result says that for a "generic" time-periodic perturbation of an integrable system of two degrees of freedom $H_\epsilon = H_0 + \epsilon H_1(\theta, p, t)$ with a strictly convex $H_0$, there exists a $\rho$-dense orbit $(\theta_{\epsilon},p_{\epsilon},t)(t)$ in $\T^2 \times B^2 \times \T$, namely, a $\rho$-neighbourhood of the orbit contains $\T^2 \times B^2 \times \T$. 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 [13], flower and simple normally hyperbolic invariant manifolds from [47] 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 [54] equipped with weak KAM theory [34], proposed by Bernard in [9].